Theorem atexcht 10308

Description: The Hilbert lattice satisfies the atom exchange property. Proposition 1(i) of [Kalmbach] p. 140. A version of this theorem related to vector analysis was originally proved by Hermann Grassmann in 1862. Also Definition 3.4-3(b) in [MegillPavicic] p. 2345 (PDF p. 8) (use atnem0 10304 to obtain atom inequality).

Assertion

Ref	Expression
atexcht	|- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> C (_ (A vH B)))

Proof of Theorem atexcht

Step	Hyp	Ref	Expression
1		chub2t 9431	. . . . . . 7 |- ((C e. CH /\ A e. CH) -> C (_ (A vH C))
2	1	ancoms 436	. . . . . 6 |- ((A e. CH /\ C e. CH) -> C (_ (A vH C))
3		atelch 10271	. . . . . 6 |- (C e. Atoms -> C e. CH)
4	2, 3	sylan2 451	. . . . 5 |- ((A e. CH /\ C e. Atoms) -> C (_ (A vH C))
5	4	3adant2 798	. . . 4 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> C (_ (A vH C))
6	5	adantr 389	. . 3 |- (((A e. CH /\ B e. Atoms /\ C e. Atoms) /\ (B (_ (A vH C) /\ (A i^i B) = 0H)) -> C (_ (A vH C))
7		cvp 10302	. . . . . . . . 9 |- ((A e. CH /\ B e. Atoms) -> ((A i^i B) = 0H <-> A <o (A vH B)))
8		chjclt 9329	. . . . . . . . . . 11 |- ((A e. CH /\ B e. CH) -> (A vH B) e. CH)
9		atelch 10271	. . . . . . . . . . 11 |- (B e. Atoms -> B e. CH)
10	8, 9	sylan2 451	. . . . . . . . . 10 |- ((A e. CH /\ B e. Atoms) -> (A vH B) e. CH)
11		cvpsst 10212	. . . . . . . . . 10 |- ((A e. CH /\ (A vH B) e. CH) -> (A <o (A vH B) -> A (. (A vH B)))
12	10, 11	syldan 467	. . . . . . . . 9 |- ((A e. CH /\ B e. Atoms) -> (A <o (A vH B) -> A (. (A vH B)))
13	7, 12	sylbid 203	. . . . . . . 8 |- ((A e. CH /\ B e. Atoms) -> ((A i^i B) = 0H -> A (. (A vH B)))
14	13	3adant3 799	. . . . . . 7 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((A i^i B) = 0H -> A (. (A vH B)))
15	14	adantld 390	. . . . . 6 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> A (. (A vH B)))
16		chub1t 9430	. . . . . . . . . . . 12 |- ((A e. CH /\ C e. CH) -> A (_ (A vH C))
17	16	3adant2 798	. . . . . . . . . . 11 |- ((A e. CH /\ B e. CH /\ C e. CH) -> A (_ (A vH C))
18	17	a1d 12	. . . . . . . . . 10 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (B (_ (A vH C) -> A (_ (A vH C)))
19	18	ancrd 299	. . . . . . . . 9 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (B (_ (A vH C) -> (A (_ (A vH C) /\ B (_ (A vH C))))
20		chlubt 9432	. . . . . . . . . 10 |- ((A e. CH /\ B e. CH /\ (A vH C) e. CH) -> ((A (_ (A vH C) /\ B (_ (A vH C)) <-> (A vH B) (_ (A vH C)))
21		chjclt 9329	. . . . . . . . . . 11 |- ((A e. CH /\ C e. CH) -> (A vH C) e. CH)
22	21	3adant2 798	. . . . . . . . . 10 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A vH C) e. CH)
23	20, 22	syld3an3 870	. . . . . . . . 9 |- ((A e. CH /\ B e. CH /\ C e. CH) -> ((A (_ (A vH C) /\ B (_ (A vH C)) <-> (A vH B) (_ (A vH C)))
24	19, 23	sylibd 202	. . . . . . . 8 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (B (_ (A vH C) -> (A vH B) (_ (A vH C)))
25		id 59	. . . . . . . 8 |- (A e. CH -> A e. CH)
26	24, 25, 9, 3	syl3an 868	. . . . . . 7 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> (B (_ (A vH C) -> (A vH B) (_ (A vH C)))
27	26	adantrd 391	. . . . . 6 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> (A vH B) (_ (A vH C)))
28	15, 27	jcad 600	. . . . 5 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> (A (. (A vH B) /\ (A vH B) (_ (A vH C))))
