Theorem cdjreu 10359

Description: A member of the sum of disjoint subspaces has a unique decomposition. Part of Lemma 5 of [Holland] p. 1520.

Hypotheses

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

Assertion

Ref	Expression
cdjreu	|- ((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 cdjreu

Step	Hyp	Ref	Expression
1		cdjreu.1	. . . . 5 |- A e. SH
2		cdjreu.2	. . . . 5 |- B e. SH
3	1, 2	shsel 9280	. . . 4 |- (C e. (A +H B) <-> E.x e. A E.y e. B C = (x +h y))
4	3	biimp 151	. . 3 |- (C e. (A +H B) -> E.x e. A E.y e. B C = (x +h y))
5		hvaddsub4t 8945	. . . . . . . . . . . . 13 |- (((x e. H~ /\ y e. H~) /\ (z e. H~ /\ w e. H~)) -> ((x +h y) = (z +h w) <-> (x -h z) = (w -h y)))
6	1	shel 9082	. . . . . . . . . . . . . 14 |- (x e. A -> x e. H~)
7	2	shel 9082	. . . . . . . . . . . . . 14 |- (y e. B -> y e. H~)
8	6, 7	anim12i 333	. . . . . . . . . . . . 13 |- ((x e. A /\ y e. B) -> (x e. H~ /\ y e. H~))
9	1	shel 9082	. . . . . . . . . . . . . 14 |- (z e. A -> z e. H~)
10	2	shel 9082	. . . . . . . . . . . . . 14 |- (w e. B -> w e. H~)
11	9, 10	anim12i 333	. . . . . . . . . . . . 13 |- ((z e. A /\ w e. B) -> (z e. H~ /\ w e. H~))
12	5, 8, 11	syl2an 454	. . . . . . . . . . . 12 |- (((x e. A /\ y e. B) /\ (z e. A /\ w e. B)) -> ((x +h y) = (z +h w) <-> (x -h z) = (w -h y)))
13	12	an4s 508	. . . . . . . . . . 11 |- (((x e. A /\ z e. A) /\ (y e. B /\ w e. B)) -> ((x +h y) = (z +h w) <-> (x -h z) = (w -h y)))
14	13	adantll 392	. . . . . . . . . 10 |- ((((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)))
15		eleq1 1534	. . . . . . . . . . . . . . . 16 |- ((x -h z) = (w -h y) -> ((x -h z) e. B <-> (w -h y) e. B))
16		shsubcltOLD 9090	. . . . . . . . . . . . . . . . . 18 |- (B e. SH -> ((w e. B /\ y e. B) -> (w -h y) e. B))
17	2, 16	ax-mp 7	. . . . . . . . . . . . . . . . 17 |- ((w e. B /\ y e. B) -> (w -h y) e. B)
18	17	ancoms 436	. . . . . . . . . . . . . . . 16 |- ((y e. B /\ w e. B) -> (w -h y) e. B)
19	15, 18	syl5cbir 211	. . . . . . . . . . . . . . 15 |- ((y e. B /\ w e. B) -> ((x -h z) = (w -h y) -> (x -h z) e. B))
20	19	adantl 388	. . . . . . . . . . . . . 14 |- (((x e. A /\ z e. A) /\ (y e. B /\ w e. B)) -> ((x -h z) = (w -h y) -> (x -h z) e. B))
21		shsubcltOLD 9090	. . . . . . . . . . . . . . . 16 |- (A e. SH -> ((x e. A /\ z e. A) -> (x -h z) e. A))
22	1, 21	ax-mp 7	. . . . . . . . . . . . . . 15 |- ((x e. A /\ z e. A) -> (x -h z) e. A)
23	22	adantr 389	. . . . . . . . . . . . . 14 |- (((x e. A /\ z e. A) /\ (y e. B /\ w e. B)) -> (x -h z) e. A)
24	20, 23	jctild 601	. . . . . . . . . . . . 13 |- (((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)))
25	24	adantll 392	. . . . . . . . . . . 12 |- ((((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)))
26		eleq2 1535	. . . . . . . . . . . . . 14 |- ((A i^i B) = 0H -> ((x -h z) e. (A i^i B) <-> (x -h z) e. 0H))
27		elin 2207	. . . . . . . . . . . . . 14 |- ((x -h z) e. (A i^i B) <-> ((x -h z) e. A /\ (x -h z) e. B))
28	26, 27	syl5bbr 534	. . . . . . . . . . . . 13 |- ((A i^i B) = 0H -> (((x -h z) e. A /\ (x -h z) e. B) <-> (x -h z) e. 0H))
29	28	ad2antrr 404	. . . . . . . . . . . 12 |- ((((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))
30	25, 29	sylibd 202	. . . . . . . . . . 11 |- ((((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))
31		hvsubeq0t 8935	. . . . . . . . . . . . . 14 |- ((x e. H~ /\ z e. H~) -> ((x -h z) = 0h <-> x = z))
32		elch0 9126	. . . . . . . . . . . . . 14 |- ((x -h z) e. 0H <-> (x -h z) = 0h)
33	31, 32	syl5bb 532	. . . . . . . . . . . . 13 |- ((x e. H~ /\ z e. H~) -> ((x -h z) e. 0H <-> x = z))
34	33, 6, 9	syl2an 454	. . . . . . . . . . . 12 |- ((x e. A /\ z e. A) -> ((x -h z) e. 0H <-> x = z))
35	34	ad2antlr 405	. . . . . . . . . . 11 |- ((((A i^i B) = 0H /\ (x e. A /\ z e. A)) /\ (y e. B /\ w e. B)) -> ((x -h z) e. 0H <-> x = z))
36	30, 35	sylibd 202	. . . . . . . . . 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 = z))
37	14, 36	sylbid 203	. . . . . . . . 9 |- ((((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))
38		eqtr2t 1493	. . . . . . . . 9 |- ((C = (x +h y) /\ C = (z +h w)) -> (x +h y) = (z +h w))
39	37, 38	syl5 21	. . . . . . . 8 |- ((((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))
40	39	ex 373	. . . . . . 7 |- (((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	r19.23advv 1749	. . . . . 6 |- (((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		reeanv 1778	. . . . . 6 |- (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)))
43	41, 42	syl5ibr 207	. . . . 5 |- (((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))
44	43	ex 373	. . . 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)))
45	44	r19.21aivv 1720	. . 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))
46	4, 45	anim12i 333	. 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)))
47		opreq1 3968	. . . . . 6 |- (x = z -> (x +h y) = (z +h y))
48	47	eqeq2d 1486	. . . . 5 |- (x = z -> (C = (x +h y) <-> C = (z +h y)))
49	48	rexbidv 1664	. . . 4 |- (x = z -> (E.y e. B C = (x +h y) <-> E.y e. B C = (z +h y)))
50		opreq2 3969	. . . . . 6 |- (y = w -> (z +h y) = (z +h w))
51	50	eqeq2d 1486	. . . . 5 |- (y = w -> (C = (z +h y) <-> C = (z +h w)))
52	51	cbvrexv 1801	. . . 4 |- (E.y e. B C = (z +h y) <-> E.w e. B C = (z +h w))
53	49, 52	syl6bb 536	. . 3 |- (x = z -> (E.y e. B C = (x +h y) <-> E.w e. B C = (z +h w)))
54	53	reu4 1934	. 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)))
55	46, 54	sylibr 200	1 |- ((C e. (A +H B) /\ (A i^i B) = 0H) -> E!x e. A E.y e. B C = (x +h y