cdlemkid2 - Mathbox for Norm Megill

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Theorem cdlemkid2 41184

Description: Lemma for cdlemkid 41196. (Contributed by NM, 24-Jul-2013.)

**Hypotheses**
Ref	Expression
cdlemk5.b	⊢ 𝐵 = (Base‘𝐾)
cdlemk5.l	⊢ ≤ = (le‘𝐾)
cdlemk5.j	⊢ ∨ = (join‘𝐾)
cdlemk5.m	⊢ ∧ = (meet‘𝐾)
cdlemk5.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemk5.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemk5.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemk5.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemk5.z	⊢ 𝑍 = ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑏 ∘ ^◡𝐹))))
cdlemk5.y	⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ^◡𝑏))))

**Assertion**
Ref	Expression
cdlemkid2	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → ⦋𝐺 / 𝑔⦌𝑌 = 𝑃)

Distinct variable groups: ∧ ,𝑔 ∨ ,𝑔 𝐵,𝑔 𝑃,𝑔 𝑅,𝑔 𝑇,𝑔 𝑔,𝑍 𝑔,𝑏 Allowed substitution hints: 𝐴(𝑔,𝑏) 𝐵(𝑏) 𝑃(𝑏) 𝑅(𝑏) 𝑇(𝑏) 𝐹(𝑔,𝑏) 𝐺(𝑔,𝑏) 𝐻(𝑔,𝑏) ∨ (𝑏) 𝐾(𝑔,𝑏) ≤ (𝑔,𝑏) ∧ (𝑏) 𝑁(𝑔,𝑏) 𝑊(𝑔,𝑏) 𝑌(𝑔,𝑏) 𝑍(𝑏)

