Theorem cdleme40m 38930

Description: Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one on 𝑃 ∨ 𝑄 line. TODO: FIX COMMENT Use proof idea from cdleme32sn1awN 38895. (Contributed by NM, 18-Mar-2013.)

Hypotheses

Ref	Expression
cdleme40.b	⊢ 𝐵 = (Base‘𝐾)
cdleme40.l	⊢ ≤ = (le‘𝐾)
cdleme40.j	⊢ ∨ = (join‘𝐾)
cdleme40.m	⊢ ∧ = (meet‘𝐾)
cdleme40.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme40.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme40.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme40.e	⊢ 𝐸 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdleme40.g	⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
cdleme40.i	⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺))
cdleme40.n	⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
cdleme40a1.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))
cdleme40a1.c	⊢ 𝐶 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝑌))
cdleme40.t	⊢ 𝑇 = ((𝑣 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑣) ∧ 𝑊)))
cdleme40.f	⊢ 𝐹 = ((𝑃 ∨ 𝑄) ∧ (𝑇 ∨ ((𝑆 ∨ 𝑣) ∧ 𝑊)))

Assertion

Ref	Expression
cdleme40m	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → ⦋𝑅 / 𝑠⦌𝑁 ≠ 𝐹)

Distinct variable groups:   𝑣,𝐴   𝑣,𝐵   𝑣,𝐻   𝑣, ∨   𝑣,𝐾   𝑣, ≤   𝑣, ∧   𝑣,𝑃   𝑣,𝑄   𝑣,𝑅   𝑣,𝑈   𝑣,𝑊,𝑠,𝑡,𝑦   𝐴,𝑠   𝑦,𝑡,𝐴   𝐵,𝑠,𝑡,𝑦   𝐸,𝑠   𝑡,𝐹   𝑡,𝐻,𝑦   ∨ ,𝑠,𝑡,𝑦   𝑡,𝐾,𝑦   ≤ ,𝑠,𝑡,𝑦   ∧ ,𝑠,𝑡,𝑦   𝑃,𝑠,𝑡,𝑦   𝑄,𝑠,𝑡,𝑦   𝑅,𝑠,𝑡,𝑦   𝑡,𝑈,𝑦   𝑊,𝑠,𝑡,𝑦   𝑦,𝑌   𝑡,𝑆,𝑣,𝑦   𝑇,𝑠,𝑡,𝑦

Allowed substitution hints:   𝐶(𝑦,𝑣,𝑡,𝑠)   𝐷(𝑦,𝑣,𝑡,𝑠)   𝑆(𝑠)   𝑇(𝑣)   𝑈(𝑠)   𝐸(𝑦,𝑣,𝑡)   𝐹(𝑦,𝑣,𝑠)   𝐺(𝑦,𝑣,𝑡,𝑠)   𝐻(𝑠)   𝐼(𝑦,𝑣,𝑡,𝑠)   𝐾(𝑠)   𝑁(𝑦,𝑣,𝑡,𝑠)   𝑌(𝑣,𝑡,𝑠)

