Theorem cdlemg7fvN 40091

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 ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐹‘(𝐺‘𝑋)) = ((𝐹‘(𝐺‘𝑃)) ∨ (𝑋 ∧ 𝑊)))

Proof of Theorem cdlemg7fvN

Step	Hyp	Ref	Expression
1		simp1 1134	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp32 1208	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝐺 ∈ 𝑇)
3		simp2l 1197	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑃 ∈ 𝐴 ∧ ¬ 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)
8	4, 5, 6, 7	ltrnel 39606	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊))
9	1, 2, 3, 8	syl3anc 1369	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊))
10		simp2r 1198	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊))
11		cdlemg7fv.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
12	4, 5, 6, 7, 11	cdlemg7fvbwN 40074	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇) → ((𝐺‘𝑋) ∈ 𝐵 ∧ ¬ (𝐺‘𝑋) ≤ 𝑊))
13	1, 10, 2, 12	syl3anc 1369	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝐺‘𝑋) ∈ 𝐵 ∧ ¬ (𝐺‘𝑋) ≤ 𝑊))
14		simp31 1207	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝐹 ∈ 𝑇)
15		simp33 1209	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)
16		cdlemg7fv.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
17		cdlemg7fv.m	. . . . . . . . 9 ⊢ ∧ = (meet‘𝐾)
18	6, 7, 4, 16, 5, 17, 11	cdlemg2fv 40066	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐺‘𝑋) = ((𝐺‘𝑃) ∨ (𝑋 ∧ 𝑊)))
19	1, 3, 10, 2, 15, 18	syl122anc 1377	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐺‘𝑋) = ((𝐺‘𝑃) ∨ (𝑋 ∧ 𝑊)))
20	19	oveq1d 7429	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝐺‘𝑋) ∧ 𝑊) = (((𝐺‘𝑃) ∨ (𝑋 ∧ 𝑊)) ∧ 𝑊))
21		simp2rl 1240	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑋 ∈ 𝐵)
22	11, 4, 16, 17, 5, 6	lhpelim 39504	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) → (((𝐺‘𝑃) ∨ (𝑋 ∧ 𝑊)) ∧ 𝑊) = (𝑋 ∧ 𝑊))
23	1, 9, 21, 22	syl3anc 1369	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (((𝐺‘𝑃) ∨ (𝑋 ∧ 𝑊)) ∧ 𝑊) = (𝑋 ∧ 𝑊))
24	20, 23	eqtrd 2768	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝐺‘𝑋) ∧ 𝑊) = (𝑋 ∧ 𝑊))
25	24	oveq2d 7430	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝐺‘𝑃) ∨ ((𝐺‘𝑋) ∧ 𝑊)) = ((𝐺‘𝑃) ∨ (𝑋 ∧ 𝑊)))
26	25, 19	eqtr4d 2771	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝐺‘𝑃) ∨ ((𝐺‘𝑋) ∧ 𝑊)) = (𝐺‘𝑋))
27	6, 7, 4, 16, 5, 17, 11	cdlemg2fv 40066	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊) ∧ ((𝐺‘𝑋) ∈ 𝐵 ∧ ¬ (𝐺‘𝑋) ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ ((𝐺‘𝑃) ∨ ((𝐺‘𝑋) ∧ 𝑊)) = (𝐺‘𝑋))) → (𝐹‘(𝐺‘𝑋)) = ((𝐹‘(𝐺‘𝑃)) ∨ ((𝐺‘𝑋) ∧ 𝑊)))
28	1, 9, 13, 14, 26, 27	syl122anc 1377	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐹‘(𝐺‘𝑋)) = ((𝐹‘(𝐺‘𝑃)) ∨ ((𝐺‘𝑋) ∧ 𝑊)))
29	24	oveq2d 7430	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝐹‘(𝐺‘𝑃)) ∨ ((𝐺‘𝑋) ∧ 𝑊)) = ((𝐹‘(𝐺‘𝑃)) ∨ (𝑋 ∧ 𝑊)))
30	28, 29	eqtrd 2768	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐹‘(𝐺‘𝑋)) = ((𝐹‘(𝐺‘𝑃)) ∨ (𝑋 ∧ 𝑊)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   class class class wbr 5142  ‘cfv 6542  (class class class)co 7414  Basecbs 17173  lecple 17233  joincjn 18296  meetcmee 18297  Atomscatm 38729  HLchlt 38816  LHypclh 39451  LTrncltrn 39568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-riotaBAD 38419

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-undef 8272  df-map 8840  df-proset 18280  df-poset 18298  df-plt 18315  df-lub 18331  df-glb 18332  df-join 18333  df-meet 18334  df-p0 18410  df-p1 18411  df-lat 18417  df-clat 18484  df-oposet 38642  df-ol 38644  df-oml 38645  df-covers 38732  df-ats 38733  df-atl 38764  df-cvlat 38788  df-hlat 38817  df-llines 38965  df-lplanes 38966  df-lvols 38967  df-lines 38968  df-psubsp 38970  df-pmap 38971  df-padd 39263  df-lhyp 39455  df-laut 39456  df-ldil 39571  df-ltrn 39572  df-trl 39626

This theorem is referenced by:  cdlemg7aN  40092

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