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Theorem hvsubsub4t 8922
Description: Hilbert vector space addition/subtraction law.
Assertion
Ref Expression
hvsubsub4t |- (((A e. H~ /\ B e. H~) /\ (C e. H~ /\ D e. H~)) -> ((A -h B) -h (C -h D)) = ((A -h C) -h (B -h D)))
Proof of Theorem hvsubsub4t
StepHypRef Expression
1 opreq1 3974 . . . 4 |- (A = if(A e. H~, A, 0h) -> (A -h B) = (if(A e. H~, A, 0h) -h B))
21opreq1d 3981 . . 3 |- (A = if(A e. H~, A, 0h) -> ((A -h B) -h (C -h D)) = ((if(A e. H~, A, 0h) -h B) -h (C -h D)))
3 opreq1 3974 . . . 4 |- (A = if(A e. H~, A, 0h) -> (A -h C) = (if(A e. H~, A, 0h) -h C))
43opreq1d 3981 . . 3 |- (A = if(A e. H~, A, 0h) -> ((A -h C) -h (B -h D)) = ((if(A e. H~, A, 0h) -h C) -h (B -h D)))
52, 4eqeq12d 1492 . 2 |- (A = if(A e. H~, A, 0h) -> (((A -h B) -h (C -h D)) = ((A -h C) -h (B -h D)) <-> ((if(A e. H~, A, 0h) -h B) -h (C -h D)) = ((if(A e. H~, A, 0h) -h C) -h (B -h D))))
6 opreq2 3975 . . . 4 |- (B = if(B e. H~, B, 0h) -> (if(A e. H~, A, 0h) -h B) = (if(A e. H~, A, 0h) -h if(B e. H~, B, 0h)))
76opreq1d 3981 . . 3 |- (B = if(B e. H~, B, 0h) -> ((if(A e. H~, A, 0h) -h B) -h (C -h D)) = ((if(A e. H~, A, 0h) -h if(B e. H~, B, 0h)) -h (C -h D)))
8 opreq1 3974 . . . 4 |- (B = if(B e. H~, B, 0h) -> (B -h D) = (if(B e. H~, B, 0h) -h D))
98opreq2d 3982 . . 3 |- (B = if(B e. H~, B, 0h) -> ((if(A e. H~, A, 0h) -h C) -h (B -h D)) = ((if(A e. H~, A, 0h) -h C) -h (if(B e. H~, B, 0h) -h D)))
107, 9eqeq12d 1492 . 2 |- (B = if(B e. H~, B, 0h) -> (((if(A e. H~, A, 0h) -h B) -h (C -h D)) = ((if(A e. H~, A, 0h) -h C) -h (B -h D)) <-> ((if(A e. H~, A, 0h) -h if(B e. H~, B, 0h)) -h (C -h D)) = ((if(A e. H~, A, 0h) -h C) -h (if(B e. H~, B, 0h) -h D))))
11 opreq1 3974 . . . 4 |- (C = if(C e. H~, C, 0h) -> (C -h D) = (if(C e. H~, C, 0h) -h D))
1211opreq2d 3982 . . 3 |- (C = if(C e. H~, C, 0h) -> ((if(A e. H~, A, 0h) -h if(B e. H~, B, 0h)) -h (C -h D)) = ((if(A e. H~, A, 0h) -h if(B e. H~, B, 0h)) -h (if(C e. H~, C, 0h) -h D)))
13 opreq2 3975 . . . 4 |- (C = if(C e. H~, C, 0h) -> (if(A e. H~, A, 0h) -h C) = (if(A e. H~, A, 0h) -h if(C e. H~, C, 0h)))
1413opreq1d 3981 . . 3 |- (C = if(C e. H~, C, 0h) -> ((if(A e. H~, A, 0h) -h C) -h (if(B e. H~, B, 0h) -h D)) = ((if(A e. H~, A, 0h) -h if(C e. H~, C, 0h)) -h (if(B e. H~, B, 0h) -h D)))
1512, 14eqeq12d 1492 . 2 |- (C = if(C e. H~, C, 0h) -> (((if(A e. H~, A, 0h) -h if(B e. H~, B, 0h)) -h (C -h D)) = ((if(A e. H~, A, 0h) -h C) -h (if(B e. H~, B, 0h) -h D)) <-> ((if(A e. H~, A, 0h) -h if(B e. H~, B, 0h)) -h (if(C e. H~, C, 0h) -h D)) = ((if(A e. H~, A, 0h) -h if(C e. H~, C, 0h)) -h (if(B e. H~, B, 0h) -h D))))
