Theorem hvaddsub4t 8866

Description: Hilbert vector space addition/subtraction law.

Assertion

Ref	Expression
hvaddsub4t	|- (((A e. H~ /\ B e. H~) /\ (C e. H~ /\ D e. H~)) -> ((A +h B) = (C +h D) <-> (A -h C) = (D -h B)))

Proof of Theorem hvaddsub4t

Step	Hyp	Ref	Expression
1		hvsubcan2t 8863	. . 3 |- (((A +h B) e. H~ /\ (C +h D) e. H~ /\ (C +h B) e. H~) -> (((A +h B) -h (C +h B)) = ((C +h D) -h (C +h B)) <-> (A +h B) = (C +h D)))
2		hvaddclt 8803	. . . 4 |- ((A e. H~ /\ B e. H~) -> (A +h B) e. H~)
3	2	adantr 389	. . 3 |- (((A e. H~ /\ B e. H~) /\ (C e. H~ /\ D e. H~)) -> (A +h B) e. H~)
4		hvaddclt 8803	. . . 4 |- ((C e. H~ /\ D e. H~) -> (C +h D) e. H~)
5	4	adantl 388	. . 3 |- (((A e. H~ /\ B e. H~) /\ (C e. H~ /\ D e. H~)) -> (C +h D) e. H~)
6		hvaddclt 8803	. . . . 5 |- ((C e. H~ /\ B e. H~) -> (C +h B) e. H~)
7	6	ancoms 436	. . . 4 |- ((B e. H~ /\ C e. H~) -> (C +h B) e. H~)
8	7	ad2ant2lr 410	. . 3 |- (((A e. H~ /\ B e. H~) /\ (C e. H~ /\ D e. H~)) -> (C +h B) e. H~)
9	1, 3, 5, 8	syl3anc 856	. 2 |- (((A e. H~ /\ B e. H~) /\ (C e. H~ /\ D e. H~)) -> (((A +h B) -h (C +h B)) = ((C +h D) -h (C +h B)) <-> (A +h B) = (C +h D)))
10		pm3.27 323	. . . . . . . 8 |- ((A e. H~ /\ B e. H~) -> B e. H~)
11	10	anim2i 335	. . . . . . 7 |- ((C e. H~ /\ (A e. H~ /\ B e. H~)) -> (C e. H~ /\ B e. H~))
12	11	ancoms 436	. . . . . 6 |- (((A e. H~ /\ B e. H~) /\ C e. H~) -> (C e. H~ /\ B e. H~))
13		hvsub4t 8827	. . . . . 6 |- (((A e. H~ /\ B e. H~) /\ (C e. H~ /\ B e. H~)) -> ((A +h B) -h (C +h B)) = ((A -h C) +h (B -h B)))
14	12, 13	syldan 467	. . . . 5 |- (((A e. H~ /\ B e. H~) /\ C e. H~) -> ((A +h B) -h (C +h B)) = ((A -h C) +h (B -h B)))
15		hvsubidt 8816	. . . . . . 7 |- (B e. H~ -> (B -h B) = 0h)
16	15	ad2antlr 405	. . . . . 6 |- (((A e. H~ /\ B e. H~) /\ C e. H~) -> (B -h B) = 0h)
17	16	opreq2d 3961	. . . . 5 |- (((A e. H~ /\ B e. H~) /\ C e. H~) -> ((A -h C) +h (B -h B)) = ((A -h C) +h 0h))
18		hvsubclt 8808	. . . . . . 7 |- ((A e. H~ /\ C e. H~) -> (A -h C) e. H~)
19		ax-hvaddid 8795	. . . . . . 7 |- ((A -h C) e. H~ -> ((A -h C) +h 0h) = (A -h C))
20	18, 19	syl 10	. . . . . 6 |- ((A e. H~ /\ C e. H~) -> ((A -h C) +h 0h) = (A -h C))
21	20	adantlr 393	. . . . 5 |- (((A e. H~ /\ B e. H~) /\ C e. H~) -> ((A -h C) +h 0h) = (A -h C))
22	14, 17, 21	3eqtrd 1503	. . . 4 |- (((A e. H~ /\ B e. H~) /\ C e. H~) -> ((A +h B) -h (C +h B)) = (A -h C))
