Theorem hvaddsub4 10578

Description: Hilbert vector space addition/subtraction law.

Assertion

Ref	Expression
hvaddsub4	|- (((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 hvaddsub4

Step	Hyp	Ref	Expression
1		hvaddcl 10514	. . . 4 |- ((A e. ~H /\ B e. ~H) -> (A +h B) e. ~H)
2	1	adantr 425	. . 3 |- (((A e. ~H /\ B e. ~H) /\ (C e. ~H /\ D e. ~H)) -> (A +h B) e. ~H)
3		hvaddcl 10514	. . . 4 |- ((C e. ~H /\ D e. ~H) -> (C +h D) e. ~H)
4	3	adantl 424	. . 3 |- (((A e. ~H /\ B e. ~H) /\ (C e. ~H /\ D e. ~H)) -> (C +h D) e. ~H)
5		hvaddcl 10514	. . . . 5 |- ((C e. ~H /\ B e. ~H) -> (C +h B) e. ~H)
6	5	ancoms 484	. . . 4 |- ((B e. ~H /\ C e. ~H) -> (C +h B) e. ~H)
7	6	ad2ant2lr 446	. . 3 |- (((A e. ~H /\ B e. ~H) /\ (C e. ~H /\ D e. ~H)) -> (C +h B) e. ~H)
8		hvsubcan2 10575	. . 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)))
9	2, 4, 7, 8	syl111anc 1100	. 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		simpr 350	. . . . . . . 8 |- ((A e. ~H /\ B e. ~H) -> B e. ~H)
11	10	anim2i 362	. . . . . . 7 |- ((C e. ~H /\ (A e. ~H /\ B e. ~H)) -> (C e. ~H /\ B e. ~H))
12	11	ancoms 484	. . . . . 6 |- (((A e. ~H /\ B e. ~H) /\ C e. ~H) -> (C e. ~H /\ B e. ~H))
13		hvsub4 10538	. . . . . 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 516	. . . . 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		hvsubid 10527	. . . . . . 7 |- (B e. ~H -> (B -h B) = 0h)
16	15	ad2antlr 441	. . . . . 6 |- (((A e. ~H /\ B e. ~H) /\ C e. ~H) -> (B -h B) = 0h)
17	16	opreq2d 4898	. . . . 5 |- (((A e. ~H /\ B e. ~H) /\ C e. ~H) -> ((A -h C) +h (B -h B)) = ((A -h C) +h 0h))
18		hvsubcl 10519	. . . . . . 7 |- ((A e. ~H /\ C e. ~H) -> (A -h C) e. ~H)
19		ax-hvaddid 10506	. . . . . . 7 |- ((A -h C) e. ~H -> ((A -h C) +h 0h) = (A -h C))
20	18, 19	syl 12	. . . . . 6 |- ((A e. ~H /\ C e. ~H) -> ((A -h C) +h 0h) = (A -h C))
21	20	adantlr 429	. . . . 5 |- (((A e. ~H /\ B e. ~H) /\ C e. ~H) -> ((A -h C) +h 0h) = (A -h C))
22	14, 17, 21	3eqtrd 1929	. . . 4 |- (((A e. ~H /\ B e. ~H) /\ C e. ~H) -> ((A +h B) -h (C +h B)) = (A -h C))
23	22	adantrr 431	. . 3 |- (((A e. ~H /\ B e. ~H) /\ (C e. ~H /\ D e. ~H)) -> ((A +h B) -h (C +h B)) = (A -h C))
24		simpl 346	. . . . . . . 8 |- ((C e. ~H /\ D e. ~H) -> C e. ~H)
25	24	anim1i 361	. . . . . . 7 |- (((C e. ~H /\ D e. ~H) /\ B e. ~H) -> (C e. ~H /\ B e. ~H))
26		hvsub4 10538	. . . . . . 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 516	. . . . . 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		hvsubid 10527	. . . . . . . 8 |- (C e. ~H -> (C -h C) = 0h)
29	28	ad2antrr 440	. . . . . . 7 |- (((C e. ~H /\ D e. ~H) /\ B e. ~H) -> (C -h C) = 0h)
30	29	opreq1d 4897	. . . . . 6 |- (((C e. ~H /\ D e. ~H) /\ B e. ~H) -> ((C -h C) +h (D -h B)) = (0h +h (D -h B)))
31		hvsubcl 10519	. . . . . . . 8 |- ((D e. ~H /\ B e. ~H) -> (D -h B) e. ~H)
32		hvaddid2 10524	. . . . . . . 8 |- ((D -h B) e. ~H -> (0h +h (D -h B)) = (D -h B))
33	31, 32	syl 12	. . . . . . 7 |- ((D e. ~H /\ B e. ~H) -> (0h +h (D -h B)) = (D -h B))
34	33	adantll 428	. . . . . 6 |- (((C e. ~H /\ D e. ~H) /\ B e. ~H) -> (0h +h (D -h B)) = (D -h B))
35	27, 30, 34	3eqtrd 1929	. . . . 5 |- (((C e. ~H /\ D e. ~H) /\ B e. ~H) -> ((C +h D) -h (C +h B)) = (D -h B))
36	35	ancoms 484	. . . 4 |- ((B e. ~H /\ (C e. ~H /\ D e. ~H)) -> ((C +h D) -h (C +h B)) = (D -h B))
37	36	adantll 428	. . 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 1899	. 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 589	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 163   /\ wa 240   = wceq 1298   e. wcel 1300  (class class class)co 4884  ~Hchil 10420   +h cva 10421  0hc0v 10423   -h cmv 10424

This theorem is referenced by:  cdjreui 12004

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731  ax-hfvadd 10502  ax-hvcom 10503  ax-hvass 10504  ax-hv0cl 10505  ax-hvaddid 10506  ax-hfvmul 10507  ax-hvmulid 10508  ax-hvmulass 10509  ax-hvdistr1 10510  ax-hvdistr2 10511  ax-hvmul0 10512

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-hvsub 10472

Copyright terms: Public domain