Theorem spansncvt 9593

Description: Hilbert space has the covering property (using spans of singletons to represent atoms). Exercise 5 of [Kalmbach] p. 153.

Assertion

Ref	Expression
spansncvt	|- ((A e. CH /\ B e. CH /\ C e. H~) -> ((A (. B /\ B (_ (A vH (span` {C}))) -> B = (A vH (span` {C}))))

Proof of Theorem spansncvt

Step	Hyp	Ref	Expression
1		psseq1 2138	. . . 4 |- (A = if(A e. CH, A, H~) -> (A (. B <-> if(A e. CH, A, H~) (. B))
2		opreq1 3974	. . . . 5 |- (A = if(A e. CH, A, H~) -> (A vH (span` {C})) = (if(A e. CH, A, H~) vH (span` {C})))
3	2	sseq2d 2092	. . . 4 |- (A = if(A e. CH, A, H~) -> (B (_ (A vH (span` {C})) <-> B (_ (if(A e. CH, A, H~) vH (span` {C}))))
4	1, 3	anbi12d 630	. . 3 |- (A = if(A e. CH, A, H~) -> ((A (. B /\ B (_ (A vH (span` {C}))) <-> (if(A e. CH, A, H~) (. B /\ B (_ (if(A e. CH, A, H~) vH (span` {C})))))
5	2	eqeq2d 1489	. . 3 |- (A = if(A e. CH, A, H~) -> (B = (A vH (span` {C})) <-> B = (if(A e. CH, A, H~) vH (span` {C}))))
6	4, 5	imbi12d 628	. 2 |- (A = if(A e. CH, A, H~) -> (((A (. B /\ B (_ (A vH (span` {C}))) -> B = (A vH (span` {C}))) <-> ((if(A e. CH, A, H~) (. B /\ B (_ (if(A e. CH, A, H~) vH (span` {C}))) -> B = (if(A e. CH, A, H~) vH (span` {C})))))
7		psseq2 2139	. . . 4 |- (B = if(B e. CH, B, H~) -> (if(A e. CH, A, H~) (. B <-> if(A e. CH, A, H~) (. if(B e. CH, B, H~)))
8		sseq1 2085	. . . 4 |- (B = if(B e. CH, B, H~) -> (B (_ (if(A e. CH, A, H~) vH (span` {C})) <-> if(B e. CH, B, H~) (_ (if(A e. CH, A, H~) vH (span` {C}))))
9	7, 8	anbi12d 630	. . 3 |- (B = if(B e. CH, B, H~) -> ((if(A e. CH, A, H~) (. B /\ B (_ (if(A e. CH, A, H~) vH (span` {C}))) <-> (if(A e. CH, A, H~) (. if(B e. CH, B, H~) /\ if(B e. CH, B, H~) (_ (if(A e. CH, A, H~) vH (span` {C})))))
10		eqeq1 1484	. . 3 |- (B = if(B e. CH, B, H~) -> (B = (if(A e. CH, A, H~) vH (span` {C})) <-> if(B e. CH, B, H~) = (if(A e. CH, A, H~) vH (span` {C}))))
11	9, 10	imbi12d 628	. 2 |- (B = if(B e. CH, B, H~) -> (((if(A e. CH, A, H~) (. B /\ B (_ (if(A e. CH, A, H~) vH (span` {C}))) -> B = (if(A e. CH, A, H~) vH (span` {C}))) <-> ((if(A e. CH, A, H~) (. if(B e. CH, B, H~) /\ if(B e. CH, B, H~) (_ (if(A e. CH, A, H~) vH (span` {C}))) -> if(B e. CH, B, H~) = (if(A e. CH, A, H~) vH (span` {C})))))
12		sneq 2421	. . . . . . 7 |- (C = if(C e. H~, C, 0h) -> {C} = {if(C e. H~, C, 0h)})
13	12	fveq2d 3734	. . . . . 6 |- (C = if(C e. H~, C, 0h) -> (span` {C}) = (span` {if(C e. H~, C, 0h)}))
14	13	opreq2d 3982	. . . . 5 |- (C = if(C e. H~, C, 0h) -> (if(A e. CH, A, H~) vH (span` {C})) = (if(A e. CH, A, H~) vH (span` {if(C e. H~, C, 0h)})))
