Theorem cdleme40n 40451

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. TODO get rid of '.<' class? (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)))
cdleme40a1.x	⊢ 𝑋 = ((𝑃 ∨ 𝑄) ∧ (𝑇 ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊)))
cdleme40.o	⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)) → 𝑧 = 𝑋))
cdleme40.v	⊢ 𝑉 = if(𝑢 ≤ (𝑃 ∨ 𝑄), 𝑂, < )
cdleme40a1.z	⊢ 𝑍 = (℩𝑧 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)) → 𝑧 = 𝐹))

Assertion

Ref	Expression
cdleme40n	⊢ ((((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)) → ⦋𝑅 / 𝑠⦌𝑁 ≠ ⦋𝑆 / 𝑢⦌𝑉)

Distinct variable groups:   𝑣,𝑢,𝑧,𝐴   𝑢,𝐵,𝑣,𝑧   𝑧,𝐹   𝑣,𝐻,𝑧   𝑢, ∨ ,𝑣,𝑧   𝑣,𝐾,𝑧   𝑢, ≤ ,𝑣,𝑧   𝑢, ∧ ,𝑣,𝑧   𝑢,𝑃,𝑣,𝑧   𝑢,𝑄,𝑣,𝑧   𝑣,𝑅,𝑧   𝑢,𝑆,𝑧   𝑢,𝑇   𝑣,𝑈,𝑧   𝑢,𝑊,𝑣,𝑧   𝑣,𝑠,𝑡,𝑦,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   𝑣,𝐷   𝑣,𝐼   𝑣,𝑁

Allowed substitution hints:   𝐶(𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝐷(𝑦,𝑧,𝑢,𝑡,𝑠)   𝑅(𝑢)   𝑆(𝑠)   < (𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝑇(𝑧,𝑣)   𝑈(𝑢,𝑠)   𝐸(𝑦,𝑧,𝑣,𝑢,𝑡)   𝐹(𝑦,𝑣,𝑢,𝑠)   𝐺(𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝐻(𝑢,𝑠)   𝐼(𝑦,𝑧,𝑢,𝑡,𝑠)   𝐾(𝑢,𝑠)   𝑁(𝑦,𝑧,𝑢,𝑡,𝑠)   𝑂(𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝑉(𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝑋(𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝑌(𝑧,𝑣,𝑢,𝑡,𝑠)   𝑍(𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)

