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Theorem cdlemg44 36021
Description: Part of proof of Lemma G of [Crawley] p. 116, fifth line of third paragraph on p. 117: "and hence fg = gf." (Contributed by NM, 3-Jun-2013.)
Hypotheses
Ref Expression
cdlemg44.h |- H = ( LHyp ` K )
cdlemg44.t |- T = ( ( LTrn ` K ) ` W )
cdlemg44.r |- R = ( ( trL ` K ) ` W )
Assertion
Ref Expression
cdlemg44 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( F o. G ) = ( G o. F ) )
Proof of Theorem cdlemg44
Dummy variable p is distinct from all other variables.
StepHypRef Expression
1 eqid 2622 . . . 4 |- ( le ` K ) = ( le ` K )
2 eqid 2622 . . . 4 |- ( Atoms ` K ) = ( Atoms ` K )
3 cdlemg44.h . . . 4 |- H = ( LHyp ` K )
41, 2, 3lhpexnle 35292 . . 3 |- ( ( K e. HL /\ W e. H ) -> E. p e. ( Atoms ` K ) -. p ( le ` K ) W )
543ad2ant1 1082 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> E. p e. ( Atoms ` K ) -. p ( le ` K ) W )
6 simp11 1091 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) /\ p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) -> ( K e. HL /\ W e. H ) )
7 simp12l 1174 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) /\ p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) -> F e. T )
8 simp12r 1175 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) /\ p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) -> G e. T )
9 cdlemg44.t . . . . . 6 |- T = ( ( LTrn ` K ) ` W )
103, 9ltrnco 36007 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( F o. G ) e. T )
116, 7, 8, 10syl3anc 1326 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) /\ p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) -> ( F o. G ) e. T )
123, 9ltrnco 36007 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ F e. T ) -> ( G o. F ) e. T )
136, 8, 7, 12syl3anc 1326 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) /\ p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) -> ( G o. F ) e. T )
14 3simpc 1060 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) /\ p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) -> ( p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) )
15 simp13 1093 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) /\ p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) -> ( R ` F ) =/= ( R ` G ) )
16 cdlemg44.r . . . . . . 7 |- R = ( ( trL ` K ) ` W )
173, 9, 16, 1, 2cdlemg44b 36020 . . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( F ` ( G ` p ) ) = ( G ` ( F ` p ) ) )
186, 7, 8, 14, 15, 17syl131anc 1339 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) /\ p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) -> ( F ` ( G ` p ) ) = ( G ` ( F ` p ) ) )
19 simp12 1092 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) /\ p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) -> ( F e. T /\ G e. T ) )
20 simp2 1062 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) /\ p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) -> p e. ( Atoms ` K ) )
211, 2, 3, 9ltrncoval 35431 . . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ p e. ( Atoms ` K ) ) -> ( ( F o. G ) ` p ) = ( F ` ( G ` p ) ) )
226, 19, 20, 21syl3anc 1326 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) /\ p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) -> ( ( F o. G ) ` p ) = ( F ` ( G ` p ) ) )
231, 2, 3, 9ltrncoval 35431 . . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( G e. T /\ F e. T ) /\ p e. ( Atoms ` K ) ) -> ( ( G o. F ) ` p ) = ( G ` ( F ` p ) ) )
246, 8, 7, 20, 23syl121anc 1331 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) /\ p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) -> ( ( G o. F ) ` p ) = ( G ` ( F ` p ) ) )
2518, 22, 243eqtr4d 2666 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) /\ p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) -> ( ( F o. G ) ` p ) = ( ( G o. F ) ` p ) )
261, 2, 3, 9cdlemd 35494 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F o. G ) e. T /\ ( G o. F ) e. T ) /\ ( p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) /\ ( ( F o. G ) ` p ) = ( ( G o. F ) ` p ) ) -> ( F o. G ) = ( G o. F ) )
276, 11, 13, 14, 25, 26syl311anc 1340 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) /\ p e. ( Atoms ` K ) /\ -. p ( le ` K ) W ) -> ( F o. G ) = ( G o. F ) )
2827rexlimdv3a 3033 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( E. p e. ( Atoms ` K ) -. p ( le ` K ) W -> ( F o. G ) = ( G o. F ) ) )
295, 28mpd 15 1 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( F o. G ) = ( G o. F ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   o. ccom 5118   ` cfv 5888   lecple 15948   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-map 7859  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446
This theorem is referenced by:  cdlemg47  36024  ltrncom  36026
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