Proof of Theorem cdleme40m

Step	Hyp	Ref	Expression
1		simp22l 1292	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → 𝑅 ∈ 𝐴)
2		simp3l1 1278	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → 𝑅 ≤ (𝑃 ∨ 𝑄))
3		cdleme40.g	. . . 4 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
4		cdleme40.i	. . . 4 ⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺))
5		cdleme40.n	. . . 4 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
6		cdleme40a1.y	. . . 4 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))
7		cdleme40a1.c	. . . 4 ⊢ 𝐶 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝑌))
8	3, 4, 5, 6, 7	cdleme31sn1c 38851	. . 3 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶)
9	1, 2, 8	syl2anc 584	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶)
10		cdleme40.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
11	10	fvexi 6856	. . 3 ⊢ 𝐵 ∈ V
12		nfv 1917	. . . 4 ⊢ Ⅎ𝑡(((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄))))
13		nfra1 3267	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝑌)
14		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑡𝐵
15	13, 14	nfriota 7326	. . . . . . 7 ⊢ Ⅎ𝑡(℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝑌))
16	7, 15	nfcxfr 2905	. . . . . 6 ⊢ Ⅎ𝑡𝐶
17		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑡𝐹
18	16, 17	nfne 3045	. . . . 5 ⊢ Ⅎ𝑡 𝐶 ≠ 𝐹
19	18	a1i 11	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → Ⅎ𝑡 𝐶 ≠ 𝐹)
20	7	a1i 11	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → 𝐶 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝑌)))
21		neeq1 3006	. . . . 5 ⊢ (𝑌 = 𝐶 → (𝑌 ≠ 𝐹 ↔ 𝐶 ≠ 𝐹))
22	21	adantl 482	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) ∧ 𝑌 = 𝐶) → (𝑌 ≠ 𝐹 ↔ 𝐶 ≠ 𝐹))
23		simpl1 1191	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) ∧ (𝑡 ∈ 𝐴 ∧ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)))) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
24		simpl2 1192	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) ∧ (𝑡 ∈ 𝐴 ∧ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)))) → (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)))
25		simpl3l 1228	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) ∧ (𝑡 ∈ 𝐴 ∧ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)))) → (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆))
26		simprl 769	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) ∧ (𝑡 ∈ 𝐴 ∧ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)))) → 𝑡 ∈ 𝐴)
27		simprrl 779	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) ∧ (𝑡 ∈ 𝐴 ∧ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑡 ≤ 𝑊)
28		simprrr 780	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) ∧ (𝑡 ∈ 𝐴 ∧ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑡 ≤ (𝑃 ∨ 𝑄))
29	26, 27, 28	jca31 515	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) ∧ (𝑡 ∈ 𝐴 ∧ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)))) → ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)))
30		simp3r1 1281	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → 𝑣 ∈ 𝐴)
31		simp3r2 1282	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑣 ≤ 𝑊)
32		simp3r3 1283	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑣 ≤ (𝑃 ∨ 𝑄))
33	30, 31, 32	jca31 515	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → ((𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊) ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))
34	33	adantr 481	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) ∧ (𝑡 ∈ 𝐴 ∧ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)))) → ((𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊) ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))
35		cdleme40.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
36		cdleme40.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
37		cdleme40.m	. . . . . . 7 ⊢ ∧ = (meet‘𝐾)
38		cdleme40.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
39		cdleme40.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
40		cdleme40.u	. . . . . . 7 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
41		cdleme40.e	. . . . . . 7 ⊢ 𝐸 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
42		cdleme40.t	. . . . . . 7 ⊢ 𝑇 = ((𝑣 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑣) ∧ 𝑊)))
43		cdleme40.f	. . . . . . 7 ⊢ 𝐹 = ((𝑃 ∨ 𝑄) ∧ (𝑇 ∨ ((𝑆 ∨ 𝑣) ∧ 𝑊)))
44	35, 36, 37, 38, 39, 40, 41, 6, 42, 43	cdleme39n 38929	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊) ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → 𝑌 ≠ 𝐹)
45	23, 24, 25, 29, 34, 44	syl113anc 1382	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) ∧ (𝑡 ∈ 𝐴 ∧ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)))) → 𝑌 ≠ 𝐹)
46	45	ex 413	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → ((𝑡 ∈ 𝐴 ∧ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄))) → 𝑌 ≠ 𝐹))
47		simp1 1136	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
48		simp22r 1293	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑅 ≤ 𝑊)
49		simp21 1206	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → 𝑃 ≠ 𝑄)
50	10, 35, 36, 37, 38, 39, 40, 41, 6, 7	cdleme25cl 38820	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝐶 ∈ 𝐵)
51	47, 1, 48, 49, 2, 50	syl122anc 1379	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → 𝐶 ∈ 𝐵)
52		simp11 1203	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
53		simp12 1204	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
54		simp13 1205	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
55	35, 36, 38, 39	cdlemb2 38504	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → ∃𝑡 ∈ 𝐴 (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)))
56	52, 53, 54, 49, 55	syl121anc 1375	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → ∃𝑡 ∈ 𝐴 (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)))
57	12, 19, 20, 22, 46, 51, 56	riotasv3d 37422	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) ∧ 𝐵 ∈ V) → 𝐶 ≠ 𝐹)
58	11, 57	mpan2 689	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → 𝐶 ≠ 𝐹)
59	9, 58	eqnetrd 3011	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))) → ⦋𝑅 / 𝑠⦌𝑁 ≠ 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  Ⅎwnf 1785   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3073  Vcvv 3445  ⦋csb 3855  ifcif 4486   class class class wbr 5105  ‘cfv 6496  ℩crio 7312  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  meetcmee 18201  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  ax-riotaBAD 37415

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  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-rmo 3353  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-iin 4957  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-mpo 7362  df-1st 7921  df-2nd 7922  df-undef 8204  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-p1 18315  df-lat 18321  df-clat 18388  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-llines 37961  df-lplanes 37962  df-lvols 37963  df-lines 37964  df-psubsp 37966  df-pmap 37967  df-padd 38259  df-lhyp 38451

This theorem is referenced by:  cdleme40n  38931