Proof of Theorem cdleme40n

Step	Hyp	Ref	Expression
1		cdleme40.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2	1	fvexi 6921	. . 3 ⊢ 𝐵 ∈ V
3		nfv 1912	. . . 4 ⊢ Ⅎ𝑣(((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))
4		nfcv 2903	. . . . . 6 ⊢ Ⅎ𝑣⦋𝑅 / 𝑠⦌𝑁
5		cdleme40a1.z	. . . . . . 7 ⊢ 𝑍 = (℩𝑧 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)) → 𝑧 = 𝐹))
6		nfra1 3282	. . . . . . . 8 ⊢ Ⅎ𝑣∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)) → 𝑧 = 𝐹)
7		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑣𝐵
8	6, 7	nfriota 7400	. . . . . . 7 ⊢ Ⅎ𝑣(℩𝑧 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)) → 𝑧 = 𝐹))
9	5, 8	nfcxfr 2901	. . . . . 6 ⊢ Ⅎ𝑣𝑍
10	4, 9	nfne 3041	. . . . 5 ⊢ Ⅎ𝑣⦋𝑅 / 𝑠⦌𝑁 ≠ 𝑍
11	10	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)) → Ⅎ𝑣⦋𝑅 / 𝑠⦌𝑁 ≠ 𝑍)
12	5	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)) → 𝑍 = (℩𝑧 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)) → 𝑧 = 𝐹)))
13		neeq2 3002	. . . . 5 ⊢ (𝐹 = 𝑍 → (⦋𝑅 / 𝑠⦌𝑁 ≠ 𝐹 ↔ ⦋𝑅 / 𝑠⦌𝑁 ≠ 𝑍))
14	13	adantl 481	. . . 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)) ∧ 𝐹 = 𝑍) → (⦋𝑅 / 𝑠⦌𝑁 ≠ 𝐹 ↔ ⦋𝑅 / 𝑠⦌𝑁 ≠ 𝑍))
15		simpl11 1247	. . . . . 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)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
16		simpl12 1248	. . . . . 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)))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
17		simpl13 1249	. . . . . 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)))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
18		simpl21 1250	. . . . . 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)))) → 𝑃 ≠ 𝑄)
19		simpl22 1251	. . . . . 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)))) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
20		simpl23 1252	. . . . . 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)))) → (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))
21		simpl3 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)))) → (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆))
22		simprl 771	. . . . . . 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)
23		simprrl 781	. . . . . . 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 ≤ 𝑊)
24		simprrr 782	. . . . . . 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 ≤ (𝑃 ∨ 𝑄))
25	22, 23, 24	3jca 1127	. . . . . 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)))) → (𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))
26		cdleme40.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
27		cdleme40.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
28		cdleme40.m	. . . . . . 7 ⊢ ∧ = (meet‘𝐾)
29		cdleme40.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
30		cdleme40.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
31		cdleme40.u	. . . . . . 7 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
32		cdleme40.e	. . . . . . 7 ⊢ 𝐸 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
33		cdleme40.g	. . . . . . 7 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
34		cdleme40.i	. . . . . . 7 ⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺))
35		cdleme40.n	. . . . . . 7 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
36		cdleme40a1.y	. . . . . . 7 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))
37		cdleme40a1.c	. . . . . . 7 ⊢ 𝐶 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝑌))
38		cdleme40.t	. . . . . . 7 ⊢ 𝑇 = ((𝑣 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑣) ∧ 𝑊)))
39		cdleme40.f	. . . . . . 7 ⊢ 𝐹 = ((𝑃 ∨ 𝑄) ∧ (𝑇 ∨ ((𝑆 ∨ 𝑣) ∧ 𝑊)))
40	1, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39	cdleme40m 40450	. . . . . 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)))) → ⦋𝑅 / 𝑠⦌𝑁 ≠ 𝐹)
41	15, 16, 17, 18, 19, 20, 21, 25, 40	syl332anc 1400	. . . . 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 / 𝑠⦌𝑁 ≠ 𝐹)
42	41	ex 412	. . . 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))) → ⦋𝑅 / 𝑠⦌𝑁 ≠ 𝐹))
43		simp1 1135	. . . . 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)) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
44		simp23l 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)) → 𝑆 ∈ 𝐴)
45		simp23r 1294	. . . . 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)) → ¬ 𝑆 ≤ 𝑊)
46		simp21 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)) → 𝑃 ≠ 𝑄)
47		simp32 1209	. . . . 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)) → 𝑆 ≤ (𝑃 ∨ 𝑄))
48	1, 26, 27, 28, 29, 30, 31, 38, 39, 5	cdleme25cl 40340	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑍 ∈ 𝐵)
49	43, 44, 45, 46, 47, 48	syl122anc 1378	. . . 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)) → 𝑍 ∈ 𝐵)
50		simp11 1202	. . . . 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)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
51		simp12 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)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
52		simp13 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)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
53	26, 27, 29, 30	cdlemb2 40024	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → ∃𝑣 ∈ 𝐴 (¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))
54	50, 51, 52, 46, 53	syl121anc 1374	. . . 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)))
55	3, 11, 12, 14, 42, 49, 54	riotasv3d 38942	. . 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)) ∧ 𝐵 ∈ V) → ⦋𝑅 / 𝑠⦌𝑁 ≠ 𝑍)
56	2, 55	mpan2 691	. 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)) → ⦋𝑅 / 𝑠⦌𝑁 ≠ 𝑍)
57		cdleme40a1.x	. . . 4 ⊢ 𝑋 = ((𝑃 ∨ 𝑄) ∧ (𝑇 ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊)))
58		cdleme40.o	. . . 4 ⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)) → 𝑧 = 𝑋))
59		cdleme40.v	. . . 4 ⊢ 𝑉 = if(𝑢 ≤ (𝑃 ∨ 𝑄), 𝑂, < )
60	57, 58, 59, 39, 5	cdleme31sn1c 40371	. . 3 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑆 / 𝑢⦌𝑉 = 𝑍)
61	44, 47, 60	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)) → ⦋𝑆 / 𝑢⦌𝑉 = 𝑍)
62	56, 61	neeqtrrd 3013	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)) → ⦋𝑅 / 𝑠⦌𝑁 ≠ ⦋𝑆 / 𝑢⦌𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537  Ⅎwnf 1780   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  Vcvv 3478  ⦋csb 3908  ifcif 4531   class class class wbr 5148  ‘cfv 6563  ℩crio 7387  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  Atomscatm 39245  HLchlt 39332  LHypclh 39967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-riotaBAD 38935

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-undef 8297  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481  df-lplanes 39482  df-lvols 39483  df-lines 39484  df-psubsp 39486  df-pmap 39487  df-padd 39779  df-lhyp 39971

This theorem is referenced by:  cdleme40w  40453