**Proof of Theorem cdlemkid2**
Step	Hyp	Ref	Expression
1		simp32 1211	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → 𝐺 = ( I ↾ 𝐵))
2	1	csbeq1d 3853	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → ⦋𝐺 / 𝑔⦌𝑌 = ⦋( I ↾ 𝐵) / 𝑔⦌𝑌)
3		cdlemk5.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
4		cdlemk5.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
5		cdlemk5.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
6	3, 4, 5	idltrn 40410	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇)
7	6	3ad2ant1 1133	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → ( I ↾ 𝐵) ∈ 𝑇)
8		cdlemk5.y	. . . . 5 ⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ^◡𝑏))))
9	8	cdlemk41 41180	. . . 4 ⊢ (( I ↾ 𝐵) ∈ 𝑇 → ⦋( I ↾ 𝐵) / 𝑔⦌𝑌 = ((𝑃 ∨ (𝑅‘( I ↾ 𝐵))) ∧ (𝑍 ∨ (𝑅‘(( I ↾ 𝐵) ∘ ^◡𝑏)))))
10	7, 9	syl 17	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → ⦋( I ↾ 𝐵) / 𝑔⦌𝑌 = ((𝑃 ∨ (𝑅‘( I ↾ 𝐵))) ∧ (𝑍 ∨ (𝑅‘(( I ↾ 𝐵) ∘ ^◡𝑏)))))
11		eqid 2736	. . . . . . . . 9 ⊢ (0.‘𝐾) = (0.‘𝐾)
12		cdlemk5.r	. . . . . . . . 9 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
13	3, 11, 4, 12	trlid0 40436	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑅‘( I ↾ 𝐵)) = (0.‘𝐾))
14	13	3ad2ant1 1133	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → (𝑅‘( I ↾ 𝐵)) = (0.‘𝐾))
15	14	oveq2d 7374	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → (𝑃 ∨ (𝑅‘( I ↾ 𝐵))) = (𝑃 ∨ (0.‘𝐾)))
16		simp1l 1198	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → 𝐾 ∈ HL)
17		hlol 39621	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
18	16, 17	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → 𝐾 ∈ OL)
19		simp31l 1297	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → 𝑃 ∈ 𝐴)
20		cdlemk5.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
21	3, 20	atbase 39549	. . . . . . . 8 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
22	19, 21	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → 𝑃 ∈ 𝐵)
23		cdlemk5.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
24	3, 23, 11	olj01 39485	. . . . . . 7 ⊢ ((𝐾 ∈ OL ∧ 𝑃 ∈ 𝐵) → (𝑃 ∨ (0.‘𝐾)) = 𝑃)
25	18, 22, 24	syl2anc 584	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → (𝑃 ∨ (0.‘𝐾)) = 𝑃)
26	15, 25	eqtrd 2771	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → (𝑃 ∨ (𝑅‘( I ↾ 𝐵))) = 𝑃)
27		simp1 1136	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
28		simp33l 1301	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → 𝑏 ∈ 𝑇)
29	4, 5	ltrncnv 40406	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑏 ∈ 𝑇) → ^◡𝑏 ∈ 𝑇)
30	27, 28, 29	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → ^◡𝑏 ∈ 𝑇)
31	3, 4, 5	ltrn1o 40384	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ^◡𝑏 ∈ 𝑇) → ^◡𝑏:𝐵–1-1-onto→𝐵)
32	27, 30, 31	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → ^◡𝑏:𝐵–1-1-onto→𝐵)
33		f1of 6774	. . . . . . . . . 10 ⊢ (^◡𝑏:𝐵–1-1-onto→𝐵 → ^◡𝑏:𝐵⟶𝐵)
34		fcoi2 6709	. . . . . . . . . 10 ⊢ (^◡𝑏:𝐵⟶𝐵 → (( I ↾ 𝐵) ∘ ^◡𝑏) = ^◡𝑏)
35	32, 33, 34	3syl 18	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → (( I ↾ 𝐵) ∘ ^◡𝑏) = ^◡𝑏)
36	35	fveq2d 6838	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → (𝑅‘(( I ↾ 𝐵) ∘ ^◡𝑏)) = (𝑅‘^◡𝑏))
37	4, 5, 12	trlcnv 40425	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑏 ∈ 𝑇) → (𝑅‘^◡𝑏) = (𝑅‘𝑏))
38	27, 28, 37	syl2anc 584	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → (𝑅‘^◡𝑏) = (𝑅‘𝑏))
39	36, 38	eqtrd 2771	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → (𝑅‘(( I ↾ 𝐵) ∘ ^◡𝑏)) = (𝑅‘𝑏))
40	39	oveq2d 7374	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → (𝑍 ∨ (𝑅‘(( I ↾ 𝐵) ∘ ^◡𝑏))) = (𝑍 ∨ (𝑅‘𝑏)))
41		simp31 1210	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
42		simp33 1212	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))
43	41, 42	jca 511	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵))))
44		cdlemk5.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
45		cdlemk5.m	. . . . . . . 8 ⊢ ∧ = (meet‘𝐾)
46		cdlemk5.z	. . . . . . . 8 ⊢ 𝑍 = ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑏 ∘ ^◡𝐹))))
47	3, 44, 23, 45, 20, 4, 5, 12, 46	cdlemkid1 41182	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → (𝑍 ∨ (𝑅‘𝑏)) = (𝑃 ∨ (𝑅‘𝑏)))
48	43, 47	syld3an3 1411	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → (𝑍 ∨ (𝑅‘𝑏)) = (𝑃 ∨ (𝑅‘𝑏)))
49	40, 48	eqtrd 2771	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → (𝑍 ∨ (𝑅‘(( I ↾ 𝐵) ∘ ^◡𝑏))) = (𝑃 ∨ (𝑅‘𝑏)))
50	26, 49	oveq12d 7376	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → ((𝑃 ∨ (𝑅‘( I ↾ 𝐵))) ∧ (𝑍 ∨ (𝑅‘(( I ↾ 𝐵) ∘ ^◡𝑏)))) = (𝑃 ∧ (𝑃 ∨ (𝑅‘𝑏))))
51	16	hllatd 39624	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → 𝐾 ∈ Lat)
52	3, 4, 5, 12	trlcl 40424	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑏 ∈ 𝑇) → (𝑅‘𝑏) ∈ 𝐵)
53	27, 28, 52	syl2anc 584	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → (𝑅‘𝑏) ∈ 𝐵)
54	3, 23, 45	latabs2 18399	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ (𝑅‘𝑏) ∈ 𝐵) → (𝑃 ∧ (𝑃 ∨ (𝑅‘𝑏))) = 𝑃)
55	51, 22, 53, 54	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → (𝑃 ∧ (𝑃 ∨ (𝑅‘𝑏))) = 𝑃)
56	50, 55	eqtrd 2771	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → ((𝑃 ∨ (𝑅‘( I ↾ 𝐵))) ∧ (𝑍 ∨ (𝑅‘(( I ↾ 𝐵) ∘ ^◡𝑏)))) = 𝑃)
57	10, 56	eqtrd 2771	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → ⦋( I ↾ 𝐵) / 𝑔⦌𝑌 = 𝑃)
58	2, 57	eqtrd 2771	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏 ∈ 𝑇 ∧ 𝑏 ≠ ( I ↾ 𝐵)))) → ⦋𝐺 / 𝑔⦌𝑌 = 𝑃)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ⦋csb 3849 class class class wbr 5098 I cid 5518 ^◡ccnv 5623 ↾ cres 5626 ∘ ccom 5628 ⟶wf 6488 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7358 Basecbs 17136 lecple 17184 joincjn 18234 meetcmee 18235 0.cp0 18344 Latclat 18354 OLcol 39434 Atomscatm 39523 HLchlt 39610 LHypclh 40244 LTrncltrn 40361 trLctrl 40418

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-riotaBAD 39213

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-undef 8215 df-map 8765 df-proset 18217 df-poset 18236 df-plt 18251 df-lub 18267 df-glb 18268 df-join 18269 df-meet 18270 df-p0 18346 df-p1 18347 df-lat 18355 df-clat 18422 df-oposet 39436 df-ol 39438 df-oml 39439 df-covers 39526 df-ats 39527 df-atl 39558 df-cvlat 39582 df-hlat 39611 df-llines 39758 df-lplanes 39759 df-lvols 39760 df-lines 39761 df-psubsp 39763 df-pmap 39764 df-padd 40056 df-lhyp 40248 df-laut 40249 df-ldil 40364 df-ltrn 40365 df-trl 40419

This theorem is referenced by: cdlemkid3N 41193 cdlemkid4 41194

W3C validator