15	14	sseq2d 2092	. . . 4 |- (C = if(C e. H~, C, 0h) -> (if(B e. CH, B, H~) (_ (if(A e. CH, A, H~) vH (span` {C})) <-> if(B e. CH, B, H~) (_ (if(A e. CH, A, H~) vH (span` {if(C e. H~, C, 0h)}))))
16	15	anbi2d 618	. . 3 |- (C = if(C e. H~, C, 0h) -> ((if(A e. CH, A, H~) (. if(B e. CH, B, H~) /\ if(B e. CH, B, H~) (_ (if(A e. CH, A, H~) vH (span` {C}))) <-> (if(A e. CH, A, H~) (. if(B e. CH, B, H~) /\ if(B e. CH, B, H~) (_ (if(A e. CH, A, H~) vH (span` {if(C e. H~, C, 0h)})))))
17	14	eqeq2d 1489	. . 3 |- (C = if(C e. H~, C, 0h) -> (if(B e. CH, B, H~) = (if(A e. CH, A, H~) vH (span` {C})) <-> if(B e. CH, B, H~) = (if(A e. CH, A, H~) vH (span` {if(C e. H~, C, 0h)}))))
18	16, 17	imbi12d 628	. 2 |- (C = if(C e. H~, C, 0h) -> (((if(A e. CH, A, H~) (. if(B e. CH, B, H~) /\ if(B e. CH, B, H~) (_ (if(A e. CH, A, H~) vH (span` {C}))) -> if(B e. CH, B, H~) = (if(A e. CH, A, H~) vH (span` {C}))) <-> ((if(A e. CH, A, H~) (. if(B e. CH, B, H~) /\ if(B e. CH, B, H~) (_ (if(A e. CH, A, H~) vH (span` {if(C e. H~, C, 0h)}))) -> if(B e. CH, B, H~) = (if(A e. CH, A, H~) vH (span` {if(C e. H~, C, 0h)})))))
19		helch 9111	. . . 4 |- H~ e. CH
20	19	elimel 2398	. . 3 |- if(A e. CH, A, H~) e. CH
21	19	elimel 2398	. . 3 |- if(B e. CH, B, H~) e. CH
22		ax-hv0cl 8868	. . . 4 |- 0h e. H~
23	22	elimel 2398	. . 3 |- if(C e. H~, C, 0h) e. H~
24	20, 21, 23	spansncv 9592	. 2 |- ((if(A e. CH, A, H~) (. if(B e. CH, B, H~) /\ if(B e. CH, B, H~) (_ (if(A e. CH, A, H~) vH (span` {if(C e. H~, C, 0h)}))) -> if(B e. CH, B, H~) = (if(A e. CH, A, H~) vH (span` {if(C e. H~, C, 0h)})))
25	6, 11, 18, 24	dedth3h 2392	1 |- ((A e. CH /\ B e. CH /\ C e. H~) -> ((A (. B /\ B (_ (A vH (span` {C}))) -> B = (A vH (span` {C}))))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960   (_ wss 2050   (. wpss 2051  ifcif 2365  {csn 2413  ` cfv 3188  (class class class)co 3969  H~chil 8783  0hc0v 8786  CHcch 8793  spancspn 8796   vH chj 8797

This theorem is referenced by:  spansncv2t 10215

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634  ax-ac 4754  ax-hilex 8864  ax-hfvadd 8865  ax-hvcom 8866  ax-hvass 8867  ax-hv0cl 8868  ax-hvaddid 8869  ax-hfvmul 8870  ax-hvmulid 8871  ax-hvmulass 8872  ax-hvdistr1 8873  ax-hvdistr2 8874  ax-hvmul0 8875  ax-hfi 8941  ax-his1 8944  ax-his2 8945  ax-his3 8946  ax-his4 8947  ax-hcompl 9066

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-iin 2573  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-map 4330  df-en 4374  df-dom 4375  df-sdom 4376  df-sup 4583  df-r1 4653  df-rank 4654  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715  df-n 5927  df-2 5972  df-3 5973  df-4 5974  df-n0 6102  df-z 6138  df-fl 6226  df-q 6257  df-seq1 6309  df-shft 6342  df-ioo