Theorem cdlemefrs29bpre0 38859

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 3065	. . 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 469	. . . . . . 7 ⊢ (((𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑)) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) ↔ (𝑠 ∈ 𝐴 ∧ ((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅)))
3	2	imbi1i 349	. . . . . 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 451	. . . . . 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 451	. . . . . 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 301	. . . . 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 1248	. . . . . . . . . . . 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 1227	. . . . . . . . . . . 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 2736	. . . . . . . . . . . . 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 38493	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑅 ∧ 𝑊) = (0.‘𝐾))
15	7, 8, 14	syl2anc 584	. . . . . . . . . . 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 7373	. . . . . . . . . 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 1284	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → 𝐾 ∈ HL)
18		hlol 37823	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
19	17, 18	syl 17	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 769	. . . . . . . . . . . 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 37751	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ 𝐵)
24	21, 23	syl 17	. . . . . . . . . . 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 37687	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OL ∧ 𝑠 ∈ 𝐵) → (𝑠 ∨ (0.‘𝐾)) = 𝑠)
27	20, 24, 26	syl2anc 584	. . . . . . . . . 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 2776	. . . . . . . . 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 2738	. . . . . . . 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 7373	. . . . . . . . . 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 1191	. . . . . . . . . . . 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 1226	. . . . . . . . . . . 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 771	. . . . . . . . . . . 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 1374	. . . . . . . . . . 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 37687	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OL ∧ 𝑁 ∈ 𝐵) → (𝑁 ∨ (0.‘𝐾)) = 𝑁)
37	20, 35, 36	syl2anc 584	. . . . . . . . . 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 2776	. . . . . . . . 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 2747	. . . . . . . 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 344	. . . . . . 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 802	. . . . . 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 451	. . . . . . 7 ⊢ ((((𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑)) ∧ 𝑠 = 𝑅) → 𝑧 = 𝑁) ↔ ((𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑)) → (𝑠 = 𝑅 → 𝑧 = 𝑁)))
43		simp2rl 1242	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → 𝑅 ∈ 𝐴)
44		simp2rr 1243	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ¬ 𝑅 ≤ 𝑊)
45		simp3 1138	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → 𝜓)
46		eleq1 2825	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑅 → (𝑠 ∈ 𝐴 ↔ 𝑅 ∈ 𝐴))
47		breq1 5108	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑅 → (𝑠 ≤ 𝑊 ↔ 𝑅 ≤ 𝑊))
48	47	notbid 317	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑅 → (¬ 𝑠 ≤ 𝑊 ↔ ¬ 𝑅 ≤ 𝑊))
49		cdlemefrs27.eq	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑅 → (𝜑 ↔ 𝜓))
50	48, 49	anbi12d 631	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑅 → ((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ↔ (¬ 𝑅 ≤ 𝑊 ∧ 𝜓)))
51	46, 50	anbi12d 631	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑅 → ((𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑)) ↔ (𝑅 ∈ 𝐴 ∧ (¬ 𝑅 ≤ 𝑊 ∧ 𝜓))))
52	51	biimprcd 249	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝐴 ∧ (¬ 𝑅 ≤ 𝑊 ∧ 𝜓)) → (𝑠 = 𝑅 → (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))))
53	43, 44, 45, 52	syl12anc 835	. . . . . . . . . 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 563	. . . . . . . . 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 341	. . . . . . . 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 2743	. . . . . . . . 9 ⊢ (𝑧 = 𝑁 ↔ 𝑁 = 𝑧)
57	56	imbi2i 335	. . . . . . . 8 ⊢ ((𝑠 = 𝑅 → 𝑧 = 𝑁) ↔ (𝑠 = 𝑅 → 𝑁 = 𝑧))
58	55, 57	bitr3di 285	. . . . . . 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 284	. . . . . 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 278	. . . . 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 282	. . . 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 1923	. . 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 282	. 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 2907	. . . . 5 ⊢ Ⅎ𝑠𝑧
65	64	csbiebg 3888	. . . 4 ⊢ (𝑅 ∈ 𝐴 → (∀𝑠(𝑠 = 𝑅 → 𝑁 = 𝑧) ↔ ⦋𝑅 / 𝑠⦌𝑁 = 𝑧))
66	43, 65	syl 17	. . 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 2743	. . 3 ⊢ (⦋𝑅 / 𝑠⦌𝑁 = 𝑧 ↔ 𝑧 = ⦋𝑅 / 𝑠⦌𝑁)
68	66, 67	bitrdi 286	. 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 278	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))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087  ∀wal 1539   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ⦋csb 3855   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  meetcmee 18201  0.cp0 18312  OLcol 37636  Atomscatm 37725  HLchlt 37812  LHypclh 38447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-lat 18321  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-lhyp 38451

This theorem is referenced by:  cdlemefrs29bpre1  38860  cdlemefrs32fva  38863  cdlemefr29bpre0N  38869  cdlemefs29bpre0N  38879