Theorem cdlemkfid3N 34927

Description: TODO: is this useful or should it be deleted? (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)

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 ) ) ) )

Assertion

Ref	Expression
cdlemkfid3N	|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> [_ G / g ]_ Y = ( G ` P ) )

Distinct variable groups:   ./\ , g   .\/ , g   B, g   P, g   R, g   T, g   g, Z   g, b   g, G

Allowed substitution hints:   A( g, b)   B( b)   P( b)   R( b)   T( b)   F( g, b)   G( b)   H( g, b)   .\/ ( b)   K( g, b)   .<_ ( g, b)   ./\ ( b)   N( g, b)   W( g, b)   Y( g, b)   Z( b)

Proof of Theorem cdlemkfid3N

Step	Hyp	Ref	Expression
1		simp22 1022	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> G e. T )
2		cdlemk5.y	. . . 4 |- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
3	2	cdlemk41 34922	. . 3 |- ( G e. T -> [_ G / g ]_ Y = ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) )
4	1, 3	syl 16	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> [_ G / g ]_ Y = ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) )
5		simp1 988	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( K e. HL /\ W e. H ) /\ F = N ) )
6		simp21l 1105	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F e. T )
7		simp21r 1106	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F =/= ( _I |` B ) )
8		simp23l 1109	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> b e. T )
9		simp31 1024	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` b ) =/= ( R ` F ) )
10		simp33 1026	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) )
11		cdlemk5.b	. . . . . 6 |- B = ( Base ` K )
12		cdlemk5.l	. . . . . 6 |- .<_ = ( le ` K )
13		cdlemk5.j	. . . . . 6 |- .\/ = ( join ` K )
14		cdlemk5.m	. . . . . 6 |- ./\ = ( meet ` K )
15		cdlemk5.a	. . . . . 6 |- A = ( Atoms ` K )
16		cdlemk5.h	. . . . . 6 |- H = ( LHyp ` K )
17		cdlemk5.t	. . . . . 6 |- T = ( ( LTrn ` K ) ` W )
18		cdlemk5.r	. . . . . 6 |- R = ( ( trL ` K ) ` W )
19		cdlemk5.z	. . . . . 6 |- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
20	11, 12, 13, 14, 15, 16, 17, 18, 19	cdlemkfid2N 34925	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ b e. T ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> Z = ( b ` P ) )
21	5, 6, 7, 8, 9, 10, 20	syl132anc 1237	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> Z = ( b ` P ) )
22	21	oveq1d 6218	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( Z .\/ ( R ` ( G o. `' b ) ) ) = ( ( b ` P ) .\/ ( R ` ( G o. `' b ) ) ) )
23	22	oveq2d 6219	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( b ` P ) .\/ ( R ` ( G o. `' b ) ) ) ) )
24		simp1l 1012	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
25		simp23r 1110	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> b =/= ( _I |` B ) )
26		simp32 1025	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` b ) =/= ( R ` G ) )
27	26	necomd 2723	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` G ) =/= ( R ` b ) )
28	11, 12, 13, 14, 15, 16, 17, 18	cdlemkfid1N 34923	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( b e. T /\ b =/= ( _I |` B ) /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` b ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( P .\/ ( R ` G ) ) ./\ ( ( b ` P ) .\/ ( R ` ( G o. `' b ) ) ) ) = ( G ` P ) )
29	24, 8, 25, 1, 27, 10, 28	syl132anc 1237	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( P .\/ ( R ` G ) ) ./\ ( ( b ` P ) .\/ ( R ` ( G o. `' b ) ) ) ) = ( G ` P ) )
30	4, 23, 29	3eqtrd 2499	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> [_ G / g ]_ Y = ( G ` P ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2648   [_csb 3398   class class class wbr 4403   _I cid 4742   `'ccnv 4950   |` cres 4953   o. ccom 4955   ` cfv 5529  (class class class)co 6203   Basecbs 14295   lecple 14367   joincjn 15236   meetcmee 15237   Atomscatm 33266   HLchlt 33353   LHypclh 33986   LTrncltrn 34103   trLctrl 34160

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 32962

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  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 15238  df-plt 15250  df-lub 15266  df-glb 15267  df-join 15268  df-meet 15269  df-p0 15331  df-p1 15332  df-lat 15338  df-clat 15400  df-oposet 33179  df-ol 33181  df-oml 33182  df-covers 33269  df-ats 33270  df-atl 33301  df-cvlat 33325  df-hlat 33354  df-llines 33500  df-lplanes 33501  df-lvols 33502  df-lines 33503  df-psubsp 33505  df-pmap 33506  df-padd 33798  df-lhyp 33990  df-laut 33991  df-ldil 34106  df-ltrn 34107  df-trl 34161

This theorem is referenced by: (None)