29	28	imp 350	. . . 4 |- (((A e. CH /\ B e. Atoms /\ C e. Atoms) /\ (B (_ (A vH C) /\ (A i^i B) = 0H)) -> (A (. (A vH B) /\ (A vH B) (_ (A vH C)))
30	14, 26	anim12d 558	. . . . . . . . 9 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> (((A i^i B) = 0H /\ B (_ (A vH C)) -> (A (. (A vH B) /\ (A vH B) (_ (A vH C))))
31	30	ancomsd 437	. . . . . . . 8 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> (A (. (A vH B) /\ (A vH B) (_ (A vH C))))
32		psssstr 2152	. . . . . . . 8 |- ((A (. (A vH B) /\ (A vH B) (_ (A vH C)) -> A (. (A vH C))
33	31, 32	syl6 22	. . . . . . 7 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> A (. (A vH C)))
34		chcv2t 10283	. . . . . . . 8 |- ((A e. CH /\ C e. Atoms) -> (A (. (A vH C) <-> A <o (A vH C)))
35	34	3adant2 798	. . . . . . 7 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> (A (. (A vH C) <-> A <o (A vH C)))
36	33, 35	sylibd 202	. . . . . 6 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> A <o (A vH C)))
37		3simp1 788	. . . . . . . . 9 |- ((A e. CH /\ B e. CH /\ C e. CH) -> A e. CH)
38	8	3adant3 799	. . . . . . . . 9 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A vH B) e. CH)
39	37, 22, 38	3jca 819	. . . . . . . 8 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A e. CH /\ (A vH C) e. CH /\ (A vH B) e. CH))
40	39, 25, 9, 3	syl3an 868	. . . . . . 7 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> (A e. CH /\ (A vH C) e. CH /\ (A vH B) e. CH))
41		cvnbtwn2t 10214	. . . . . . 7 |- ((A e. CH /\ (A vH C) e. CH /\ (A vH B) e. CH) -> (A <o (A vH C) -> ((A (. (A vH B) /\ (A vH B) (_ (A vH C)) -> (A vH B) = (A vH C))))
42	40, 41	syl 10	. . . . . 6 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> (A <o (A vH C) -> ((A (. (A vH B) /\ (A vH B) (_ (A vH C)) -> (A vH B) = (A vH C))))
43	36, 42	syld 27	. . . . 5 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> ((A (. (A vH B) /\ (A vH B) (_ (A vH C)) -> (A vH B) = (A vH C))))
44	43	imp 350	. . . 4 |- (((A e. CH /\ B e. Atoms /\ C e. Atoms) /\ (B (_ (A vH C) /\ (A i^i B) = 0H)) -> ((A (. (A vH B) /\ (A vH B) (_ (A vH C)) -> (A vH B) = (A vH C)))
45	29, 44	mpd 26	. . 3 |- (((A e. CH /\ B e. Atoms /\ C e. Atoms) /\ (B (_ (A vH C) /\ (A i^i B) = 0H)) -> (A vH B) = (A vH C))
46	6, 45	sseqtr4d 2098	. 2 |- (((A e. CH /\ B e. Atoms /\ C e. Atoms) /\ (B (_ (A vH C) /\ (A i^i B) = 0H)) -> C (_ (A vH B))
47	46	ex 373	1 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> C (_ (A vH B)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958   i^i cin 2046   (_ wss 2047   (. wpss 2048   class class class wbr 2619  (class class class)co 3963  CHcch 8798   vH chj 8802  0Hc0h 8804  Atomscat 8833   <o ccv 8834

This theorem is referenced by:  atoml 10309  atcvatlem 10312  atcvat4 10324  mdsymlem3 10332  mdsymlem5 10334

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625  ax-ac 4744  ax-hilex 8869  ax-hfvadd 8870  ax-hvcom 8871  ax-hvass 8872  ax-hv0cl 8873  ax-hvaddid 8874  ax-hfvmul 8875  ax-hvmulid 8876  ax-hvmulass 8877  ax-hvdistr1 8878  ax-hvdistr2 8879  ax-hvmul0 8880  ax-hfi 8946  ax-his1 8949  ax-his2 8950  ax-his3 8951  ax-his4 8952  ax-hcompl 9071

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-iin 2569  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er