Theorem cdleme27b 35656

Description: Lemma for cdleme27N 35657. (Contributed by NM, 3-Feb-2013.)

Hypotheses

Ref	Expression
cdleme26.b	|- B = ( Base ` K )
cdleme26.l	|- .<_ = ( le ` K )
cdleme26.j	|- .\/ = ( join ` K )
cdleme26.m	|- ./\ = ( meet ` K )
cdleme26.a	|- A = ( Atoms ` K )
cdleme26.h	|- H = ( LHyp ` K )
cdleme27.u	|- U = ( ( P .\/ Q ) ./\ W )
cdleme27.f	|- F = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdleme27.z	|- Z = ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) )
cdleme27.n	|- N = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( s .\/ z ) ./\ W ) ) )
cdleme27.d	|- D = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) )
cdleme27.c	|- C = if ( s .<_ ( P .\/ Q ) , D , F )
cdleme27.g	|- G = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdleme27.o	|- O = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) )
cdleme27.e	|- E = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = O ) )
cdleme27.y	|- Y = if ( t .<_ ( P .\/ Q ) , E , G )

Assertion

Ref	Expression
cdleme27b	|- ( s = t -> C = Y )

Distinct variable groups:   t, s, u, z, A   B, s, t, u, z   u, F   u, G   H, s, t, z   .\/ , s, t, u, z   K, s, t, z   .<_ , s, t, u, z   ./\ , s, t, u, z   t, N, u   O, s, u   P, s, t, u, z   Q, s, t, u, z   U, s, t, u, z   W, s, t, u, z

Allowed substitution hints:   C( z, u, t, s)   D( z, u, t, s)   E( z, u, t, s)   F( z, t, s)   G( z, t, s)   H( u)   K( u)   N( z, s)   O( z, t)   Y( z, u, t, s)   Z( z, u, t, s)

Proof of Theorem cdleme27b

Step	Hyp	Ref	Expression
1		breq1 4656	. . 3 |- ( s = t -> ( s .<_ ( P .\/ Q ) <-> t .<_ ( P .\/ Q ) ) )
2		oveq1 6657	. . . . . . . . . . . 12 |- ( s = t -> ( s .\/ z ) = ( t .\/ z ) )
3	2	oveq1d 6665	. . . . . . . . . . 11 |- ( s = t -> ( ( s .\/ z ) ./\ W ) = ( ( t .\/ z ) ./\ W ) )
4	3	oveq2d 6666	. . . . . . . . . 10 |- ( s = t -> ( Z .\/ ( ( s .\/ z ) ./\ W ) ) = ( Z .\/ ( ( t .\/ z ) ./\ W ) ) )
5	4	oveq2d 6666	. . . . . . . . 9 |- ( s = t -> ( ( P .\/ Q ) ./\ ( Z .\/ ( ( s .\/ z ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) )
6		cdleme27.n	. . . . . . . . 9 |- N = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( s .\/ z ) ./\ W ) ) )
7		cdleme27.o	. . . . . . . . 9 |- O = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) )
8	5, 6, 7	3eqtr4g 2681	. . . . . . . 8 |- ( s = t -> N = O )
9	8	eqeq2d 2632	. . . . . . 7 |- ( s = t -> ( u = N <-> u = O ) )
10	9	imbi2d 330	. . . . . 6 |- ( s = t -> ( ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) <-> ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = O ) ) )
11	10	ralbidv 2986	. . . . 5 |- ( s = t -> ( A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) <-> A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = O ) ) )
12	11	riotabidv 6613	. . . 4 |- ( s = t -> ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) ) = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = O ) ) )
13		cdleme27.d	. . . 4 |- D = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) )
14		cdleme27.e	. . . 4 |- E = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = O ) )
15	12, 13, 14	3eqtr4g 2681	. . 3 |- ( s = t -> D = E )
16		oveq1 6657	. . . . 5 |- ( s = t -> ( s .\/ U ) = ( t .\/ U ) )
17		oveq2 6658	. . . . . . 7 |- ( s = t -> ( P .\/ s ) = ( P .\/ t ) )
18	17	oveq1d 6665	. . . . . 6 |- ( s = t -> ( ( P .\/ s ) ./\ W ) = ( ( P .\/ t ) ./\ W ) )
19	18	oveq2d 6666	. . . . 5 |- ( s = t -> ( Q .\/ ( ( P .\/ s ) ./\ W ) ) = ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
20	16, 19	oveq12d 6668	. . . 4 |- ( s = t -> ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) ) = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) )
21		cdleme27.f	. . . 4 |- F = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
22		cdleme27.g	. . . 4 |- G = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
23	20, 21, 22	3eqtr4g 2681	. . 3 |- ( s = t -> F = G )
24	1, 15, 23	ifbieq12d 4113	. 2 |- ( s = t -> if ( s .<_ ( P .\/ Q ) , D , F ) = if ( t .<_ ( P .\/ Q ) , E , G ) )
25		cdleme27.c	. 2 |- C = if ( s .<_ ( P .\/ Q ) , D , F )
26		cdleme27.y	. 2 |- Y = if ( t .<_ ( P .\/ Q ) , E , G )
27	24, 25, 26	3eqtr4g 2681	1 |- ( s = t -> C = Y )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   A.wral 2912   ifcif 4086   class class class wbr 4653   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Atomscatm 34550   LHypclh 35270

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-iota 5851  df-fv 5896  df-riota 6611  df-ov 6653

This theorem is referenced by:  cdleme27N  35657  cdleme28c  35660