Theorem cdlemk11t 31428

Description: Part of proof of Lemma K of [Crawley] p. 118. Eq. 5, line 36, p. 119. G, I stand for g, h. X represents tau. (Contributed by NM, 21-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
cdlemk11t	|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( [_ G / g ]_ X ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' 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   I, b, g, z

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

Proof of Theorem cdlemk11t

Step	Hyp	Ref	Expression
1		simp11l 1068	. . 3 |- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> K e. HL )
2		simp11r 1069	. . 3 |- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> W e. H )
3		cdlemk5.b	. . . 4 |- B = ( Base ` K )
4		cdlemk5.h	. . . 4 |- H = ( LHyp ` K )
5		cdlemk5.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
6		cdlemk5.r	. . . 4 |- R = ( ( trL ` K ) ` W )
7	3, 4, 5, 6	cdlemftr3 31047	. . 3 |- ( ( K e. HL /\ W e. H ) -> E. b e. T ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) )
8	1, 2, 7	syl2anc 643	. 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> E. b e. T ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) )
9		nfv 1626	. . 3 |- F/ b ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) )
10		nfcv 2540	. . . . . 6 |- F/_ b G
11		cdlemk5.x	. . . . . . 7 |- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
12		nfra1 2716	. . . . . . . 8 |- F/ b A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y )
13		nfcv 2540	. . . . . . . 8 |- F/_ b T
14	12, 13	nfriota 6518	. . . . . . 7 |- F/_ b ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
15	11, 14	nfcxfr 2537	. . . . . 6 |- F/_ b X
16	10, 15	nfcsb 3245	. . . . 5 |- F/_ b [_ G / g ]_ X
17		nfcv 2540	. . . . 5 |- F/_ b P
18	16, 17	nffv 5694	. . . 4 |- F/_ b ( [_ G / g ]_ X ` P )
19		nfcv 2540	. . . 4 |- F/_ b .<_
20		nfcv 2540	. . . . . . 7 |- F/_ b I
21	20, 15	nfcsb 3245	. . . . . 6 |- F/_ b [_ I / g ]_ X
22	21, 17	nffv 5694	. . . . 5 |- F/_ b ( [_ I / g ]_ X ` P )
23		nfcv 2540	. . . . 5 |- F/_ b .\/
24		nfcv 2540	. . . . 5 |- F/_ b ( R ` ( I o. `' G ) )
25	22, 23, 24	nfov 6063	. . . 4 |- F/_ b ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' G ) ) )
26	18, 19, 25	nfbr 4216	. . 3 |- F/ b ( [_ G / g ]_ X ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' G ) ) )
27		simp11 987	. . . . 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) )
28		simp12 988	. . . . 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) )
29		simp2 958	. . . . 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> b e. T )
30		simp3l 985	. . . . . 6 |- ( ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> b =/= ( _I |` B ) )
31		simp3r1 1065	. . . . . 6 |- ( ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( R ` b ) =/= ( R ` F ) )
32		simp3r2 1066	. . . . . 6 |- ( ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( R ` b ) =/= ( R ` G ) )
33	30, 31, 32	3jca 1134	. . . . 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) )
34		simp13l 1072	. . . . . 6 |- ( ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> I e. T )
35		simp13r 1073	. . . . . 6 |- ( ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> I =/= ( _I |` B ) )
36		simp3r3 1067	. . . . . 6 |- ( ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( R ` b ) =/= ( R ` I ) )
37	34, 35, 36	3jca 1134	. . . . 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) )
38		cdlemk5.l	. . . . . 6 |- .<_ = ( le ` K )
39		cdlemk5.j	. . . . . 6 |- .\/ = ( join ` K )
40		cdlemk5.m	. . . . . 6 |- ./\ = ( meet ` K )
41		cdlemk5.a	. . . . . 6 |- A = ( Atoms ` K )
42		cdlemk5.z	. . . . . 6 |- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
43		cdlemk5.y	. . . . . 6 |- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
44	3, 38, 39, 40, 41, 4, 5, 6, 42, 43, 11	cdlemk11tc 31427	. . . . 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 ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( [_ G / g ]_ X ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' G ) ) ) )
45	27, 28, 29, 33, 37, 44	syl113anc 1196	. . . 4 |- ( ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( [_ G / g ]_ X ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' G ) ) ) )
46	45	3exp 1152	. . 3 |- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( b e. T -> ( ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) -> ( [_ G / g ]_ X ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' G ) ) ) ) ) )
47	9, 26, 46	rexlimd 2787	. 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( E. b e. T ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) -> ( [_ G / g ]_ X ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' G ) ) ) ) )
48	8, 47	mpd 15	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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( [_ G / g ]_ X ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' G ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   A.wral 2666   E.wrex 2667   [_csb 3211   class class class wbr 4172   _I cid 4453   `'ccnv 4836   |` cres 4839   o. ccom 4841   ` cfv 5413  (class class class)co 6040   iota_crio 6501   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640

This theorem is referenced by:  cdlemk45  31429

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641