Theorem cdleme32fva 41022

Description: Part of proof of Lemma D in [Crawley] p. 113. Value of 𝐹 at an atom not under 𝑊. (Contributed by NM, 2-Mar-2013.)

Hypotheses

Ref	Expression
cdleme32.b	⊢ 𝐵 = (Base‘𝐾)
cdleme32.l	⊢ ≤ = (le‘𝐾)
cdleme32.j	⊢ ∨ = (join‘𝐾)
cdleme32.m	⊢ ∧ = (meet‘𝐾)
cdleme32.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme32.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme32.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme32.c	⊢ 𝐶 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
cdleme32.d	⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdleme32.e	⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
cdleme32.i	⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸))
cdleme32.n	⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐶)
cdleme32.o	⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))))
cdleme32.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥))

Assertion

Ref	Expression
cdleme32fva	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → ⦋𝑅 / 𝑥⦌𝑂 = ⦋𝑅 / 𝑠⦌𝑁)

Distinct variable groups:   𝑡,𝑠,𝑥,𝑦,𝑧,𝐴   𝐵,𝑠,𝑡,𝑥,𝑦,𝑧   𝑦,𝐶   𝐷,𝑠,𝑦,𝑧   𝑦,𝐸   𝐻,𝑠,𝑡   ∨ ,𝑠,𝑡,𝑥,𝑦,𝑧   𝐾,𝑠,𝑡   ≤ ,𝑠,𝑡,𝑥,𝑦,𝑧   ∧ ,𝑠,𝑡,𝑥,𝑦,𝑧   𝑥,𝑁,𝑧   𝑃,𝑠,𝑡,𝑥,𝑦,𝑧   𝑄,𝑠,𝑡,𝑥,𝑦,𝑧   𝑈,𝑠,𝑡,𝑥,𝑦,𝑧   𝑊,𝑠,𝑡,𝑥,𝑦,𝑧   𝑅,𝑠,𝑡,𝑦   𝑦,𝐻   𝑦,𝐾   𝑥,𝑅,𝑧   𝑧,𝐻   𝑧,𝐾

Allowed substitution hints:   𝐶(𝑥,𝑧,𝑡,𝑠)   𝐷(𝑥,𝑡)   𝐸(𝑥,𝑧,𝑡,𝑠)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐻(𝑥)   𝐼(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐾(𝑥)   𝑁(𝑦,𝑡,𝑠)   𝑂(𝑥,𝑦,𝑧,𝑡,𝑠)

