Theorem cdlemefrs29cpre1 39866

Description: TODO: FIX COMMENT. (Contributed by NM, 29-Mar-2013.)

Hypotheses

Ref	Expression
cdlemefrs27.b	⊢ 𝐵 = (Base‘𝐾)
cdlemefrs27.l	⊢ ≤ = (le‘𝐾)
cdlemefrs27.j	⊢ ∨ = (join‘𝐾)
cdlemefrs27.m	⊢ ∧ = (meet‘𝐾)
cdlemefrs27.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemefrs27.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemefrs27.eq	⊢ (𝑠 = 𝑅 → (𝜑 ↔ 𝜓))
cdlemefrs27.nb	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))) → 𝑁 ∈ 𝐵)
cdlemefrs27.rnb	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ⦋𝑅 / 𝑠⦌𝑁 ∈ 𝐵)

Assertion

Ref	Expression
cdlemefrs29cpre1	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ∃!𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))))

Distinct variable groups:   𝑧,𝑠,𝐴   𝐻,𝑠   ∨ ,𝑠   𝐾,𝑠   ≤ ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠   𝑊,𝑠   𝜓,𝑠   𝑧,𝐴   𝑧,𝐵   𝑧,𝐻   𝑧,𝐾   𝑧, ≤   𝑧,𝑁   𝑧,𝑃   𝑧,𝑄   𝑧,𝑅   𝑧,𝑊   𝜓,𝑧   𝐵,𝑠   𝑧, ∨   ∧ ,𝑠,𝑧   𝜑,𝑧

Allowed substitution hints:   𝜑(𝑠)   𝑁(𝑠)

Proof of Theorem cdlemefrs29cpre1

Step	Hyp	Ref	Expression
1		cdlemefrs27.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		cdlemefrs27.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		cdlemefrs27.j	. . 3 ⊢ ∨ = (join‘𝐾)
4		cdlemefrs27.m	. . 3 ⊢ ∧ = (meet‘𝐾)
5		cdlemefrs27.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemefrs27.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemefrs27.eq	. . 3 ⊢ (𝑠 = 𝑅 → (𝜑 ↔ 𝜓))
8		cdlemefrs27.nb	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))) → 𝑁 ∈ 𝐵)
9		cdlemefrs27.rnb	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ⦋𝑅 / 𝑠⦌𝑁 ∈ 𝐵)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	cdlemefrs29bpre1 39865	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ∃𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))))
11		simp11 1201	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		simp2rl 1240	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → 𝑅 ∈ 𝐴)
13	1, 5	atbase 38756	. . . . . 6 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ 𝐵)
14	12, 13	syl 17	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → 𝑅 ∈ 𝐵)
15		simp2rr 1241	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ¬ 𝑅 ≤ 𝑊)
16	1, 2, 3, 4, 5, 6	lhpmcvr2 39492	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐵 ∧ ¬ 𝑅 ≤ 𝑊)) → ∃𝑠 ∈ 𝐴 (¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅))
17	11, 14, 15, 16	syl12anc 836	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ∃𝑠 ∈ 𝐴 (¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅))
18		simpl3 1191	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ 𝑠 ∈ 𝐴) → 𝜓)
19	7	pm5.32ri 575	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 = 𝑅) ↔ (𝜓 ∧ 𝑠 = 𝑅))
20	19	baibr 536	. . . . . . . . 9 ⊢ (𝜓 → (𝑠 = 𝑅 ↔ (𝜑 ∧ 𝑠 = 𝑅)))
21	18, 20	syl 17	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ 𝑠 ∈ 𝐴) → (𝑠 = 𝑅 ↔ (𝜑 ∧ 𝑠 = 𝑅)))
22		simp2r 1198	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
23		eqid 2728	. . . . . . . . . . . . . 14 ⊢ (0.‘𝐾) = (0.‘𝐾)
24	2, 4, 23, 5, 6	lhpmat 39498	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑅 ∧ 𝑊) = (0.‘𝐾))
25	11, 22, 24	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → (𝑅 ∧ 𝑊) = (0.‘𝐾))
26	25	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ 𝑠 ∈ 𝐴) → (𝑅 ∧ 𝑊) = (0.‘𝐾))
27	26	oveq2d 7431	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ 𝑠 ∈ 𝐴) → (𝑠 ∨ (𝑅 ∧ 𝑊)) = (𝑠 ∨ (0.‘𝐾)))
28		simp11l 1282	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → 𝐾 ∈ HL)
29		hlol 38828	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
30	28, 29	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → 𝐾 ∈ OL)
31	1, 5	atbase 38756	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ 𝐵)
32	1, 3, 23	olj01 38692	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OL ∧ 𝑠 ∈ 𝐵) → (𝑠 ∨ (0.‘𝐾)) = 𝑠)
33	30, 31, 32	syl2an 595	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ 𝑠 ∈ 𝐴) → (𝑠 ∨ (0.‘𝐾)) = 𝑠)
34	27, 33	eqtrd 2768	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ 𝑠 ∈ 𝐴) → (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑠)
35	34	eqeq1d 2730	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ 𝑠 ∈ 𝐴) → ((𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅 ↔ 𝑠 = 𝑅))
36	35	anbi2d 629	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ 𝑠 ∈ 𝐴) → ((𝜑 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) ↔ (𝜑 ∧ 𝑠 = 𝑅)))
37	21, 35, 36	3bitr4d 311	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ 𝑠 ∈ 𝐴) → ((𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅 ↔ (𝜑 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅)))
38	37	anbi2d 629	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ 𝑠 ∈ 𝐴) → ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) ↔ (¬ 𝑠 ≤ 𝑊 ∧ (𝜑 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅))))
39		anass 468	. . . . . 6 ⊢ (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) ↔ (¬ 𝑠 ≤ 𝑊 ∧ (𝜑 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅)))
40	38, 39	bitr4di 289	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ 𝑠 ∈ 𝐴) → ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) ↔ ((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅)))
41	40	rexbidva 3172	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → (∃𝑠 ∈ 𝐴 (¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) ↔ ∃𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅)))
42	17, 41	mpbid 231	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ∃𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅))
43		reusv1 5392	. . 3 ⊢ (∃𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → (∃!𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ ∃𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))))
44	42, 43	syl 17	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → (∃!𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ ∃𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))))
45	10, 44	mpbird 257	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ∃!𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ≠ wne 2936  ∀wral 3057  ∃wrex 3066  ∃!wreu 3370  ⦋csb 3890   class class class wbr 5143  ‘cfv 6543  (class class class)co 7415  Basecbs 17174  lecple 17234  joincjn 18297  meetcmee 18298  0.cp0 18409  OLcol 38641  Atomscatm 38730  HLchlt 38817  LHypclh 39452

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 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  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 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7371  df-ov 7418  df-oprab 7419  df-proset 18281  df-poset 18299  df-plt 18316  df-lub 18332  df-glb 18333  df-join 18334  df-meet 18335  df-p0 18411  df-p1 18412  df-lat 18418  df-clat 18485  df-oposet 38643  df-ol 38645  df-oml 38646  df-covers 38733  df-ats 38734  df-atl 38765  df-cvlat 38789  df-hlat 38818  df-lhyp 39456

This theorem is referenced by:  cdlemefrs29clN  39867  cdlemefrs32fva  39868  cdlemefs29cpre1N  39886

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