cdleme32fva - Mathbox for Norm Megill

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Theorem cdleme32fva 39798

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 1196	. . . 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 38649	. . . 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 2724	. . . 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 39740	. . 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 1133	. . . 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 1135	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → 𝑃 ≠ 𝑄)
12		simp2 1134	. . . 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 39797	. . . 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 834	. . 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 1909	. . . . . . . . 9 ⊢ Ⅎ𝑠 ¬ 𝑅 ≤ 𝑊
26		nfcsb1v 3910	. . . . . . . . . 10 ⊢ Ⅎ𝑠⦋𝑅 / 𝑠⦌𝑁
27	26	nfeq2 2912	. . . . . . . . 9 ⊢ Ⅎ𝑠 𝑧 = ⦋𝑅 / 𝑠⦌𝑁
28	25, 27	nfim 1891	. . . . . . . 8 ⊢ Ⅎ𝑠(¬ 𝑅 ≤ 𝑊 → 𝑧 = ⦋𝑅 / 𝑠⦌𝑁)
29		breq1 5141	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑅 → (𝑠 ≤ 𝑊 ↔ 𝑅 ≤ 𝑊))
30	29	notbid 318	. . . . . . . . . 10 ⊢ (𝑠 = 𝑅 → (¬ 𝑠 ≤ 𝑊 ↔ ¬ 𝑅 ≤ 𝑊))
31		csbeq1a 3899	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑅 → 𝑁 = ⦋𝑅 / 𝑠⦌𝑁)
32	31	eqeq2d 2735	. . . . . . . . . 10 ⊢ (𝑠 = 𝑅 → (𝑧 = 𝑁 ↔ 𝑧 = ⦋𝑅 / 𝑠⦌𝑁))
33	30, 32	imbi12d 344	. . . . . . . . 9 ⊢ (𝑠 = 𝑅 → ((¬ 𝑠 ≤ 𝑊 → 𝑧 = 𝑁) ↔ (¬ 𝑅 ≤ 𝑊 → 𝑧 = ⦋𝑅 / 𝑠⦌𝑁)))
34	33	ax-gen 1789	. . . . . . . 8 ⊢ ∀𝑠(𝑠 = 𝑅 → ((¬ 𝑠 ≤ 𝑊 → 𝑧 = 𝑁) ↔ (¬ 𝑅 ≤ 𝑊 → 𝑧 = ⦋𝑅 / 𝑠⦌𝑁)))
35		ceqsralt 3499	. . . . . . . 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 1447	. . . . . . 7 ⊢ (𝑅 ∈ 𝐴 → (∀𝑠 ∈ 𝐴 (𝑠 = 𝑅 → (¬ 𝑠 ≤ 𝑊 → 𝑧 = 𝑁)) ↔ (¬ 𝑅 ≤ 𝑊 → 𝑧 = ⦋𝑅 / 𝑠⦌𝑁)))
37	36	adantr 480	. . . . . 6 ⊢ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) → (∀𝑠 ∈ 𝐴 (𝑠 = 𝑅 → (¬ 𝑠 ≤ 𝑊 → 𝑧 = 𝑁)) ↔ (¬ 𝑅 ≤ 𝑊 → 𝑧 = ⦋𝑅 / 𝑠⦌𝑁)))
38	37	3ad2ant2 1131	. . . . 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 1200	. . . . . . . . . . . . . . 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 2724	. . . . . . . . . . . . . . . 16 ⊢ (0.‘𝐾) = (0.‘𝐾)
41	13, 15, 40, 3, 16	lhpmat 39391	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑅 ∧ 𝑊) = (0.‘𝐾))
42	39, 12, 41	syl2anc 583	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 7417	. . . . . . . . . . . 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 1281	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → 𝐾 ∈ HL)
46	45	adantr 480	. . . . . . . . . . . . . 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 38721	. . . . . . . . . . . . . 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 38649	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ 𝐵)
50	49	ad2antrl 725	. . . . . . . . . . . . 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 38585	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ OL ∧ 𝑠 ∈ 𝐵) → (𝑠 ∨ (0.‘𝐾)) = 𝑠)
52	48, 50, 51	syl2anc 583	. . . . . . . . . . . 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 2764	. . . . . . . . . . 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 2726	. . . . . . . . . 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 7417	. . . . . . . . . . . 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 1245	. . . . . . . . . . . . . 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 1246	. . . . . . . . . . . . . 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 1247	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 1190	. . . . . . . . . . . . . 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 39727	. . . . . . . . . . . . . 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 1376	. . . . . . . . . . . . 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 38585	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ OL ∧ 𝑁 ∈ 𝐵) → (𝑁 ∨ (0.‘𝐾)) = 𝑁)
64	48, 62, 63	syl2anc 583	. . . . . . . . . . . 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 2764	. . . . . . . . . . 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 2735	. . . . . . . . . 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 344	. . . . . . . . 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 456	. . . . . . . 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 273	. . . . . . 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 450	. . . . . . 7 ⊢ (((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ (¬ 𝑠 ≤ 𝑊 → ((𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅 → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))))
71		bi2.04 387	. . . . . . 7 ⊢ ((𝑠 = 𝑅 → (¬ 𝑠 ≤ 𝑊 → 𝑧 = 𝑁)) ↔ (¬ 𝑠 ≤ 𝑊 → (𝑠 = 𝑅 → 𝑧 = 𝑁)))
72	69, 70, 71	3bitr4g 314	. . . . . 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 3167	. . . . 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 1197	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) → ¬ 𝑅 ≤ 𝑊)
75		biimt 360	. . . . . 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 311	. . . 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 480	. . 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 7387	. 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 2764	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)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 ∀wal 1531 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ⦋csb 3885 ifcif 4520 class class class wbr 5138 ↦ cmpt 5221 ‘cfv 6533 ℩crio 7356 (class class class)co 7401 Basecbs 17143 lecple 17203 joincjn 18266 meetcmee 18267 0.cp0 18378 OLcol 38534 Atomscatm 38623 HLchlt 38710 LHypclh 39345

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-riotaBAD 38313

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-undef 8253 df-proset 18250 df-poset 18268 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18387 df-clat 18454 df-oposet 38536 df-ol 38538 df-oml 38539 df-covers 38626 df-ats 38627 df-atl 38658 df-cvlat 38682 df-hlat 38711 df-llines 38859 df-lplanes 38860 df-lvols 38861 df-lines 38862 df-psubsp 38864 df-pmap 38865 df-padd 39157 df-lhyp 39349

This theorem is referenced by: cdleme32fva1 39799

W3C validator