Theorem cdlemk39s 34865

Description: Substitution version of cdlemk39 34842. TODO: Can any commonality with cdlemk35s 34863 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 1107	. 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 34842	. . . . 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 3285	. . . 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 3312	. . . 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 2520	. . . . . . 7 |- ( g = G -> ( g e. T <-> G e. T ) )
18		neeq1 2726	. . . . . . 7 |- ( g = G -> ( g =/= ( _I |` B ) <-> G =/= ( _I |` B ) ) )
19	17, 18	anbi12d 710	. . . . . 6 |- ( g = G -> ( ( g e. T /\ g =/= ( _I |` B ) ) <-> ( G e. T /\ G =/= ( _I |` B ) ) ) )
20	19	3anbi2d 1295	. . . . 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 1295	. . . 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 3303	. . 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 4430	. . . 4 |- ( G e. T -> ( [. G / g ]. ( R ` X ) .<_ ( R ` g ) <-> [_ G / g ]_ ( R ` X ) .<_ [_ G / g ]_ ( R ` g ) ) )
24		csbfv2g 5812	. . . . 5 |- ( G e. T -> [_ G / g ]_ ( R ` X ) = ( R ` [_ G / g ]_ X ) )
25		csbfv 5813	. . . . . 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 4389	. . . 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 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 369   /\ w3a 965   = wceq 1370   e. wcel 1757   =/= wne 2641   A.wral 2792   [.wsbc 3270   [_csb 3372   class class class wbr 4376   _I cid 4715   `'ccnv 4923   |` cres 4926   o. ccom 4928   ` cfv 5502   iota_crio 6136  (class class class)co 6176   Basecbs 14262   lecple 14333   joincjn 15202   meetcmee 15203   Atomscatm 33190   HLchlt 33277   LHypclh 33910   LTrncltrn 34027   trLctrl 34084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-riotaBAD 32886

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-iin 4258  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-1st 6663  df-2nd 6664  df-undef 6878  df-map 7302  df-poset 15204  df-plt 15216  df-lub 15232  df-glb 15233  df-join 15234  df-meet 15235  df-p0 15297  df-p1 15298  df-lat 15304  df-clat 15366  df-oposet 33103  df-ol 33105  df-oml 33106  df-covers 33193  df-ats 33194  df-atl 33225  df-cvlat 33249  df-hlat 33278  df-llines 33424  df-lplanes 33425  df-lvols 33426  df-lines 33427  df-psubsp 33429  df-pmap 33430  df-padd 33722  df-lhyp 33914  df-laut 33915  df-ldil 34030  df-ltrn 34031  df-trl 34085

This theorem is referenced by:  cdlemk39s-id  34866  cdlemk51  34879