Theorem cdlemg4a 38359

Description: TODO: FIX COMMENT If fg(p) = p, then tr f = tr g. (Contributed by NM, 23-Apr-2013.)

Hypotheses

Ref	Expression
cdlemg4.l	⊢ ≤ = (le‘𝐾)
cdlemg4.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg4.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg4.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg4.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

Assertion

Ref	Expression
cdlemg4a	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → (𝑅‘𝐹) = (𝑅‘𝐺))

Proof of Theorem cdlemg4a

Step	Hyp	Ref	Expression
1		simp3 1140	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → (𝐹‘(𝐺‘𝑃)) = 𝑃)
2	1	oveq2d 7229	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → ((𝐺‘𝑃)(join‘𝐾)(𝐹‘(𝐺‘𝑃))) = ((𝐺‘𝑃)(join‘𝐾)𝑃))
3		simp1l 1199	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → 𝐾 ∈ HL)
4		simp1 1138	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
5		simp23 1210	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → 𝐺 ∈ 𝑇)
6		simp21 1208	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
7		cdlemg4.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
8		cdlemg4.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
9		cdlemg4.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
10		cdlemg4.t	. . . . . . . 8 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
11	7, 8, 9, 10	ltrnel 37890	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊))
12	11	simpld 498	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺‘𝑃) ∈ 𝐴)
13	4, 5, 6, 12	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → (𝐺‘𝑃) ∈ 𝐴)
14		simp21l 1292	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → 𝑃 ∈ 𝐴)
15		eqid 2737	. . . . . 6 ⊢ (join‘𝐾) = (join‘𝐾)
16	15, 8	hlatjcom 37119	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝐺‘𝑃) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → ((𝐺‘𝑃)(join‘𝐾)𝑃) = (𝑃(join‘𝐾)(𝐺‘𝑃)))
17	3, 13, 14, 16	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → ((𝐺‘𝑃)(join‘𝐾)𝑃) = (𝑃(join‘𝐾)(𝐺‘𝑃)))
18	2, 17	eqtrd 2777	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → ((𝐺‘𝑃)(join‘𝐾)(𝐹‘(𝐺‘𝑃))) = (𝑃(join‘𝐾)(𝐺‘𝑃)))
19	18	oveq1d 7228	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → (((𝐺‘𝑃)(join‘𝐾)(𝐹‘(𝐺‘𝑃)))(meet‘𝐾)𝑊) = ((𝑃(join‘𝐾)(𝐺‘𝑃))(meet‘𝐾)𝑊))
20		simp22 1209	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → 𝐹 ∈ 𝑇)
21	4, 5, 6, 11	syl3anc 1373	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊))
22		eqid 2737	. . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾)
23		cdlemg4.r	. . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
24	7, 15, 22, 8, 9, 10, 23	trlval2 37914	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊)) → (𝑅‘𝐹) = (((𝐺‘𝑃)(join‘𝐾)(𝐹‘(𝐺‘𝑃)))(meet‘𝐾)𝑊))
25	4, 20, 21, 24	syl3anc 1373	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → (𝑅‘𝐹) = (((𝐺‘𝑃)(join‘𝐾)(𝐹‘(𝐺‘𝑃)))(meet‘𝐾)𝑊))
26	7, 15, 22, 8, 9, 10, 23	trlval2 37914	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐺) = ((𝑃(join‘𝐾)(𝐺‘𝑃))(meet‘𝐾)𝑊))
27	4, 5, 6, 26	syl3anc 1373	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → (𝑅‘𝐺) = ((𝑃(join‘𝐾)(𝐺‘𝑃))(meet‘𝐾)𝑊))
28	19, 25, 27	3eqtr4d 2787	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → (𝑅‘𝐹) = (𝑅‘𝐺))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   class class class wbr 5053  ‘cfv 6380  (class class class)co 7213  lecple 16809  joincjn 17818  meetcmee 17819  Atomscatm 37014  HLchlt 37101  LHypclh 37735  LTrncltrn 37852  trLctrl 37909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-map 8510  df-proset 17802  df-poset 17820  df-plt 17836  df-lub 17852  df-glb 17853  df-join 17854  df-meet 17855  df-p0 17931  df-lat 17938  df-oposet 36927  df-ol 36929  df-oml 36930  df-covers 37017  df-ats 37018  df-atl 37049  df-cvlat 37073  df-hlat 37102  df-lhyp 37739  df-laut 37740  df-ldil 37855  df-ltrn 37856  df-trl 37910

This theorem is referenced by:  cdlemg4f  38366