23	22	adantrr 395	. . 3 |- (((A e. H~ /\ B e. H~) /\ (C e. H~ /\ D e. H~)) -> ((A +h B) -h (C +h B)) = (A -h C))
24		pm3.26 319	. . . . . . . 8 |- ((C e. H~ /\ D e. H~) -> C e. H~)
25	24	anim1i 334	. . . . . . 7 |- (((C e. H~ /\ D e. H~) /\ B e. H~) -> (C e. H~ /\ B e. H~))
26		hvsub4t 8827	. . . . . . 7 |- (((C e. H~ /\ D e. H~) /\ (C e. H~ /\ B e. H~)) -> ((C +h D) -h (C +h B)) = ((C -h C) +h (D -h B)))
27	25, 26	syldan 467	. . . . . 6 |- (((C e. H~ /\ D e. H~) /\ B e. H~) -> ((C +h D) -h (C +h B)) = ((C -h C) +h (D -h B)))
28		hvsubidt 8816	. . . . . . . 8 |- (C e. H~ -> (C -h C) = 0h)
29	28	ad2antrr 404	. . . . . . 7 |- (((C e. H~ /\ D e. H~) /\ B e. H~) -> (C -h C) = 0h)
30	29	opreq1d 3960	. . . . . 6 |- (((C e. H~ /\ D e. H~) /\ B e. H~) -> ((C -h C) +h (D -h B)) = (0h +h (D -h B)))
31		hvsubclt 8808	. . . . . . . 8 |- ((D e. H~ /\ B e. H~) -> (D -h B) e. H~)
32		hvaddid2t 8813	. . . . . . . 8 |- ((D -h B) e. H~ -> (0h +h (D -h B)) = (D -h B))
33	31, 32	syl 10	. . . . . . 7 |- ((D e. H~ /\ B e. H~) -> (0h +h (D -h B)) = (D -h B))
34	33	adantll 392	. . . . . 6 |- (((C e. H~ /\ D e. H~) /\ B e. H~) -> (0h +h (D -h B)) = (D -h B))
35	27, 30, 34	3eqtrd 1503	. . . . 5 |- (((C e. H~ /\ D e. H~) /\ B e. H~) -> ((C +h D) -h (C +h B)) = (D -h B))
36	35	ancoms 436	. . . 4 |- ((B e. H~ /\ (C e. H~ /\ D e. H~)) -> ((C +h D) -h (C +h B)) = (D -h B))
37	36	adantll 392	. . 3 |- (((A e. H~ /\ B e. H~) /\ (C e. H~ /\ D e. H~)) -> ((C +h D) -h (C +h B)) = (D -h B))
38	23, 37	eqeq12d 1481	. 2 |- (((A e. H~ /\ B e. H~) /\ (C e. H~ /\ D e. H~)) -> (((A +h B) -h (C +h B)) = ((C +h D) -h (C +h B)) <-> (A -h C) = (D -h B)))
39	9, 38	bitr3d 528	1 |- (((A e. H~ /\ B e. H~) /\ (C e. H~ /\ D e. H~)) -> ((A +h B) = (C +h D) <-> (A -h C) = (D -h B)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  (class class class)co 3948  H~chil 8727   +h cva 8728  0hc0v 8730   -h cmv 8731

This theorem is referenced by:  cdjreu 10264

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597  ax-hfvadd 8791  ax-hvcom 8792  ax-hvass 8793  ax-hv0cl 8794  ax-hvaddid 8795  ax-hfvmul 8796  ax-hvmulid 8797  ax-hvmulass 8798  ax-hvdistr1 8799  ax-hvdistr2 8800  ax-hvmul0 8801

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-hvsub 8779

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