Theorem basis1t 7599

Description: Property of a basis.

Assertion

Ref	Expression
basis1t	|- ((B e. Bases /\ C e. B /\ D e. B) -> (C i^i D) (_ U.(B i^i P~(C i^i D)))

Proof of Theorem basis1t

Step	Hyp	Ref	Expression
1		ineq1 2210	. . . . 5 |- (x = C -> (x i^i y) = (C i^i y))
2		pweq 2403	. . . . . . . 8 |- ((x i^i y) = (C i^i y) -> P~(x i^i y) = P~(C i^i y))
3	1, 2	syl 10	. . . . . . 7 |- (x = C -> P~(x i^i y) = P~(C i^i y))
4	3	ineq2d 2217	. . . . . 6 |- (x = C -> (B i^i P~(x i^i y)) = (B i^i P~(C i^i y)))
5	4	unieqd 2512	. . . . 5 |- (x = C -> U.(B i^i P~(x i^i y)) = U.(B i^i P~(C i^i y)))
6	1, 5	sseq12d 2090	. . . 4 |- (x = C -> ((x i^i y) (_ U.(B i^i P~(x i^i y)) <-> (C i^i y) (_ U.(B i^i P~(C i^i y))))
7		ineq2 2211	. . . . 5 |- (y = D -> (C i^i y) = (C i^i D))
8		pweq 2403	. . . . . . . 8 |- ((C i^i y) = (C i^i D) -> P~(C i^i y) = P~(C i^i D))
9	7, 8	syl 10	. . . . . . 7 |- (y = D -> P~(C i^i y) = P~(C i^i D))
10	9	ineq2d 2217	. . . . . 6 |- (y = D -> (B i^i P~(C i^i y)) = (B i^i P~(C i^i D)))
11	10	unieqd 2512	. . . . 5 |- (y = D -> U.(B i^i P~(C i^i y)) = U.(B i^i P~(C i^i D)))
12	7, 11	sseq12d 2090	. . . 4 |- (y = D -> ((C i^i y) (_ U.(B i^i P~(C i^i y)) <-> (C i^i D) (_ U.(B i^i P~(C i^i D))))
13	6, 12	rcla42v 1880	. . 3 |- ((C e. B /\ D e. B) -> (A.x e. B A.y e. B (x i^i y) (_ U.(B i^i P~(x i^i y)) -> (C i^i D) (_ U.(B i^i P~(C i^i D))))
14		isbasisg 7596	. . . 4 |- (B e. Bases -> (B e. Bases <-> A.x e. B A.y e. B (x i^i y) (_ U.(B i^i P~(x i^i y))))
15	14	ibi 592	. . 3 |- (B e. Bases -> A.x e. B A.y e. B (x i^i y) (_ U.(B i^i P~(x i^i y)))
16	13, 15	syl5com 52	. 2 |- (B e. Bases -> ((C e. B /\ D e. B) -> (C i^i D) (_ U.(B i^i P~(C i^i D))))
17	16	3impib 831	1 |- ((B e. Bases /\ C e. B /\ D e. B) -> (C i^i D) (_ U.(B i^i P~(C i^i D)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645   i^i cin 2046   (_ wss 2047  P~cpw 2401  U.cuni 2503  Basesctb 7575

This theorem is referenced by:  bastgt 7607

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-pw 2402  df-uni 2504  df-bases 7579

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