Theorem cdjreui 27552

Description: A member of the sum of disjoint subspaces has a unique decomposition. Part of Lemma 5 of [Holland] p. 1520. (Contributed by NM, 20-May-2005.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdjreu.1	|- A e. SH
cdjreu.2	|- B e. SH

Assertion

Ref	Expression
cdjreui	|- ( ( C e. ( A +H B ) /\ ( A i^i B ) = 0H ) -> E! x e. A E. y e. B C = ( x +h y ) )

Distinct variable groups:   x, y, A   x, B, y   x, C, y

Proof of Theorem cdjreui

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cdjreu.1	. . . . 5 |- A e. SH
2		cdjreu.2	. . . . 5 |- B e. SH
3	1, 2	shseli 26435	. . . 4 |- ( C e. ( A +H B ) <-> E. x e. A E. y e. B C = ( x +h y ) )
4	3	biimpi 194	. . 3 |- ( C e. ( A +H B ) -> E. x e. A E. y e. B C = ( x +h y ) )
5		reeanv 3022	. . . . 5 |- ( E. y e. B E. w e. B ( C = ( x +h y ) /\ C = ( z +h w ) ) <-> ( E. y e. B C = ( x +h y ) /\ E. w e. B C = ( z +h w ) ) )
6		eqtr2 2481	. . . . . . 7 |- ( ( C = ( x +h y ) /\ C = ( z +h w ) ) -> ( x +h y ) = ( z +h w ) )
7	1	sheli 26332	. . . . . . . . . . . 12 |- ( x e. A -> x e. ~H )
8	2	sheli 26332	. . . . . . . . . . . 12 |- ( y e. B -> y e. ~H )
9	7, 8	anim12i 564	. . . . . . . . . . 11 |- ( ( x e. A /\ y e. B ) -> ( x e. ~H /\ y e. ~H ) )
10	1	sheli 26332	. . . . . . . . . . . 12 |- ( z e. A -> z e. ~H )
11	2	sheli 26332	. . . . . . . . . . . 12 |- ( w e. B -> w e. ~H )
12	10, 11	anim12i 564	. . . . . . . . . . 11 |- ( ( z e. A /\ w e. B ) -> ( z e. ~H /\ w e. ~H ) )
13		hvaddsub4 26196	. . . . . . . . . . 11 |- ( ( ( x e. ~H /\ y e. ~H ) /\ ( z e. ~H /\ w e. ~H ) ) -> ( ( x +h y ) = ( z +h w ) <-> ( x -h z ) = ( w -h y ) ) )
14	9, 12, 13	syl2an 475	. . . . . . . . . 10 |- ( ( ( x e. A /\ y e. B ) /\ ( z e. A /\ w e. B ) ) -> ( ( x +h y ) = ( z +h w ) <-> ( x -h z ) = ( w -h y ) ) )
15	14	an4s 824	. . . . . . . . 9 |- ( ( ( x e. A /\ z e. A ) /\ ( y e. B /\ w e. B ) ) -> ( ( x +h y ) = ( z +h w ) <-> ( x -h z ) = ( w -h y ) ) )
16	15	adantll 711	. . . . . . . 8 |- ( ( ( ( A i^i B ) = 0H /\ ( x e. A /\ z e. A ) ) /\ ( y e. B /\ w e. B ) ) -> ( ( x +h y ) = ( z +h w ) <-> ( x -h z ) = ( w -h y ) ) )
17		shsubcl 26339	. . . . . . . . . . . . . . . 16 |- ( ( B e. SH /\ w e. B /\ y e. B ) -> ( w -h y ) e. B )
18	2, 17	mp3an1 1309	. . . . . . . . . . . . . . 15 |- ( ( w e. B /\ y e. B ) -> ( w -h y ) e. B )
19	18	ancoms 451	. . . . . . . . . . . . . 14 |- ( ( y e. B /\ w e. B ) -> ( w -h y ) e. B )
20		eleq1 2526	. . . . . . . . . . . . . 14 |- ( ( x -h z ) = ( w -h y ) -> ( ( x -h z ) e. B <-> ( w -h y ) e. B ) )
