Theorem cdlemk39s 33958

Description: Substitution version of cdlemk39 33935. TODO: Can any commonality with cdlemk35s 33956 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 1116	. 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 33935	. . . . 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 3292	. . . 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 3319	. . . 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 210	. . 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 2474	. . . . . . 7 |- ( g = G -> ( g e. T <-> G e. T ) )
18		neeq1 2684	. . . . . . 7 |- ( g = G -> ( g =/= ( _I |` B ) <-> G =/= ( _I |` B ) ) )
19	17, 18	anbi12d 709	. . . . . 6 |- ( g = G -> ( ( g e. T /\ g =/= ( _I |` B ) ) <-> ( G e. T /\ G =/= ( _I |` B ) ) ) )
20	19	3anbi2d 1306	. . . . 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 1306	. . . 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 3310	. . 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 4448	. . . 4 |- ( G e. T -> ( [. G / g ]. ( R ` X ) .<_ ( R ` g ) <-> [_ G / g ]_ ( R ` X ) .<_ [_ G / g ]_ ( R ` g ) ) )
24		csbfv2g 5885	. . . . 5 |- ( G e. T -> [_ G / g ]_ ( R ` X ) = ( R ` [_ G / g ]_ X ) )
25		csbfv 5886	. . . . . 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 253	. . 3 |- ( G e. T -> ( [. G / g ]. ( R ` X ) .<_ ( R ` g ) <-> ( R ` [_ G / g ]_ X ) .<_ ( R ` G ) ) )
29	16, 22, 28	3imtr3d 267	. 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 34	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 367   /\ w3a 974   = wceq 1405   e. wcel 1842   =/= wne 2598   A.wral 2754   [.wsbc 3277   [_csb 3373   class class class wbr 4395   _I cid 4733   `'ccnv 4822   |` cres 4825   o. ccom 4827   ` cfv 5569   iota_crio 6239  (class class class)co 6278   Basecbs 14841   lecple 14916   joincjn 15897   meetcmee 15898   Atomscatm 32281   HLchlt 32368   LHypclh 33001   LTrncltrn 33118   trLctrl 33176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-riotaBAD 31977

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-undef 7005  df-map 7459  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-p1 15994  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-llines 32515  df-lplanes 32516  df-lvols 32517  df-lines 32518  df-psubsp 32520  df-pmap 32521  df-padd 32813  df-lhyp 33005  df-laut 33006  df-ldil 33121  df-ltrn 33122  df-trl 33177

This theorem is referenced by:  cdlemk39s-id  33959  cdlemk51  33972