Theorem cdlemk39s 34577

Description: Substitution version of cdlemk39 34554. TODO: Can any commonality with cdlemk35s 34575 be exploited? (Contributed by NM, 23-Jul-2013.)

Hypotheses

Ref	Expression
cdlemk5.b	|- B = ( Base ` K )
cdlemk5.l	|- .<_ = ( le ` K )
cdlemk5.j	|- .\/ = ( join ` K )
cdlemk5.m	|- ./\ = ( meet ` K )
cdlemk5.a	|- A = ( Atoms ` K )
cdlemk5.h	|- H = ( LHyp ` K )
cdlemk5.t	|- T = ( ( LTrn ` K ) ` W )
cdlemk5.r	|- R = ( ( trL ` K ) ` W )
cdlemk5.z	|- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
cdlemk5.y	|- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
cdlemk5.x	|- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )

Assertion

Ref	Expression
cdlemk39s	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( R ` [_ G / g ]_ X ) .<_ ( R ` G ) )

Distinct variable groups:   ./\ , g   .\/ , g   B, g   P, g   R, g   T, g   g, Z   g, b, G, z   ./\ , b, z   .<_ , b   z, g, .<_   .\/ , b, z   A, b, g, z   B, b, z   F, b, g, z   z, G   H, b, g, z   K, b, g, z   N, b, g, z   P, b, z   R, b, z   T, b, z   W, b, g, z   z, Y   G, b

Allowed substitution hints:   X( z, g, b)   Y( g, b)   Z( z, b)

Proof of Theorem cdlemk39s

Step	Hyp	Ref	Expression
1		simp22l 1149	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> G e. T )
2		cdlemk5.b	. . . . . 6 |- B = ( Base ` K )
3		cdlemk5.l	. . . . . 6 |- .<_ = ( le ` K )
4		cdlemk5.j	. . . . . 6 |- .\/ = ( join ` K )
5		cdlemk5.m	. . . . . 6 |- ./\ = ( meet ` K )
6		cdlemk5.a	. . . . . 6 |- A = ( Atoms ` K )
7		cdlemk5.h	. . . . . 6 |- H = ( LHyp ` K )
8		cdlemk5.t	. . . . . 6 |- T = ( ( LTrn ` K ) ` W )
9		cdlemk5.r	. . . . . 6 |- R = ( ( trL ` K ) ` W )
10		cdlemk5.z	. . . . . 6 |- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
11		cdlemk5.y	. . . . . 6 |- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
12		cdlemk5.x	. . . . . 6 |- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
13	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	cdlemk39 34554	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( g e. T /\ g =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( R ` X ) .<_ ( R ` g ) )
14	13	sbcth 3270	. . . 4 |- ( G e. T -> [. G / g ]. ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( g e. T /\ g =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( R ` X ) .<_ ( R ` g ) ) )
15		sbcimg 3297	. . . 4 |- ( G e. T -> ( [. G / g ]. ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( g e. T /\ g =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( R ` X ) .<_ ( R ` g ) ) <-> ( [. G / g ]. ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( g e. T /\ g =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> [. G / g ]. ( R ` X ) .<_ ( R ` g ) ) ) )
16	14, 15	mpbid 215	. . 3 |- ( G e. T -> ( [. G / g ]. ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( g e. T /\ g =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> [. G / g ]. ( R ` X ) .<_ ( R ` g ) ) )
17		eleq1 2537	. . . . . . 7 |- ( g = G -> ( g e. T <-> G e. T ) )
18		neeq1 2705	. . . . . . 7 |- ( g = G -> ( g =/= ( _I |` B ) <-> G =/= ( _I |` B ) ) )
19	17, 18	anbi12d 725	. . . . . 6 |- ( g = G -> ( ( g e. T /\ g =/= ( _I |` B ) ) <-> ( G e. T /\ G =/= ( _I |` B ) ) ) )
20	19	3anbi2d 1370	. . . . 5 |- ( g = G -> ( ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( g e. T /\ g =/= ( _I |` B ) ) /\ N e. T ) <-> ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) ) )
21	20	3anbi2d 1370	. . . 4 |- ( g = G -> ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( g e. T /\ g =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) <-> ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) ) )
22	21	sbcieg 3288	. . 3 |- ( G e. T -> ( [. G / g ]. ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( g e. T /\ g =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) <-> ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) ) )
23		sbcbr12g 4449	. . . 4 |- ( G e. T -> ( [. G / g ]. ( R ` X ) .<_ ( R ` g ) <-> [_ G / g ]_ ( R ` X ) .<_ [_ G / g ]_ ( R ` g ) ) )
24		csbfv2g 5915	. . . . 5 |- ( G e. T -> [_ G / g ]_ ( R ` X ) = ( R ` [_ G / g ]_ X ) )
25		csbfv 5916	. . . . . 6 |- [_ G / g ]_ ( R ` g ) = ( R ` G )
26	25	a1i 11	. . . . 5 |- ( G e. T -> [_ G / g ]_ ( R ` g ) = ( R ` G ) )
27	24, 26	breq12d 4408	. . . 4 |- ( G e. T -> ( [_ G / g ]_ ( R ` X ) .<_ [_ G / g ]_ ( R ` g ) <-> ( R ` [_ G / g ]_ X ) .<_ ( R ` G ) ) )
28	23, 27	bitrd 261	. . 3 |- ( G e. T -> ( [. G / g ]. ( R ` X ) .<_ ( R ` g ) <-> ( R ` [_ G / g ]_ X ) .<_ ( R ` G ) ) )
29	16, 22, 28	3imtr3d 275	. 2 |- ( G e. T -> ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( R ` [_ G / g ]_ X ) .<_ ( R ` G ) ) )
30	1, 29	mpcom 36	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( R ` [_ G / g ]_ X ) .<_ ( R ` G ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   [.wsbc 3255   [_csb 3349   class class class wbr 4395   _I cid 4749   `'ccnv 4838   |` cres 4841   o. ccom 4843   ` cfv 5589   iota_crio 6269  (class class class)co 6308   Basecbs 15199   lecple 15275   joincjn 16267   meetcmee 16268   Atomscatm 32900   HLchlt 32987   LHypclh 33620   LTrncltrn 33737   trLctrl 33795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-riotaBAD 32589

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-undef 7038  df-map 7492  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-p1 16364  df-lat 16370  df-clat 16432  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-llines 33134  df-lplanes 33135  df-lvols 33136  df-lines 33137  df-psubsp 33139  df-pmap 33140  df-padd 33432  df-lhyp 33624  df-laut 33625  df-ldil 33740  df-ltrn 33741  df-trl 33796

This theorem is referenced by:  cdlemk39s-id  34578  cdlemk51  34591