21	19, 20	syl5ibrcom 222	. . . . . . . . . . . . 13 |- ( ( y e. B /\ w e. B ) -> ( ( x -h z ) = ( w -h y ) -> ( x -h z ) e. B ) )
22	21	adantl 464	. . . . . . . . . . . 12 |- ( ( ( x e. A /\ z e. A ) /\ ( y e. B /\ w e. B ) ) -> ( ( x -h z ) = ( w -h y ) -> ( x -h z ) e. B ) )
23		shsubcl 26339	. . . . . . . . . . . . . 14 |- ( ( A e. SH /\ x e. A /\ z e. A ) -> ( x -h z ) e. A )
24	1, 23	mp3an1 1309	. . . . . . . . . . . . 13 |- ( ( x e. A /\ z e. A ) -> ( x -h z ) e. A )
25	24	adantr 463	. . . . . . . . . . . 12 |- ( ( ( x e. A /\ z e. A ) /\ ( y e. B /\ w e. B ) ) -> ( x -h z ) e. A )
26	22, 25	jctild 541	. . . . . . . . . . 11 |- ( ( ( x e. A /\ z e. A ) /\ ( y e. B /\ w e. B ) ) -> ( ( x -h z ) = ( w -h y ) -> ( ( x -h z ) e. A /\ ( x -h z ) e. B ) ) )
27	26	adantll 711	. . . . . . . . . 10 |- ( ( ( ( A i^i B ) = 0H /\ ( x e. A /\ z e. A ) ) /\ ( y e. B /\ w e. B ) ) -> ( ( x -h z ) = ( w -h y ) -> ( ( x -h z ) e. A /\ ( x -h z ) e. B ) ) )
28		elin 3673	. . . . . . . . . . . 12 |- ( ( x -h z ) e. ( A i^i B ) <-> ( ( x -h z ) e. A /\ ( x -h z ) e. B ) )
29		eleq2 2527	. . . . . . . . . . . 12 |- ( ( A i^i B ) = 0H -> ( ( x -h z ) e. ( A i^i B ) <-> ( x -h z ) e. 0H ) )
30	28, 29	syl5bbr 259	. . . . . . . . . . 11 |- ( ( A i^i B ) = 0H -> ( ( ( x -h z ) e. A /\ ( x -h z ) e. B ) <-> ( x -h z ) e. 0H ) )
31	30	ad2antrr 723	. . . . . . . . . 10 |- ( ( ( ( A i^i B ) = 0H /\ ( x e. A /\ z e. A ) ) /\ ( y e. B /\ w e. B ) ) -> ( ( ( x -h z ) e. A /\ ( x -h z ) e. B ) <-> ( x -h z ) e. 0H ) )
32	27, 31	sylibd 214	. . . . . . . . 9 |- ( ( ( ( A i^i B ) = 0H /\ ( x e. A /\ z e. A ) ) /\ ( y e. B /\ w e. B ) ) -> ( ( x -h z ) = ( w -h y ) -> ( x -h z ) e. 0H ) )
33		elch0 26373	. . . . . . . . . . . 12 |- ( ( x -h z ) e. 0H <-> ( x -h z ) = 0h )
34		hvsubeq0 26186	. . . . . . . . . . . 12 |- ( ( x e. ~H /\ z e. ~H ) -> ( ( x -h z ) = 0h <-> x = z ) )
35	33, 34	syl5bb 257	. . . . . . . . . . 11 |- ( ( x e. ~H /\ z e. ~H ) -> ( ( x -h z ) e. 0H <-> x = z ) )
36	7, 10, 35	syl2an 475	. . . . . . . . . 10 |- ( ( x e. A /\ z e. A ) -> ( ( x -h z ) e. 0H <-> x = z ) )
37	36	ad2antlr 724	. . . . . . . . 9 |- ( ( ( ( A i^i B ) = 0H /\ ( x e. A /\ z e. A ) ) /\ ( y e. B /\ w e. B ) ) -> ( ( x -h z ) e. 0H <-> x = z ) )
38	32, 37	sylibd 214	. . . . . . . 8 |- ( ( ( ( A i^i B ) = 0H /\ ( x e. A /\ z e. A ) ) /\ ( y e. B /\ w e. B ) ) -> ( ( x -h z ) = ( w -h y ) -> x = z ) )
