Theorem wemaplem1 8451

Description: Value of the lexicographic order on a sequence space. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Hypothesis

Ref	Expression
wemapso.t	|- T = { <. x , y >. | E. z e. A ( ( x ` z ) S ( y ` z ) /\ A. w e. A ( w R z -> ( x ` w ) = ( y ` w ) ) ) }

Assertion

Ref	Expression
wemaplem1	|- ( ( P e. V /\ Q e. W ) -> ( P T Q <-> E. a e. A ( ( P ` a ) S ( Q ` a ) /\ A. b e. A ( b R a -> ( P ` b ) = ( Q ` b ) ) ) ) )

Distinct variable groups:   a, b, x   T, a, b   w, a, y, z, b, x, A   P, a, b, w, x, y, z   Q, a, b, w, x, y, z   R, a, b, w, x, y, z   S, a, b, w, x, y, z

Allowed substitution hints:   T( x, y, z, w)   V( x, y, z, w, a, b)   W( x, y, z, w, a, b)

Proof of Theorem wemaplem1

Step	Hyp	Ref	Expression
1		fveq1 6190	. . . . . 6 |- ( x = P -> ( x ` z ) = ( P ` z ) )
2		fveq1 6190	. . . . . 6 |- ( y = Q -> ( y ` z ) = ( Q ` z ) )
3	1, 2	breqan12d 4669	. . . . 5 |- ( ( x = P /\ y = Q ) -> ( ( x ` z ) S ( y ` z ) <-> ( P ` z ) S ( Q ` z ) ) )
4		fveq1 6190	. . . . . . . 8 |- ( x = P -> ( x ` w ) = ( P ` w ) )
5		fveq1 6190	. . . . . . . 8 |- ( y = Q -> ( y ` w ) = ( Q ` w ) )
6	4, 5	eqeqan12d 2638	. . . . . . 7 |- ( ( x = P /\ y = Q ) -> ( ( x ` w ) = ( y ` w ) <-> ( P ` w ) = ( Q ` w ) ) )
7	6	imbi2d 330	. . . . . 6 |- ( ( x = P /\ y = Q ) -> ( ( w R z -> ( x ` w ) = ( y ` w ) ) <-> ( w R z -> ( P ` w ) = ( Q ` w ) ) ) )
8	7	ralbidv 2986	. . . . 5 |- ( ( x = P /\ y = Q ) -> ( A. w e. A ( w R z -> ( x ` w ) = ( y ` w ) ) <-> A. w e. A ( w R z -> ( P ` w ) = ( Q ` w ) ) ) )
9	3, 8	anbi12d 747	. . . 4 |- ( ( x = P /\ y = Q ) -> ( ( ( x ` z ) S ( y ` z ) /\ A. w e. A ( w R z -> ( x ` w ) = ( y ` w ) ) ) <-> ( ( P ` z ) S ( Q ` z ) /\ A. w e. A ( w R z -> ( P ` w ) = ( Q ` w ) ) ) ) )
10	9	rexbidv 3052	. . 3 |- ( ( x = P /\ y = Q ) -> ( E. z e. A ( ( x ` z ) S ( y ` z ) /\ A. w e. A ( w R z -> ( x ` w ) = ( y ` w ) ) ) <-> E. z e. A ( ( P ` z ) S ( Q ` z ) /\ A. w e. A ( w R z -> ( P ` w ) = ( Q ` w ) ) ) ) )
11		fveq2 6191	. . . . . 6 |- ( z = a -> ( P ` z ) = ( P ` a ) )
12		fveq2 6191	. . . . . 6 |- ( z = a -> ( Q ` z ) = ( Q ` a ) )
13	11, 12	breq12d 4666	. . . . 5 |- ( z = a -> ( ( P ` z ) S ( Q ` z ) <-> ( P ` a ) S ( Q ` a ) ) )
14		breq2 4657	. . . . . . . 8 |- ( z = a -> ( w R z <-> w R a ) )
15	14	imbi1d 331	. . . . . . 7 |- ( z = a -> ( ( w R z -> ( P ` w ) = ( Q ` w ) ) <-> ( w R a -> ( P ` w ) = ( Q ` w ) ) ) )
16	15	ralbidv 2986	. . . . . 6 |- ( z = a -> ( A. w e. A ( w R z -> ( P ` w ) = ( Q ` w ) ) <-> A. w e. A ( w R a -> ( P ` w ) = ( Q ` w ) ) ) )
17		breq1 4656	. . . . . . . 8 |- ( w = b -> ( w R a <-> b R a ) )
18		fveq2 6191	. . . . . . . . 9 |- ( w = b -> ( P ` w ) = ( P ` b ) )
19		fveq2 6191	. . . . . . . . 9 |- ( w = b -> ( Q ` w ) = ( Q ` b ) )
20	18, 19	eqeq12d 2637	. . . . . . . 8 |- ( w = b -> ( ( P ` w ) = ( Q ` w ) <-> ( P ` b ) = ( Q ` b ) ) )
21	17, 20	imbi12d 334	. . . . . . 7 |- ( w = b -> ( ( w R a -> ( P ` w ) = ( Q ` w ) ) <-> ( b R a -> ( P ` b ) = ( Q ` b ) ) ) )
22	21	cbvralv 3171	. . . . . 6 |- ( A. w e. A ( w R a -> ( P ` w ) = ( Q ` w ) ) <-> A. b e. A ( b R a -> ( P ` b ) = ( Q ` b ) ) )
23	16, 22	syl6bb 276	. . . . 5 |- ( z = a -> ( A. w e. A ( w R z -> ( P ` w ) = ( Q ` w ) ) <-> A. b e. A ( b R a -> ( P ` b ) = ( Q ` b ) ) ) )
24	13, 23	anbi12d 747	. . . 4 |- ( z = a -> ( ( ( P ` z ) S ( Q ` z ) /\ A. w e. A ( w R z -> ( P ` w ) = ( Q ` w ) ) ) <-> ( ( P ` a ) S ( Q ` a ) /\ A. b e. A ( b R a -> ( P ` b ) = ( Q ` b ) ) ) ) )
25	24	cbvrexv 3172	. . 3 |- ( E. z e. A ( ( P ` z ) S ( Q ` z ) /\ A. w e. A ( w R z -> ( P ` w ) = ( Q ` w ) ) ) <-> E. a e. A ( ( P ` a ) S ( Q ` a ) /\ A. b e. A ( b R a -> ( P ` b ) = ( Q ` b ) ) ) )
26	10, 25	syl6bb 276	. 2 |- ( ( x = P /\ y = Q ) -> ( E. z e. A ( ( x ` z ) S ( y ` z ) /\ A. w e. A ( w R z -> ( x ` w ) = ( y ` w ) ) ) <-> E. a e. A ( ( P ` a ) S ( Q ` a ) /\ A. b e. A ( b R a -> ( P ` b ) = ( Q ` b ) ) ) ) )
27		wemapso.t	. 2 |- T = { <. x , y >. | E. z e. A ( ( x ` z ) S ( y ` z ) /\ A. w e. A ( w R z -> ( x ` w ) = ( y ` w ) ) ) }
28	26, 27	brabga 4989	1 |- ( ( P e. V /\ Q e. W ) -> ( P T Q <-> E. a e. A ( ( P ` a ) S ( Q ` a ) /\ A. b e. A ( b R a -> ( P ` b ) = ( Q ` b ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   {copab 4712   ` cfv 5888

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-eu 2474  df-mo 2475  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

This theorem is referenced by:  wemaplem2  8452  wemaplem3  8453  wemappo  8454  wemapsolem  8455