Theorem basis2t 7614

Description: Property of a basis.

Assertion

Ref	Expression
basis2t	|- (((B e. Bases /\ C e. B) /\ (D e. B /\ A e. (C i^i D))) -> E.x e. B (A e. x /\ x (_ (C i^i D)))

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

Proof of Theorem basis2t

Step	Hyp	Ref	Expression
1		isbasis2g 7611	. . . . 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 (_ (y i^i z))))
2	1	ibi 594	. . . 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 (_ (y i^i z)))
3		ineq1 2213	. . . . . . 7 |- (y = C -> (y i^i z) = (C i^i z))
4		sseq2 2086	. . . . . . . . . 10 |- ((y i^i z) = (C i^i z) -> (x (_ (y i^i z) <-> x (_ (C i^i z)))
5	4	anbi2d 618	. . . . . . . . 9 |- ((y i^i z) = (C i^i z) -> ((w e. x /\ x (_ (y i^i z)) <-> (w e. x /\ x (_ (C i^i z))))
6	5	rexbidv 1667	. . . . . . . 8 |- ((y i^i z) = (C i^i z) -> (E.x e. B (w e. x /\ x (_ (y i^i z)) <-> E.x e. B (w e. x /\ x (_ (C i^i z))))
7	6	raleqd 1794	. . . . . . 7 |- ((y i^i z) = (C i^i z) -> (A.w e. (y i^i z)E.x e. B (w e. x /\ x (_ (y i^i z)) <-> A.w e. (C i^i z)E.x e. B (w e. x /\ x (_ (C i^i z))))
8	3, 7	syl 10	. . . . . 6 |- (y = C -> (A.w e. (y i^i z)E.x e. B (w e. x /\ x (_ (y i^i z)) <-> A.w e. (C i^i z)E.x e. B (w e. x /\ x (_ (C i^i z))))
9		ineq2 2214	. . . . . . 7 |- (z = D -> (C i^i z) = (C i^i D))
10		sseq2 2086	. . . . . . . . . 10 |- ((C i^i z) = (C i^i D) -> (x (_ (C i^i z) <-> x (_ (C i^i D)))
11	10	anbi2d 618	. . . . . . . . 9 |- ((C i^i z) = (C i^i D) -> ((w e. x /\ x (_ (C i^i z)) <-> (w e. x /\ x (_ (C i^i D))))
12	11	rexbidv 1667	. . . . . . . 8 |- ((C i^i z) = (C i^i D) -> (E.x e. B (w e. x /\ x (_ (C i^i z)) <-> E.x e. B (w e. x /\ x (_ (C i^i D))))
13	12	raleqd 1794	. . . . . . 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 i^i z)) <-> A.w e. (C i^i D)E.x e. B (w e. x /\ x (_ (C i^i D))))
14	9, 13	syl 10	. . . . . 6 |- (z = D -> (A.w e. (C i^i z)E.x e. B (w e. x /\ x (_ (C i^i z)) <-> A.w e. (C i^i D)E.x e. B (w e. x /\ x (_ (C i^i D))))
15	8, 14	rcla42v 1883	. . . . 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 (_ (y i^i z)) -> A.w e. (C i^i D)E.x e. B (w e. x /\ x (_ (C i^i D))))
16		eleq1 1537	. . . . . . . 8 |- (w = A -> (w e. x <-> A e. x))
17	16	anbi1d 619	. . . . . . 7 |- (w = A -> ((w e. x /\ x (_ (C i^i D)) <-> (A e. x /\ x (_ (C i^i D))))
18	17	rexbidv 1667	. . . . . 6 |- (w = A -> (E.x e. B (w e. x /\ x (_ (C i^i D)) <-> E.x e. B (A e. x /\ x (_ (C i^i D))))
19	18	rcla4cv 1877	. . . . 5 |- (A.w e. (C i^i D)E.x e. B (w e. x /\ x (_ (C i^i D)) -> (A e. (C i^i D) -> E.x e. B (A e. x /\ x (_ (C i^i D))))
20	15, 19	syl6com 53	. . . 4 |- (A.y e. B A.z e. B A.w e. (y i^i z)E.x e. B (w e. x /\ x (_ (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 i^i D)))))
21	2, 20	syl 10	. . 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 i^i D)))))
22	21	exp3a 376	. 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 i^i D))))))
23	22	imp43 370	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 i^i D)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  A.wral 1648  E.wrex 1649   i^i cin 2049   (_ wss 2050  Basesctb 7592

This theorem is referenced by:  tgclt 7623

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-10 968  ax-12 970  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

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-v 1815  df-in 2054  df-ss 2056  df-pw 2406  df-uni 2508  df-bases 7596

Copyright terms: Public domain