Proof of Theorem cdleme32fva

Step	Hyp	Ref	Expression
1		simp2l 1212	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → 𝑅 ∈ 𝐴)
2		cdleme32.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
3		cdleme32.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
4	2, 3	atbase 39874	. . . 4 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ 𝐵)
5	1, 4	syl 17	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → 𝑅 ∈ 𝐵)
6		cdleme32.o	. . . 4 ⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))))
7		eqid 2761	. . . 4 ⊢ (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))) = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))))
8	6, 7	cdleme31so 40964	. . 3 ⊢ (𝑅 ∈ 𝐵 → ⦋𝑅 / 𝑥⦌𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))))
9	5, 8	syl 17	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → ⦋𝑅 / 𝑥⦌𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))))
10		simp1 1148	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
11		simp3 1150	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → 𝑃 ≠ 𝑄)
12		simp2 1149	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
13		cdleme32.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
14		cdleme32.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
15		cdleme32.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
16		cdleme32.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
17		cdleme32.u	. . . . 5 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
18		cdleme32.c	. . . . 5 ⊢ 𝐶 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
19		cdleme32.d	. . . . 5 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
20		cdleme32.e	. . . . 5 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
21		cdleme32.i	. . . . 5 ⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸))
22		cdleme32.n	. . . . 5 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐶)
23	2, 13, 14, 15, 3, 16, 17, 18, 19, 20, 21, 22	cdleme32snb 41021	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → ⦋𝑅 / 𝑠⦌𝑁 ∈ 𝐵)
24	10, 11, 12, 23	syl12anc 847	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → ⦋𝑅 / 𝑠⦌𝑁 ∈ 𝐵)
25		nfv 1933	. . . . . . . . 9 ⊢ Ⅎ𝑠 ¬ 𝑅 ≤ 𝑊
26		nfcsb1v 3874	. . . . . . . . . 10 ⊢ Ⅎ𝑠⦋𝑅 / 𝑠⦌𝑁
27	26	nfeq2 2940	. . . . . . . . 9 ⊢ Ⅎ𝑠 𝑧 = ⦋𝑅 / 𝑠⦌𝑁
28	25, 27	nfim 1915	. . . . . . . 8 ⊢ Ⅎ𝑠(¬ 𝑅 ≤ 𝑊 → 𝑧 = ⦋𝑅 / 𝑠⦌𝑁)
29		breq1 5100	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑅 → (𝑠 ≤ 𝑊 ↔ 𝑅 ≤ 𝑊))
30	29	notbid 320	. . . . . . . . . 10 ⊢ (𝑠 = 𝑅 → (¬ 𝑠 ≤ 𝑊 ↔ ¬ 𝑅 ≤ 𝑊))
31		csbeq1a 3864	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑅 → 𝑁 = ⦋𝑅 / 𝑠⦌𝑁)
32	31	eqeq2d 2772	. . . . . . . . . 10 ⊢ (𝑠 = 𝑅 → (𝑧 = 𝑁 ↔ 𝑧 = ⦋𝑅 / 𝑠⦌𝑁))
33	30, 32	imbi12d 346	. . . . . . . . 9 ⊢ (𝑠 = 𝑅 → ((¬ 𝑠 ≤ 𝑊 → 𝑧 = 𝑁) ↔ (¬ 𝑅 ≤ 𝑊 → 𝑧 = ⦋𝑅 / 𝑠⦌𝑁)))
34	33	ax-gen 1814	. . . . . . . 8 ⊢ ∀𝑠(𝑠 = 𝑅 → ((¬ 𝑠 ≤ 𝑊 → 𝑧 = 𝑁) ↔ (¬ 𝑅 ≤ 𝑊 → 𝑧 = ⦋𝑅 / 𝑠⦌𝑁)))
35		ceqsralt 3487	. . . . . . . 8 ⊢ ((Ⅎ𝑠(¬ 𝑅 ≤ 𝑊 → 𝑧 = ⦋𝑅 / 𝑠⦌𝑁) ∧ ∀𝑠(𝑠 = 𝑅 → ((¬ 𝑠 ≤ 𝑊 → 𝑧 = 𝑁) ↔ (¬ 𝑅 ≤ 𝑊 → 𝑧 = ⦋𝑅 / 𝑠⦌𝑁))) ∧ 𝑅 ∈ 𝐴) → (∀𝑠 ∈ 𝐴 (𝑠 = 𝑅 → (¬ 𝑠 ≤ 𝑊 → 𝑧 = 𝑁)) ↔ (¬ 𝑅 ≤ 𝑊 → 𝑧 = ⦋𝑅 / 𝑠⦌𝑁)))
36	28, 34, 35	mp3an12 1471	. . . . . . 7 ⊢ (𝑅 ∈ 𝐴 → (∀𝑠 ∈ 𝐴 (𝑠 = 𝑅 → (¬ 𝑠 ≤ 𝑊 → 𝑧 = 𝑁)) ↔ (¬ 𝑅 ≤ 𝑊 → 𝑧 = ⦋𝑅 / 𝑠⦌𝑁)))
37	36	adantr 484	. . . . . 6 ⊢ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) → (∀𝑠 ∈ 𝐴 (𝑠 = 𝑅 → (¬ 𝑠 ≤ 𝑊 → 𝑧 = 𝑁)) ↔ (¬ 𝑅 ≤ 𝑊 → 𝑧 = ⦋𝑅 / 𝑠⦌𝑁)))
38	37	3ad2ant2 1146	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → (∀𝑠 ∈ 𝐴 (𝑠 = 𝑅 → (¬ 𝑠 ≤ 𝑊 → 𝑧 = 𝑁)) ↔ (¬ 𝑅 ≤ 𝑊 → 𝑧 = ⦋𝑅 / 𝑠⦌𝑁)))
39		simp11 1216	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
40		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ (0.‘𝐾) = (0.‘𝐾)
41	13, 15, 40, 3, 16	lhpmat 40615	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑅 ∧ 𝑊) = (0.‘𝐾))
42	39, 12, 41	syl2anc 593	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → (𝑅 ∧ 𝑊) = (0.‘𝐾))
43	42	adantr 484	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → (𝑅 ∧ 𝑊) = (0.‘𝐾))
44	43	oveq2d 7407	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → (𝑠 ∨ (𝑅 ∧ 𝑊)) = (𝑠 ∨ (0.‘𝐾)))
45		simp11l 1297	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → 𝐾 ∈ HL)
46	45	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → 𝐾 ∈ HL)
47		hlol 39946	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
48	46, 47	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → 𝐾 ∈ OL)
49	2, 3	atbase 39874	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ 𝐵)
50	49	ad2antrl 738	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → 𝑠 ∈ 𝐵)
51	2, 14, 40	olj01 39810	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ OL ∧ 𝑠 ∈ 𝐵) → (𝑠 ∨ (0.‘𝐾)) = 𝑠)
52	48, 50, 51	syl2anc 593	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → (𝑠 ∨ (0.‘𝐾)) = 𝑠)
