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Theorem cdlemg7fvN 35392
Description: Value of a translation composition in terms of an associated atom. (Contributed by NM, 28-Apr-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemg7fv.b 𝐵 = (Base‘𝐾)
cdlemg7fv.l = (le‘𝐾)
cdlemg7fv.j = (join‘𝐾)
cdlemg7fv.m = (meet‘𝐾)
cdlemg7fv.a 𝐴 = (Atoms‘𝐾)
cdlemg7fv.h 𝐻 = (LHyp‘𝐾)
cdlemg7fv.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
cdlemg7fvN (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → (𝐹‘(𝐺𝑋)) = ((𝐹‘(𝐺𝑃)) (𝑋 𝑊)))

Proof of Theorem cdlemg7fvN
StepHypRef Expression
1 simp1 1059 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp32 1096 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → 𝐺𝑇)
3 simp2l 1085 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
4 cdlemg7fv.l . . . . 5 = (le‘𝐾)
5 cdlemg7fv.a . . . . 5 𝐴 = (Atoms‘𝐾)
6 cdlemg7fv.h . . . . 5 𝐻 = (LHyp‘𝐾)
7 cdlemg7fv.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
84, 5, 6, 7ltrnel 34905 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐺𝑃) ∈ 𝐴 ∧ ¬ (𝐺𝑃) 𝑊))
91, 2, 3, 8syl3anc 1323 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → ((𝐺𝑃) ∈ 𝐴 ∧ ¬ (𝐺𝑃) 𝑊))
10 simp2r 1086 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → (𝑋𝐵 ∧ ¬ 𝑋 𝑊))
11 cdlemg7fv.b . . . . 5 𝐵 = (Base‘𝐾)
124, 5, 6, 7, 11cdlemg7fvbwN 35375 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐺𝑇) → ((𝐺𝑋) ∈ 𝐵 ∧ ¬ (𝐺𝑋) 𝑊))
131, 10, 2, 12syl3anc 1323 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → ((𝐺𝑋) ∈ 𝐵 ∧ ¬ (𝐺𝑋) 𝑊))
14 simp31 1095 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → 𝐹𝑇)
15 simp33 1097 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → (𝑃 (𝑋 𝑊)) = 𝑋)
16 cdlemg7fv.j . . . . . . . . 9 = (join‘𝐾)
17 cdlemg7fv.m . . . . . . . . 9 = (meet‘𝐾)
186, 7, 4, 16, 5, 17, 11cdlemg2fv 35367 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → (𝐺𝑋) = ((𝐺𝑃) (𝑋 𝑊)))
191, 3, 10, 2, 15, 18syl122anc 1332 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → (𝐺𝑋) = ((𝐺𝑃) (𝑋 𝑊)))
2019oveq1d 6619 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → ((𝐺𝑋) 𝑊) = (((𝐺𝑃) (𝑋 𝑊)) 𝑊))
21 simp2rl 1128 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → 𝑋𝐵)
2211, 4, 16, 17, 5, 6lhpelim 34803 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐺𝑃) ∈ 𝐴 ∧ ¬ (𝐺𝑃) 𝑊) ∧ 𝑋𝐵) → (((𝐺𝑃) (𝑋 𝑊)) 𝑊) = (𝑋 𝑊))
231, 9, 21, 22syl3anc 1323 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → (((𝐺𝑃) (𝑋 𝑊)) 𝑊) = (𝑋 𝑊))
2420, 23eqtrd 2655 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → ((𝐺𝑋) 𝑊) = (𝑋 𝑊))
2524oveq2d 6620 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → ((𝐺𝑃) ((𝐺𝑋) 𝑊)) = ((𝐺𝑃) (𝑋 𝑊)))
2625, 19eqtr4d 2658 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → ((𝐺𝑃) ((𝐺𝑋) 𝑊)) = (𝐺𝑋))
276, 7, 4, 16, 5, 17, 11cdlemg2fv 35367 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (((𝐺𝑃) ∈ 𝐴 ∧ ¬ (𝐺𝑃) 𝑊) ∧ ((𝐺𝑋) ∈ 𝐵 ∧ ¬ (𝐺𝑋) 𝑊)) ∧ (𝐹𝑇 ∧ ((𝐺𝑃) ((𝐺𝑋) 𝑊)) = (𝐺𝑋))) → (𝐹‘(𝐺𝑋)) = ((𝐹‘(𝐺𝑃)) ((𝐺𝑋) 𝑊)))
281, 9, 13, 14, 26, 27syl122anc 1332 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → (𝐹‘(𝐺𝑋)) = ((𝐹‘(𝐺𝑃)) ((𝐺𝑋) 𝑊)))
2924oveq2d 6620 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → ((𝐹‘(𝐺𝑃)) ((𝐺𝑋) 𝑊)) = ((𝐹‘(𝐺𝑃)) (𝑋 𝑊)))
3028, 29eqtrd 2655 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → (𝐹‘(𝐺𝑋)) = ((𝐹‘(𝐺𝑃)) (𝑋 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987   class class class wbr 4613  cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  joincjn 16865  meetcmee 16866  Atomscatm 34030  HLchlt 34117  LHypclh 34750  LTrncltrn 34867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-riotaBAD 33719
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-undef 7344  df-map 7804  df-preset 16849  df-poset 16867  df-plt 16879  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-p0 16960  df-p1 16961  df-lat 16967  df-clat 17029  df-oposet 33943  df-ol 33945  df-oml 33946  df-covers 34033  df-ats 34034  df-atl 34065  df-cvlat 34089  df-hlat 34118  df-llines 34264  df-lplanes 34265  df-lvols 34266  df-lines 34267  df-psubsp 34269  df-pmap 34270  df-padd 34562  df-lhyp 34754  df-laut 34755  df-ldil 34870  df-ltrn 34871  df-trl 34926
This theorem is referenced by:  cdlemg7aN  35393
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