Theorem cdleme31sn1 35669

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)

Hypotheses

Ref	Expression
cdleme31sn1.i	|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) )
cdleme31sn1.n	|- N = if ( s .<_ ( P .\/ Q ) , I , D )
cdleme31sn1.c	|- C = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = [_ R / s ]_ G ) )

Assertion

Ref	Expression
cdleme31sn1	|- ( ( R e. A /\ R .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N = C )

Distinct variable groups:   t, s, y, A   B, s   .\/ , s   .<_ , s   P, s   Q, s   R, s, t, y   W, s

Allowed substitution hints:   B( y, t)   C( y, t, s)   D( y, t, s)   P( y, t)   Q( y, t)   G( y, t, s)   I( y, t, s)   .\/ ( y, t)   .<_ ( y, t)   N( y, t, s)   W( y, t)

Proof of Theorem cdleme31sn1

Step	Hyp	Ref	Expression
1		cdleme31sn1.n	. . . 4 |- N = if ( s .<_ ( P .\/ Q ) , I , D )
2		eqid 2622	. . . 4 |- if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) = if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D )
3	1, 2	cdleme31sn 35668	. . 3 |- ( R e. A -> [_ R / s ]_ N = if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) )
4	3	adantr 481	. 2 |- ( ( R e. A /\ R .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N = if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) )
5		iftrue 4092	. . . . 5 |- ( R .<_ ( P .\/ Q ) -> if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) = [_ R / s ]_ I )
6		cdleme31sn1.i	. . . . . 6 |- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) )
7	6	csbeq2i 3993	. . . . 5 |- [_ R / s ]_ I = [_ R / s ]_ ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) )
8	5, 7	syl6eq 2672	. . . 4 |- ( R .<_ ( P .\/ Q ) -> if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) = [_ R / s ]_ ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) ) )
9		nfcv 2764	. . . . . . . 8 |- F/_ s A
10		nfv 1843	. . . . . . . . 9 |- F/ s ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) )
11		nfcsb1v 3549	. . . . . . . . . 10 |- F/_ s [_ R / s ]_ G
12	11	nfeq2 2780	. . . . . . . . 9 |- F/ s y = [_ R / s ]_ G
13	10, 12	nfim 1825	. . . . . . . 8 |- F/ s ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = [_ R / s ]_ G )
14	9, 13	nfral 2945	. . . . . . 7 |- F/ s A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = [_ R / s ]_ G )
15		nfcv 2764	. . . . . . 7 |- F/_ s B
16	14, 15	nfriota 6620	. . . . . 6 |- F/_ s ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = [_ R / s ]_ G ) )
17	16	a1i 11	. . . . 5 |- ( R e. A -> F/_ s ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = [_ R / s ]_ G ) ) )
18		csbeq1a 3542	. . . . . . . . 9 |- ( s = R -> G = [_ R / s ]_ G )
19	18	eqeq2d 2632	. . . . . . . 8 |- ( s = R -> ( y = G <-> y = [_ R / s ]_ G ) )
20	19	imbi2d 330	. . . . . . 7 |- ( s = R -> ( ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) <-> ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = [_ R / s ]_ G ) ) )
21	20	ralbidv 2986	. . . . . 6 |- ( s = R -> ( A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) <-> A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = [_ R / s ]_ G ) ) )
22	21	riotabidv 6613	. . . . 5 |- ( s = R -> ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) ) = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = [_ R / s ]_ G ) ) )
23	17, 22	csbiegf 3557	. . . 4 |- ( R e. A -> [_ R / s ]_ ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) ) = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = [_ R / s ]_ G ) ) )
24	8, 23	sylan9eqr 2678	. . 3 |- ( ( R e. A /\ R .<_ ( P .\/ Q ) ) -> if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = [_ R / s ]_ G ) ) )
25		cdleme31sn1.c	. . 3 |- C = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = [_ R / s ]_ G ) )
26	24, 25	syl6eqr 2674	. 2 |- ( ( R e. A /\ R .<_ ( P .\/ Q ) ) -> if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) = C )
27	4, 26	eqtrd 2656	1 |- ( ( R e. A /\ R .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   A.wral 2912   [_csb 3533   ifcif 4086   class class class wbr 4653   iota_crio 6610  (class class class)co 6650

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-sbc 3436  df-csb 3534  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-iota 5851  df-riota 6611

This theorem is referenced by:  cdleme31sn1c  35676