Theorem axcontlem1 25844

Description: Lemma for axcont 25856. Change bound variables for later use. (Contributed by Scott Fenton, 20-Jun-2013.)

Hypothesis

Ref	Expression
axcontlem1.1	|- F = { <. x , t >. | ( x e. D /\ ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) ) }

Assertion

Ref	Expression
axcontlem1	|- F = { <. y , s >. | ( y e. D /\ ( s e. ( 0 [,) +oo ) /\ A. j e. ( 1 ... N ) ( y ` j ) = ( ( ( 1 - s ) x. ( Z ` j ) ) + ( s x. ( U ` j ) ) ) ) ) }

Distinct variable groups:   D, s, t, x, y   i, j, s, t, x, y, N   U, i, j, s, t, x, y   i, Z, j, s, t, x, y

Allowed substitution hints:   D( i, j)   F( x, y, t, i, j, s)

Proof of Theorem axcontlem1

Step	Hyp	Ref	Expression
1		axcontlem1.1	. 2 |- F = { <. x , t >. | ( x e. D /\ ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) ) }
2		eleq1 2689	. . . . 5 |- ( x = y -> ( x e. D <-> y e. D ) )
3	2	adantr 481	. . . 4 |- ( ( x = y /\ t = s ) -> ( x e. D <-> y e. D ) )
4		eleq1 2689	. . . . . 6 |- ( t = s -> ( t e. ( 0 [,) +oo ) <-> s e. ( 0 [,) +oo ) ) )
5	4	adantl 482	. . . . 5 |- ( ( x = y /\ t = s ) -> ( t e. ( 0 [,) +oo ) <-> s e. ( 0 [,) +oo ) ) )
6		fveq1 6190	. . . . . . . 8 |- ( x = y -> ( x ` i ) = ( y ` i ) )
7		oveq2 6658	. . . . . . . . . 10 |- ( t = s -> ( 1 - t ) = ( 1 - s ) )
8	7	oveq1d 6665	. . . . . . . . 9 |- ( t = s -> ( ( 1 - t ) x. ( Z ` i ) ) = ( ( 1 - s ) x. ( Z ` i ) ) )
9		oveq1 6657	. . . . . . . . 9 |- ( t = s -> ( t x. ( U ` i ) ) = ( s x. ( U ` i ) ) )
10	8, 9	oveq12d 6668	. . . . . . . 8 |- ( t = s -> ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) = ( ( ( 1 - s ) x. ( Z ` i ) ) + ( s x. ( U ` i ) ) ) )
11	6, 10	eqeqan12d 2638	. . . . . . 7 |- ( ( x = y /\ t = s ) -> ( ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) <-> ( y ` i ) = ( ( ( 1 - s ) x. ( Z ` i ) ) + ( s x. ( U ` i ) ) ) ) )
12	11	ralbidv 2986	. . . . . 6 |- ( ( x = y /\ t = s ) -> ( A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) <-> A. i e. ( 1 ... N ) ( y ` i ) = ( ( ( 1 - s ) x. ( Z ` i ) ) + ( s x. ( U ` i ) ) ) ) )
13		fveq2 6191	. . . . . . . 8 |- ( i = j -> ( y ` i ) = ( y ` j ) )
14		fveq2 6191	. . . . . . . . . 10 |- ( i = j -> ( Z ` i ) = ( Z ` j ) )
15	14	oveq2d 6666	. . . . . . . . 9 |- ( i = j -> ( ( 1 - s ) x. ( Z ` i ) ) = ( ( 1 - s ) x. ( Z ` j ) ) )
16		fveq2 6191	. . . . . . . . . 10 |- ( i = j -> ( U ` i ) = ( U ` j ) )
17	16	oveq2d 6666	. . . . . . . . 9 |- ( i = j -> ( s x. ( U ` i ) ) = ( s x. ( U ` j ) ) )
18	15, 17	oveq12d 6668	. . . . . . . 8 |- ( i = j -> ( ( ( 1 - s ) x. ( Z ` i ) ) + ( s x. ( U ` i ) ) ) = ( ( ( 1 - s ) x. ( Z ` j ) ) + ( s x. ( U ` j ) ) ) )
19	13, 18	eqeq12d 2637	. . . . . . 7 |- ( i = j -> ( ( y ` i ) = ( ( ( 1 - s ) x. ( Z ` i ) ) + ( s x. ( U ` i ) ) ) <-> ( y ` j ) = ( ( ( 1 - s ) x. ( Z ` j ) ) + ( s x. ( U ` j ) ) ) ) )
20	19	cbvralv 3171	. . . . . 6 |- ( A. i e. ( 1 ... N ) ( y ` i ) = ( ( ( 1 - s ) x. ( Z ` i ) ) + ( s x. ( U ` i ) ) ) <-> A. j e. ( 1 ... N ) ( y ` j ) = ( ( ( 1 - s ) x. ( Z ` j ) ) + ( s x. ( U ` j ) ) ) )
21	12, 20	syl6bb 276	. . . . 5 |- ( ( x = y /\ t = s ) -> ( A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) <-> A. j e. ( 1 ... N ) ( y ` j ) = ( ( ( 1 - s ) x. ( Z ` j ) ) + ( s x. ( U ` j ) ) ) ) )
22	5, 21	anbi12d 747	. . . 4 |- ( ( x = y /\ t = s ) -> ( ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) <-> ( s e. ( 0 [,) +oo ) /\ A. j e. ( 1 ... N ) ( y ` j ) = ( ( ( 1 - s ) x. ( Z ` j ) ) + ( s x. ( U ` j ) ) ) ) ) )
23	3, 22	anbi12d 747	. . 3 |- ( ( x = y /\ t = s ) -> ( ( x e. D /\ ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) ) <-> ( y e. D /\ ( s e. ( 0 [,) +oo ) /\ A. j e. ( 1 ... N ) ( y ` j ) = ( ( ( 1 - s ) x. ( Z ` j ) ) + ( s x. ( U ` j ) ) ) ) ) ) )
24	23	cbvopabv 4722	. 2 |- { <. x , t >. | ( x e. D /\ ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) ) } = { <. y , s >. | ( y e. D /\ ( s e. ( 0 [,) +oo ) /\ A. j e. ( 1 ... N ) ( y ` j ) = ( ( ( 1 - s ) x. ( Z ` j ) ) + ( s x. ( U ` j ) ) ) ) ) }
25	1, 24	eqtri 2644	1 |- F = { <. y , s >. | ( y e. D /\ ( s e. ( 0 [,) +oo ) /\ A. j e. ( 1 ... N ) ( y ` j ) = ( ( ( 1 - s ) x. ( Z ` j ) ) + ( s x. ( U ` j ) ) ) ) ) }

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {copab 4712   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   +oocpnf 10071   - cmin 10266   [,)cico 12177   ...cfz 12326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  axcontlem6  25849  axcontlem11  25854