Theorem cdlemefrs29bpre0 41656

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)

Assertion

Ref	Expression
cdlemefrs29bpre0	⊢ ((((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) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ 𝑧 = ⦋𝑅 / 𝑠⦌𝑁))

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

Allowed substitution hints:   𝜑(𝑧, 𝑠)   𝜓(𝑧)   𝐴(𝑧)   𝐵(𝑧, 𝑠)   𝑃(𝑧)   𝑄(𝑧)   𝑅(𝑧)   𝐻(𝑧)   ∨ (𝑧)   𝐾(𝑧)   ≤ (𝑧)   ∧ (𝑧, 𝑠)   𝑁(𝑧, 𝑠)   𝑊(𝑧)

Proof of Theorem cdlemefrs29bpre0

Step	Hyp	Ref	Expression
1		df-ral 3081	. . 3 ⊢ (∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ ∀𝑠(𝑠 ∈ 𝐴 → (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))))
2		anass 474	. . . . . . 7 ⊢ (((𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑)) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) ↔ (𝑠 ∈ 𝐴 ∧ ((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅)))
3	2	imbi1i 352	. . . . . 6 ⊢ ((((𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑)) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ ((𝑠 ∈ 𝐴 ∧ ((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅)) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))))
4		impexp 456	. . . . . 6 ⊢ ((((𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑)) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ ((𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑)) → ((𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅 → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))))
5		impexp 456	. . . . . 6 ⊢ (((𝑠 ∈ 𝐴 ∧ ((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅)) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ (𝑠 ∈ 𝐴 → (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))))
6	3, 4, 5	3bitr3ri 305	. . . . 5 ⊢ ((𝑠 ∈ 𝐴 → (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))) ↔ ((𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑)) → ((𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅 → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))))
7		simpl11 1267	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8		simpl2r 1246	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
9		cdlemefrs27.l	. . . . . . . . . . . . 13 ⊢ ≤ = (le‘𝐾)
10		cdlemefrs27.m	. . . . . . . . . . . . 13 ⊢ ∧ = (meet‘𝐾)
11		eqid 2763	. . . . . . . . . . . . 13 ⊢ (0.‘𝐾) = (0.‘𝐾)
12		cdlemefrs27.a	. . . . . . . . . . . . 13 ⊢ 𝐴 = (Atoms‘𝐾)
13		cdlemefrs27.h	. . . . . . . . . . . . 13 ⊢ 𝐻 = (LHyp‘𝐾)
14	9, 10, 11, 12, 13	lhpmat 41290	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑅 ∧ 𝑊) = (0.‘𝐾))
15	7, 8, 14	syl2anc 596	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))) → (𝑅 ∧ 𝑊) = (0.‘𝐾))
16	15	oveq2d 7462	. . . . . . . . . 10 ⊢ (((((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)) = (𝑠 ∨ (0.‘𝐾)))
17		simp11l 1303	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → 𝐾 ∈ HL)
18		hlol 40621	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
19	17, 18	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → 𝐾 ∈ OL)
20	19	adantr 486	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))) → 𝐾 ∈ OL)
21		simprl 783	. . . . . . . . . . . 12 ⊢ (((((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 ∈ 𝐴)
22		cdlemefrs27.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝐾)
23	22, 12	atbase 40549	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ 𝐵)
24	21, 23	syl 18	. . . . . . . . . . 11 ⊢ (((((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 ∈ 𝐵)
25		cdlemefrs27.j	. . . . . . . . . . . 12 ⊢ ∨ = (join‘𝐾)
26	22, 25, 11	olj01 40485	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OL ∧ 𝑠 ∈ 𝐵) → (𝑠 ∨ (0.‘𝐾)) = 𝑠)
27	20, 24, 26	syl2anc 596	. . . . . . . . . 10 ⊢ (((((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 ∨ (0.‘𝐾)) = 𝑠)
28	16, 27	eqtrd 2798	. . . . . . . . 9 ⊢ (((((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)) = 𝑠)
29	28	eqeq1d 2765	. . . . . . . 8 ⊢ (((((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 ↔ 𝑠 = 𝑅))
30	15	oveq2d 7462	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))) → (𝑁 ∨ (𝑅 ∧ 𝑊)) = (𝑁 ∨ (0.‘𝐾)))
31		simpl1 1210	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
32		simpl2l 1245	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))) → 𝑃 ≠ 𝑄)
33		simprr 785	. . . . . . . . . . . 12 ⊢ (((((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 ≤ 𝑊 ∧ 𝜑))
34		cdlemefrs27.nb	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))) → 𝑁 ∈ 𝐵)
35	31, 32, 21, 33, 34	syl112anc 1402	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))) → 𝑁 ∈ 𝐵)
36	22, 25, 11	olj01 40485	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OL ∧ 𝑁 ∈ 𝐵) → (𝑁 ∨ (0.‘𝐾)) = 𝑁)
37	20, 35, 36	syl2anc 596	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))) → (𝑁 ∨ (0.‘𝐾)) = 𝑁)
