Theorem cdlemg7fvN 40627

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 1136	. . 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 1210	. . . 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 1199	. . . 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 40142	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊))
9	1, 2, 3, 8	syl3anc 1372	. . 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 1200	. . . 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 40610	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇) → ((𝐺‘𝑋) ∈ 𝐵 ∧ ¬ (𝐺‘𝑋) ≤ 𝑊))
13	1, 10, 2, 12	syl3anc 1372	. . 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 1209	. . 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 1211	. . . . . . . 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 40602	. . . . . . . 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 1380	. . . . . . 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 7447	. . . . . 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 1242	. . . . . . 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 40040	. . . . . . 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 1372	. . . . . 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 2776	. . . . 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 7448	. . . 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 2779	. . 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 40602	. . 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 1380	. 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 7448	. 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 2776	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 1086   = wceq 1539   ∈ wcel 2107   class class class wbr 5142  ‘cfv 6560  (class class class)co 7432  Basecbs 17248  lecple 17305  joincjn 18358  meetcmee 18359  Atomscatm 39265  HLchlt 39352  LHypclh 39987  LTrncltrn 40104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-riotaBAD 38955

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-undef 8299  df-map 8869  df-proset 18341  df-poset 18360  df-plt 18376  df-lub 18392  df-glb 18393  df-join 18394  df-meet 18395  df-p0 18471  df-p1 18472  df-lat 18478  df-clat 18545  df-oposet 39178  df-ol 39180  df-oml 39181  df-covers 39268  df-ats 39269  df-atl 39300  df-cvlat 39324  df-hlat 39353  df-llines 39501  df-lplanes 39502  df-lvols 39503  df-lines 39504  df-psubsp 39506  df-pmap 39507  df-padd 39799  df-lhyp 39991  df-laut 39992  df-ldil 40107  df-ltrn 40108  df-trl 40162

This theorem is referenced by:  cdlemg7aN  40628