39	16, 38	sylbid 215	. . . . . . 7 |- ( ( ( ( A i^i B ) = 0H /\ ( x e. A /\ z e. A ) ) /\ ( y e. B /\ w e. B ) ) -> ( ( x +h y ) = ( z +h w ) -> x = z ) )
40	6, 39	syl5 32	. . . . . 6 |- ( ( ( ( A i^i B ) = 0H /\ ( x e. A /\ z e. A ) ) /\ ( y e. B /\ w e. B ) ) -> ( ( C = ( x +h y ) /\ C = ( z +h w ) ) -> x = z ) )
41	40	rexlimdvva 2953	. . . . 5 |- ( ( ( A i^i B ) = 0H /\ ( x e. A /\ z e. A ) ) -> ( E. y e. B E. w e. B ( C = ( x +h y ) /\ C = ( z +h w ) ) -> x = z ) )
42	5, 41	syl5bir 218	. . . 4 |- ( ( ( A i^i B ) = 0H /\ ( x e. A /\ z e. A ) ) -> ( ( E. y e. B C = ( x +h y ) /\ E. w e. B C = ( z +h w ) ) -> x = z ) )
43	42	ralrimivva 2875	. . 3 |- ( ( A i^i B ) = 0H -> A. x e. A A. z e. A ( ( E. y e. B C = ( x +h y ) /\ E. w e. B C = ( z +h w ) ) -> x = z ) )
44	4, 43	anim12i 564	. 2 |- ( ( C e. ( A +H B ) /\ ( A i^i B ) = 0H ) -> ( E. x e. A E. y e. B C = ( x +h y ) /\ A. x e. A A. z e. A ( ( E. y e. B C = ( x +h y ) /\ E. w e. B C = ( z +h w ) ) -> x = z ) ) )
45		oveq1 6277	. . . . . 6 |- ( x = z -> ( x +h y ) = ( z +h y ) )
46	45	eqeq2d 2468	. . . . 5 |- ( x = z -> ( C = ( x +h y ) <-> C = ( z +h y ) ) )
47	46	rexbidv 2965	. . . 4 |- ( x = z -> ( E. y e. B C = ( x +h y ) <-> E. y e. B C = ( z +h y ) ) )
48		oveq2 6278	. . . . . 6 |- ( y = w -> ( z +h y ) = ( z +h w ) )
49	48	eqeq2d 2468	. . . . 5 |- ( y = w -> ( C = ( z +h y ) <-> C = ( z +h w ) ) )
50	49	cbvrexv 3082	. . . 4 |- ( E. y e. B C = ( z +h y ) <-> E. w e. B C = ( z +h w ) )
51	47, 50	syl6bb 261	. . 3 |- ( x = z -> ( E. y e. B C = ( x +h y ) <-> E. w e. B C = ( z +h w ) ) )
52	51	reu4 3290	. 2 |- ( E! x e. A E. y e. B C = ( x +h y ) <-> ( E. x e. A E. y e. B C = ( x +h y ) /\ A. x e. A A. z e. A ( ( E. y e. B C = ( x +h y ) /\ E. w e. B C = ( z +h w ) ) -> x = z ) ) )
53	44, 52	sylibr 212	1 |- ( ( C e. ( A +H B ) /\ ( A i^i B ) = 0H ) -> E! x e. A E. y e. B C = ( x +h y ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   A.wral 2804   E.wrex 2805   E!wreu 2806   i^i cin 3460  (class class class)co 6270   ~Hchil 26037   +h cva 26038   0hc0v 26042   -h cmv 26043   SHcsh 26046   +H cph 26049   0Hc0h 26053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-hilex 26117  ax-hfvadd 26118  ax-hvcom 26119  ax-hvass 26120  ax-hv0cl 26121  ax-hvaddid 26122  ax-hfvmul 26123  ax-hvmulid 26124  ax-hvmulass 26125  ax-hvdistr1 26126  ax-hvdistr2 26127  ax-hvmul0 26128

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-grpo 25394  df-ablo 25485  df-hvsub 26089  df-sh 26325  df-ch0 26372  df-shs 26427

This theorem is referenced by:  cdj3lem2  27555