Theorem basis2 8884

Description: Property of a basis.

Assertion

Ref	Expression
basis2	|- (((B e. Bases /\ C e. B) /\ (D e. B /\ A e. (C i^i D))) -> E.x e. B (A e. x /\ x C_ (C i^i D)))

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

Proof of Theorem basis2

Step	Hyp	Ref	Expression
1		isbasis2g 8881	. . . . 5 |- (B e. Bases -> (B e. Bases <-> A.y e. B A.z e. B A.w e. (y i^i z)E.x e. B (w e. x /\ x C_ (y i^i z))))
2	1	ibi 652	. . . 4 |- (B e. Bases -> A.y e. B A.z e. B A.w e. (y i^i z)E.x e. B (w e. x /\ x C_ (y i^i z)))
3		ineq1 2789	. . . . . . 7 |- (y = C -> (y i^i z) = (C i^i z))
4		sseq2 2639	. . . . . . . . . 10 |- ((y i^i z) = (C i^i z) -> (x C_ (y i^i z) <-> x C_ (C i^i z)))
5	4	anbi2d 678	. . . . . . . . 9 |- ((y i^i z) = (C i^i z) -> ((w e. x /\ x C_ (y i^i z)) <-> (w e. x /\ x C_ (C i^i z))))
6	5	rexbidv 2124	. . . . . . . 8 |- ((y i^i z) = (C i^i z) -> (E.x e. B (w e. x /\ x C_ (y i^i z)) <-> E.x e. B (w e. x /\ x C_ (C i^i z))))
7	6	raleqbi1dv 2271	. . . . . . 7 |- ((y i^i z) = (C i^i z) -> (A.w e. (y i^i z)E.x e. B (w e. x /\ x C_ (y i^i z)) <-> A.w e. (C i^i z)E.x e. B (w e. x /\ x C_ (C i^i z))))
8	3, 7	syl 12	. . . . . 6 |- (y = C -> (A.w e. (y i^i z)E.x e. B (w e. x /\ x C_ (y i^i z)) <-> A.w e. (C i^i z)E.x e. B (w e. x /\ x C_ (C i^i z))))
9		ineq2 2790	. . . . . . 7 |- (z = D -> (C i^i z) = (C i^i D))
10		sseq2 2639	. . . . . . . . . 10 |- ((C i^i z) = (C i^i D) -> (x C_ (C i^i z) <-> x C_ (C i^i D)))
11	10	anbi2d 678	. . . . . . . . 9 |- ((C i^i z) = (C i^i D) -> ((w e. x /\ x C_ (C i^i z)) <-> (w e. x /\ x C_ (C i^i D))))
12	11	rexbidv 2124	. . . . . . . 8 |- ((C i^i z) = (C i^i D) -> (E.x e. B (w e. x /\ x C_ (C i^i z)) <-> E.x e. B (w e. x /\ x C_ (C i^i D))))
13	12	raleqbi1dv 2271	. . . . . . 7 |- ((C i^i z) = (C i^i D) -> (A.w e. (C i^i z)E.x e. B (w e. x /\ x C_ (C i^i z)) <-> A.w e. (C i^i D)E.x e. B (w e. x /\ x C_ (C i^i D))))
14	9, 13	syl 12	. . . . . 6 |- (z = D -> (A.w e. (C i^i z)E.x e. B (w e. x /\ x C_ (C i^i z)) <-> A.w e. (C i^i D)E.x e. B (w e. x /\ x C_ (C i^i D))))
15	8, 14	rcla42v 2384	. . . . 5 |- ((C e. B /\ D e. B) -> (A.y e. B A.z e. B A.w e. (y i^i z)E.x e. B (w e. x /\ x C_ (y i^i z)) -> A.w e. (C i^i D)E.x e. B (w e. x /\ x C_ (C i^i D))))
16		eleq1 1957	. . . . . . . 8 |- (w = A -> (w e. x <-> A e. x))
17	16	anbi1d 679	. . . . . . 7 |- (w = A -> ((w e. x /\ x C_ (C i^i D)) <-> (A e. x /\ x C_ (C i^i D))))
18	17	rexbidv 2124	. . . . . 6 |- (w = A -> (E.x e. B (w e. x /\ x C_ (C i^i D)) <-> E.x e. B (A e. x /\ x C_ (C i^i D))))
19	18	rcla4cv 2377	. . . . 5 |- (A.w e. (C i^i D)E.x e. B (w e. x /\ x C_ (C i^i D)) -> (A e. (C i^i D) -> E.x e. B (A e. x /\ x C_ (C i^i D))))
20	15, 19	syl6com 64	. . . 4 |- (A.y e. B A.z e. B A.w e. (y i^i z)E.x e. B (w e. x /\ x C_ (y i^i z)) -> ((C e. B /\ D e. B) -> (A e. (C i^i D) -> E.x e. B (A e. x /\ x C_ (C i^i D)))))
21	2, 20	syl 12	. . 3 |- (B e. Bases -> ((C e. B /\ D e. B) -> (A e. (C i^i D) -> E.x e. B (A e. x /\ x C_ (C i^i D)))))
22	21	exp3a 405	. 2 |- (B e. Bases -> (C e. B -> (D e. B -> (A e. (C i^i D) -> E.x e. B (A e. x /\ x C_ (C i^i D))))))
23	22	imp43 397	1 |- (((B e. Bases /\ C e. B) /\ (D e. B /\ A e. (C i^i D))) -> E.x e. B (A e. x /\ x C_ (C i^i D)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   i^i cin 2592   C_ wss 2593  Basesctb 8859

This theorem is referenced by:  tgcl 8894

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294  df-in 2603  df-ss 2605  df-pw 3035  df-uni 3178  df-bases 8863

Copyright terms: Public domain