Theorem cdlemk11t 34953

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 1099	. . 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 1100	. . 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 34572	. . 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 661	. 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 1674	. . 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 2616	. . . . . 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 2810	. . . . . . . 8 |- F/ b A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y )
13		nfcv 2616	. . . . . . . 8 |- F/_ b T
14	12, 13	nfriota 6173	. . . . . . 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 2614	. . . . . 6 |- F/_ b X
16	10, 15	nfcsb 3416	. . . . 5 |- F/_ b [_ G / g ]_ X
17		nfcv 2616	. . . . 5 |- F/_ b P
18	16, 17	nffv 5809	. . . 4 |- F/_ b ( [_ G / g ]_ X ` P )
19		nfcv 2616	. . . 4 |- F/_ b .<_
20		nfcv 2616	. . . . . . 7 |- F/_ b I
21	20, 15	nfcsb 3416	. . . . . 6 |- F/_ b [_ I / g ]_ X
22	21, 17	nffv 5809	. . . . 5 |- F/_ b ( [_ I / g ]_ X ` P )
23		nfcv 2616	. . . . 5 |- F/_ b .\/
24		nfcv 2616	. . . . 5 |- F/_ b ( R ` ( I o. `' G ) )
25	22, 23, 24	nfov 6226	. . . 4 |- F/_ b ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' G ) ) )
26	18, 19, 25	nfbr 4447	. . 3 |- F/ b ( [_ G / g ]_ X ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' G ) ) )
27		simp11 1018	. . . . 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 1019	. . . . 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 989	. . . . 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 1016	. . . . . 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 1096	. . . . . 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 1097	. . . . . 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 1168	. . . . 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 1103	. . . . . 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 1104	. . . . . 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 1098	. . . . . 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 1168	. . . . 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 34952	. . . . 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 1231	. . . 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 1187	. . 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 2944	. 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 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2648   A.wral 2799   E.wrex 2800   [_csb 3398   class class class wbr 4403   _I cid 4742   `'ccnv 4950   |` cres 4953   o. ccom 4955   ` cfv 5529   iota_crio 6163  (class class class)co 6203   Basecbs 14296   lecple 14368   joincjn 15237   meetcmee 15238   Atomscatm 33271   HLchlt 33358   LHypclh 33991   LTrncltrn 34108   trLctrl 34165

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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-riotaBAD 32967

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 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-undef 6905  df-map 7329  df-poset 15239  df-plt 15251  df-lub 15267  df-glb 15268  df-join 15269  df-meet 15270  df-p0 15332  df-p1 15333  df-lat 15339  df-clat 15401  df-oposet 33184  df-ol 33186  df-oml 33187  df-covers 33274  df-ats 33275  df-atl 33306  df-cvlat 33330  df-hlat 33359  df-llines 33505  df-lplanes 33506  df-lvols 33507  df-lines 33508  df-psubsp 33510  df-pmap 33511  df-padd 33803  df-lhyp 33995  df-laut 33996  df-ldil 34111  df-ltrn 34112  df-trl 34166

This theorem is referenced by:  cdlemk45  34954