Theorem cdleme31sde 35673

Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.)

Hypotheses

Ref	Expression
cdleme31sde.c	|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdleme31sde.e	|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdleme31sde.x	|- Y = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme31sde.z	|- Z = ( ( P .\/ Q ) ./\ ( Y .\/ ( ( R .\/ S ) ./\ W ) ) )

Assertion

Ref	Expression
cdleme31sde	|- ( ( R e. A /\ S e. A ) -> [_ R / s ]_ [_ S / t ]_ E = Z )

Distinct variable groups:   t, s, A   .\/ , s, t   ./\ , s, t   P, s, t   Q, s, t   R, s   S, s, t   W, s, t   Y, s, t

Allowed substitution hints:   D( t, s)   R( t)   U( t, s)   E( t, s)   Z( t, s)

Proof of Theorem cdleme31sde

Step	Hyp	Ref	Expression
1		cdleme31sde.e	. . . . 5 |- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
2	1	csbeq2i 3993	. . . 4 |- [_ S / t ]_ E = [_ S / t ]_ ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
3		nfcvd 2765	. . . . 5 |- ( S e. A -> F/_ t ( ( P .\/ Q ) ./\ ( Y .\/ ( ( s .\/ S ) ./\ W ) ) ) )
4		oveq1 6657	. . . . . . . . 9 |- ( t = S -> ( t .\/ U ) = ( S .\/ U ) )
5		oveq2 6658	. . . . . . . . . . 11 |- ( t = S -> ( P .\/ t ) = ( P .\/ S ) )
6	5	oveq1d 6665	. . . . . . . . . 10 |- ( t = S -> ( ( P .\/ t ) ./\ W ) = ( ( P .\/ S ) ./\ W ) )
7	6	oveq2d 6666	. . . . . . . . 9 |- ( t = S -> ( Q .\/ ( ( P .\/ t ) ./\ W ) ) = ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8	4, 7	oveq12d 6668	. . . . . . . 8 |- ( t = S -> ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) )
9		cdleme31sde.c	. . . . . . . 8 |- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
10		cdleme31sde.x	. . . . . . . 8 |- Y = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
11	8, 9, 10	3eqtr4g 2681	. . . . . . 7 |- ( t = S -> D = Y )
12		oveq2 6658	. . . . . . . 8 |- ( t = S -> ( s .\/ t ) = ( s .\/ S ) )
13	12	oveq1d 6665	. . . . . . 7 |- ( t = S -> ( ( s .\/ t ) ./\ W ) = ( ( s .\/ S ) ./\ W ) )
14	11, 13	oveq12d 6668	. . . . . 6 |- ( t = S -> ( D .\/ ( ( s .\/ t ) ./\ W ) ) = ( Y .\/ ( ( s .\/ S ) ./\ W ) ) )
15	14	oveq2d 6666	. . . . 5 |- ( t = S -> ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( Y .\/ ( ( s .\/ S ) ./\ W ) ) ) )
16	3, 15	csbiegf 3557	. . . 4 |- ( S e. A -> [_ S / t ]_ ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( Y .\/ ( ( s .\/ S ) ./\ W ) ) ) )
17	2, 16	syl5eq 2668	. . 3 |- ( S e. A -> [_ S / t ]_ E = ( ( P .\/ Q ) ./\ ( Y .\/ ( ( s .\/ S ) ./\ W ) ) ) )
18	17	csbeq2dv 3992	. 2 |- ( S e. A -> [_ R / s ]_ [_ S / t ]_ E = [_ R / s ]_ ( ( P .\/ Q ) ./\ ( Y .\/ ( ( s .\/ S ) ./\ W ) ) ) )
19		eqid 2622	. . 3 |- ( ( P .\/ Q ) ./\ ( Y .\/ ( ( s .\/ S ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( Y .\/ ( ( s .\/ S ) ./\ W ) ) )
20		cdleme31sde.z	. . 3 |- Z = ( ( P .\/ Q ) ./\ ( Y .\/ ( ( R .\/ S ) ./\ W ) ) )
21	19, 20	cdleme31se 35670	. 2 |- ( R e. A -> [_ R / s ]_ ( ( P .\/ Q ) ./\ ( Y .\/ ( ( s .\/ S ) ./\ W ) ) ) = Z )
22	18, 21	sylan9eqr 2678	1 |- ( ( R e. A /\ S e. A ) -> [_ R / s ]_ [_ S / t ]_ E = Z )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   [_csb 3533  (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:  cdlemefs44  35714  cdlemefs45ee  35718  cdleme17d2  35783