38	30, 37	eqtrd 2798	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))) → (𝑁 ∨ (𝑅 ∧ 𝑊)) = 𝑁)
39	38	eqeq2d 2774	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))) → (𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)) ↔ 𝑧 = 𝑁))
40	29, 39	imbi12d 347	. . . . . . 7 ⊢ (((((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 → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ (𝑠 = 𝑅 → 𝑧 = 𝑁)))
41	40	pm5.74da 816	. . . . . 6 ⊢ ((((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 → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))) ↔ ((𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑)) → (𝑠 = 𝑅 → 𝑧 = 𝑁))))
42		impexp 456	. . . . . . 7 ⊢ ((((𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑)) ∧ 𝑠 = 𝑅) → 𝑧 = 𝑁) ↔ ((𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑)) → (𝑠 = 𝑅 → 𝑧 = 𝑁)))
43		simp2rl 1261	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → 𝑅 ∈ 𝐴)
44		simp2rr 1262	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ¬ 𝑅 ≤ 𝑊)
45		simp3 1156	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → 𝜓)
46		eleq1 2852	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑅 → (𝑠 ∈ 𝐴 ↔ 𝑅 ∈ 𝐴))
47		breq1 5109	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑅 → (𝑠 ≤ 𝑊 ↔ 𝑅 ≤ 𝑊))
48	47	notbid 321	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑅 → (¬ 𝑠 ≤ 𝑊 ↔ ¬ 𝑅 ≤ 𝑊))
49		cdlemefrs27.eq	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑅 → (𝜑 ↔ 𝜓))
50	48, 49	anbi12d 644	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑅 → ((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ↔ (¬ 𝑅 ≤ 𝑊 ∧ 𝜓)))
51	46, 50	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑅 → ((𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑)) ↔ (𝑅 ∈ 𝐴 ∧ (¬ 𝑅 ≤ 𝑊 ∧ 𝜓))))
52	51	biimprcd 253	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝐴 ∧ (¬ 𝑅 ≤ 𝑊 ∧ 𝜓)) → (𝑠 = 𝑅 → (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))))
53	43, 44, 45, 52	syl12anc 850	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → (𝑠 = 𝑅 → (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))))
54	53	pm4.71rd 572	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → (𝑠 = 𝑅 ↔ ((𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑)) ∧ 𝑠 = 𝑅)))
55	54	imbi1d 344	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ((𝑠 = 𝑅 → 𝑧 = 𝑁) ↔ (((𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑)) ∧ 𝑠 = 𝑅) → 𝑧 = 𝑁)))
56		eqcom 2770	. . . . . . . . 9 ⊢ (𝑧 = 𝑁 ↔ 𝑁 = 𝑧)
57	56	imbi2i 339	. . . . . . . 8 ⊢ ((𝑠 = 𝑅 → 𝑧 = 𝑁) ↔ (𝑠 = 𝑅 → 𝑁 = 𝑧))
58	55, 57	bitr3di 289	. . . . . . 7 ⊢ ((((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) → 𝑧 = 𝑁) ↔ (𝑠 = 𝑅 → 𝑁 = 𝑧)))
59	42, 58	bitr3id 288	. . . . . 6 ⊢ ((((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 → 𝑧 = 𝑁)) ↔ (𝑠 = 𝑅 → 𝑁 = 𝑧)))
60	41, 59	bitrd 282	. . . . 5 ⊢ ((((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 → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))) ↔ (𝑠 = 𝑅 → 𝑁 = 𝑧)))
61	6, 60	bitrid 286	. . . 4 ⊢ ((((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) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))) ↔ (𝑠 = 𝑅 → 𝑁 = 𝑧)))
62	61	albidv 1955	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → (∀𝑠(𝑠 ∈ 𝐴 → (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))) ↔ ∀𝑠(𝑠 = 𝑅 → 𝑁 = 𝑧)))
63	1, 62	bitrid 286	. 2 ⊢ ((((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) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ ∀𝑠(𝑠 = 𝑅 → 𝑁 = 𝑧)))
64		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑠𝑧
65	64	csbiebg 3883	. . . 4 ⊢ (𝑅 ∈ 𝐴 → (∀𝑠(𝑠 = 𝑅 → 𝑁 = 𝑧) ↔ ⦋𝑅 / 𝑠⦌𝑁 = 𝑧))
66	43, 65	syl 18	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → (∀𝑠(𝑠 = 𝑅 → 𝑁 = 𝑧) ↔ ⦋𝑅 / 𝑠⦌𝑁 = 𝑧))
67		eqcom 2770	. . 3 ⊢ (⦋𝑅 / 𝑠⦌𝑁 = 𝑧 ↔ 𝑧 = ⦋𝑅 / 𝑠⦌𝑁)
68	66, 67	bitrdi 290	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → (∀𝑠(𝑠 = 𝑅 → 𝑁 = 𝑧) ↔ 𝑧 = ⦋𝑅 / 𝑠⦌𝑁))
69	63, 68	bitrd 282	1 ⊢ ((((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) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ 𝑧 = ⦋𝑅 / 𝑠⦌𝑁))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ⦋csb 3851   class class class wbr 5106  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  lecple 17467  joincjn 18517  meetcmee 18518  0.cp0 18627  OLcol 40434  Atomscatm 40523  HLchlt 40610  LHypclh 41244

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-proset 18500  df-poset 18519  df-plt 18534  df-lub 18550  df-glb 18551  df-join 18552  df-meet 18553  df-p0 18629  df-lat 18638  df-oposet 40436  df-ol 40438  df-oml 40439  df-covers 40526  df-ats 40527  df-atl 40558  df-cvlat 40582  df-hlat 40611  df-lhyp 41248

This theorem is used by:  cdlemefrs29bpre1  41657  cdlemefrs32fva  41660  cdlemefr29bpre0N  41666  cdlemefs29bpre0N  41676