16 opreq2 3975 . . . 4 |- (D = if(D e. H~, D, 0h) -> (if(C e. H~, C, 0h) -h D) = (if(C e. H~, C, 0h) -h if(D e. H~, D, 0h)))
1716opreq2d 3982 . . 3 |- (D = if(D e. H~, D, 0h) -> ((if(A e. H~, A, 0h) -h if(B e. H~, B, 0h)) -h (if(C e. H~, C, 0h) -h D)) = ((if(A e. H~, A, 0h) -h if(B e. H~, B, 0h)) -h (if(C e. H~, C, 0h) -h if(D e. H~, D, 0h))))
18 opreq2 3975 . . . 4 |- (D = if(D e. H~, D, 0h) -> (if(B e. H~, B, 0h) -h D) = (if(B e. H~, B, 0h) -h if(D e. H~, D, 0h)))
1918opreq2d 3982 . . 3 |- (D = if(D e. H~, D, 0h) -> ((if(A e. H~, A, 0h) -h if(C e. H~, C, 0h)) -h (if(B e. H~, B, 0h) -h D)) = ((if(A e. H~, A, 0h) -h if(C e. H~, C, 0h)) -h (if(B e. H~, B, 0h) -h if(D e. H~, D, 0h))))
2017, 19eqeq12d 1492 . 2 |- (D = if(D e. H~, D, 0h) -> (((if(A e. H~, A, 0h) -h if(B e. H~, B, 0h)) -h (if(C e. H~, C, 0h) -h D)) = ((if(A e. H~, A, 0h) -h if(C e. H~, C, 0h)) -h (if(B e. H~, B, 0h) -h D)) <-> ((if(A e. H~, A, 0h) -h if(B e. H~, B, 0h)) -h (if(C e. H~, C, 0h) -h if(D e. H~, D, 0h))) = ((if(A e. H~, A, 0h) -h if(C e. H~, C, 0h)) -h (if(B e. H~, B, 0h) -h if(D e. H~, D, 0h)))))
21 ax-hv0cl 8868 . . . 4 |- 0h e. H~
2221elimel 2398 . . 3 |- if(A e. H~, A, 0h) e. H~
2321elimel 2398 . . 3 |- if(B e. H~, B, 0h) e. H~
2421elimel 2398 . . 3 |- if(C e. H~, C, 0h) e. H~
2521elimel 2398 . . 3 |- if(D e. H~, D, 0h) e. H~
2622, 23, 24, 25hvsubsub4 8921 . 2 |- ((if(A e. H~, A, 0h) -h if(B e. H~, B, 0h)) -h (if(C e. H~, C, 0h) -h if(D e. H~, D, 0h))) = ((if(A e. H~, A, 0h) -h if(C e. H~, C, 0h)) -h (if(B e. H~, B, 0h) -h if(D e. H~, D, 0h)))
275, 10, 15, 20, 26dedth4h 2393 1 |- (((A e. H~ /\ B e. H~) /\ (C e. H~ /\ D e. H~)) -> ((A -h B) -h (C -h D)) = ((A -h C) -h (B -h D)))
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  ifcif 2365  (class class class)co 3969  H~chil 8783  0hc0v 8786   -h cmv 8787
This theorem is referenced by:  5oalem3 9596  5oalem5 9598
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-inf2 4634  ax-hfvadd 8865  ax-hvcom 8866  ax-hvass 8867  ax-hv0cl 8868  ax-hfvmul 8870  ax-hvdistr1 8873
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-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-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-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-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-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-sub 5368  df-neg 5370  df-hvsub 8835
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