53	44, 52	eqtrd 2796	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑠)
54	53	eqeq1d 2763	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → ((𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅 ↔ 𝑠 = 𝑅))
55	43	oveq2d 7407	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → (𝑁 ∨ (𝑅 ∧ 𝑊)) = (𝑁 ∨ (0.‘𝐾)))
56		simpl11 1261	. . . . . . . . . . . . . 14 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
57		simpl12 1262	. . . . . . . . . . . . . 14 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
58		simpl13 1263	. . . . . . . . . . . . . 14 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
59		simpr 488	. . . . . . . . . . . . . 14 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊))
60		simpl3 1206	. . . . . . . . . . . . . 14 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → 𝑃 ≠ 𝑄)
61	2, 13, 14, 15, 3, 16, 17, 18, 19, 20, 21, 22	cdleme27cl 40951	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄)) → 𝑁 ∈ 𝐵)
62	56, 57, 58, 59, 60, 61	syl122anc 1397	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → 𝑁 ∈ 𝐵)
63	2, 14, 40	olj01 39810	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ OL ∧ 𝑁 ∈ 𝐵) → (𝑁 ∨ (0.‘𝐾)) = 𝑁)
64	48, 62, 63	syl2anc 593	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → (𝑁 ∨ (0.‘𝐾)) = 𝑁)
65	55, 64	eqtrd 2796	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → (𝑁 ∨ (𝑅 ∧ 𝑊)) = 𝑁)
66	65	eqeq2d 2772	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → (𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)) ↔ 𝑧 = 𝑁))
67	54, 66	imbi12d 346	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → (((𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅 → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ (𝑠 = 𝑅 → 𝑧 = 𝑁)))
68	67	expr 460	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ 𝑠 ∈ 𝐴) → (¬ 𝑠 ≤ 𝑊 → (((𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅 → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ (𝑠 = 𝑅 → 𝑧 = 𝑁))))
69	68	pm5.74d 275	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ 𝑠 ∈ 𝐴) → ((¬ 𝑠 ≤ 𝑊 → ((𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅 → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))) ↔ (¬ 𝑠 ≤ 𝑊 → (𝑠 = 𝑅 → 𝑧 = 𝑁))))
70		impexp 454	. . . . . . 7 ⊢ (((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ (¬ 𝑠 ≤ 𝑊 → ((𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅 → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))))
71		bi2.04 390	. . . . . . 7 ⊢ ((𝑠 = 𝑅 → (¬ 𝑠 ≤ 𝑊 → 𝑧 = 𝑁)) ↔ (¬ 𝑠 ≤ 𝑊 → (𝑠 = 𝑅 → 𝑧 = 𝑁)))
72	69, 70, 71	3bitr4g 316	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ 𝑠 ∈ 𝐴) → (((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ (𝑠 = 𝑅 → (¬ 𝑠 ≤ 𝑊 → 𝑧 = 𝑁))))
73	72	ralbidva 3182	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → (∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ ∀𝑠 ∈ 𝐴 (𝑠 = 𝑅 → (¬ 𝑠 ≤ 𝑊 → 𝑧 = 𝑁))))
74		simp2r 1213	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → ¬ 𝑅 ≤ 𝑊)
75		biimt 362	. . . . . 6 ⊢ (¬ 𝑅 ≤ 𝑊 → (𝑧 = ⦋𝑅 / 𝑠⦌𝑁 ↔ (¬ 𝑅 ≤ 𝑊 → 𝑧 = ⦋𝑅 / 𝑠⦌𝑁)))
76	74, 75	syl 17	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → (𝑧 = ⦋𝑅 / 𝑠⦌𝑁 ↔ (¬ 𝑅 ≤ 𝑊 → 𝑧 = ⦋𝑅 / 𝑠⦌𝑁)))
77	38, 73, 76	3bitr4d 313	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → (∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ 𝑧 = ⦋𝑅 / 𝑠⦌𝑁))
78	77	adantr 484	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ 𝑧 ∈ 𝐵) → (∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ 𝑧 = ⦋𝑅 / 𝑠⦌𝑁))
79	24, 78	riota5 7377	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))) = ⦋𝑅 / 𝑠⦌𝑁)
80	9, 79	eqtrd 2796	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → ⦋𝑅 / 𝑥⦌𝑂 = ⦋𝑅 / 𝑠⦌𝑁)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097  ∀wal 1557   = wceq 1559  Ⅎwnf 1802   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  ⦋csb 3850  ifcif 4477   class class class wbr 5097   ↦ cmpt 5178  ‘cfv 6516  ℩crio 7347  (class class class)co 7391  Basecbs 17236  lecple 17284  joincjn 18334  meetcmee 18335  0.cp0 18444  OLcol 39759  Atomscatm 39848  HLchlt 39935  LHypclh 40569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-riotaBAD 39538

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-iin 4949  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-undef 8247  df-proset 18317  df-poset 18336  df-plt 18351  df-lub 18367  df-glb 18368  df-join 18369  df-meet 18370  df-p0 18446  df-p1 18447  df-lat 18455  df-clat 18522  df-oposet 39761  df-ol 39763  df-oml 39764  df-covers 39851  df-ats 39852  df-atl 39883  df-cvlat 39907  df-hlat 39936  df-llines 40083  df-lplanes 40084  df-lvols 40085  df-lines 40086  df-psubsp 40088  df-pmap 40089  df-padd 40381  df-lhyp 40573

This theorem is referenced by:  cdleme32fva1  41023