Theorem cdleme31sn2 35677

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

Hypotheses

Ref	Expression
cdleme32sn2.d	|- D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdleme31sn2.n	|- N = if ( s .<_ ( P .\/ Q ) , I , D )
cdleme31sn2.c	|- C = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )

Assertion

Ref	Expression
cdleme31sn2	|- ( ( R e. A /\ -. R .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N = C )

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

Allowed substitution hints:   C( s)   D( s)   I( s)   N( s)

Proof of Theorem cdleme31sn2

Step	Hyp	Ref	Expression
1		cdleme31sn2.n	. . . . 5 |- N = if ( s .<_ ( P .\/ Q ) , I , D )
2		eqid 2622	. . . . 5 |- 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	. . . 4 |- ( R e. A -> [_ R / s ]_ N = if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) )
4	3	adantr 481	. . 3 |- ( ( R e. A /\ -. R .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N = if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) )
5		iffalse 4095	. . . . 5 |- ( -. R .<_ ( P .\/ Q ) -> if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) = [_ R / s ]_ D )
6		cdleme32sn2.d	. . . . . 6 |- D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
7	6	csbeq2i 3993	. . . . 5 |- [_ R / s ]_ D = [_ R / s ]_ ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
8	5, 7	syl6eq 2672	. . . 4 |- ( -. R .<_ ( P .\/ Q ) -> if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) = [_ R / s ]_ ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) ) )
9		nfcvd 2765	. . . . 5 |- ( R e. A -> F/_ s ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
10		oveq1 6657	. . . . . 6 |- ( s = R -> ( s .\/ U ) = ( R .\/ U ) )
11		oveq2 6658	. . . . . . . 8 |- ( s = R -> ( P .\/ s ) = ( P .\/ R ) )
12	11	oveq1d 6665	. . . . . . 7 |- ( s = R -> ( ( P .\/ s ) ./\ W ) = ( ( P .\/ R ) ./\ W ) )
13	12	oveq2d 6666	. . . . . 6 |- ( s = R -> ( Q .\/ ( ( P .\/ s ) ./\ W ) ) = ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
14	10, 13	oveq12d 6668	. . . . 5 |- ( s = R -> ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) ) = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
15	9, 14	csbiegf 3557	. . . 4 |- ( R e. A -> [_ R / s ]_ ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) ) = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
16	8, 15	sylan9eqr 2678	. . 3 |- ( ( R e. A /\ -. R .<_ ( P .\/ Q ) ) -> if ( R .<_ ( P .\/ Q ) , [_ R / s ]_ I , [_ R / s ]_ D ) = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
17	4, 16	eqtrd 2656	. 2 |- ( ( R e. A /\ -. R .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
18		cdleme31sn2.c	. 2 |- C = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
19	17, 18	syl6eqr 2674	1 |- ( ( R e. A /\ -. R .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   [_csb 3533   ifcif 4086   class class class wbr 4653  (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-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-fv 5896  df-ov 6653

This theorem is referenced by:  cdlemefr32sn2aw  35692  cdleme43frv1snN  35696  cdlemefr31fv1  35699  cdleme35sn2aw  35746  cdleme35sn3a  35747