Theorem cdlemeg46c 40632

Description: TODO FIX COMMENT. (Contributed by NM, 1-Apr-2013.)

Hypotheses

Ref	Expression
cdlemef46g.b	⊢ 𝐵 = (Base‘𝐾)
cdlemef46g.l	⊢ ≤ = (le‘𝐾)
cdlemef46g.j	⊢ ∨ = (join‘𝐾)
cdlemef46g.m	⊢ ∧ = (meet‘𝐾)
cdlemef46g.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemef46g.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemef46g.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdlemef46g.d	⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdlemefs46g.e	⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
cdlemef46g.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃 ∨ 𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥))
cdlemef46.v	⊢ 𝑉 = ((𝑄 ∨ 𝑃) ∧ 𝑊)
cdlemef46.n	⊢ 𝑁 = ((𝑣 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊)))
cdlemefs46.o	⊢ 𝑂 = ((𝑄 ∨ 𝑃) ∧ (𝑁 ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊)))
cdlemef46.g	⊢ 𝐺 = (𝑎 ∈ 𝐵 ↦ if((𝑄 ≠ 𝑃 ∧ ¬ 𝑎 ≤ 𝑊), (℩𝑐 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ((¬ 𝑢 ≤ 𝑊 ∧ (𝑢 ∨ (𝑎 ∧ 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 ≤ (𝑄 ∨ 𝑃), (℩𝑏 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑃)) → 𝑏 = 𝑂)), ⦋𝑢 / 𝑣⦌𝑁) ∨ (𝑎 ∧ 𝑊)))), 𝑎))

Assertion

Ref	Expression
cdlemeg46c	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (𝐹‘(𝐺‘𝑆)) = ⦋𝑆 / 𝑣⦌⦋𝑁 / 𝑡⦌𝐷)

Distinct variable groups:   𝑡,𝑠,𝑥,𝑦,𝑧,𝐴   𝐵,𝑠,𝑡,𝑥,𝑦,𝑧   𝐷,𝑠,𝑥,𝑦,𝑧   𝑥,𝐸,𝑦,𝑧   𝐻,𝑠,𝑡,𝑥,𝑦,𝑧   ∨ ,𝑠,𝑡,𝑥,𝑦,𝑧   𝐾,𝑠,𝑡,𝑥,𝑦,𝑧   ≤ ,𝑠,𝑡,𝑥,𝑦,𝑧   ∧ ,𝑠,𝑡,𝑥,𝑦,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   𝑆,𝑠,𝑡,𝑥,𝑦,𝑧   𝑎,𝑏,𝑐,𝑢,𝑣,𝐴   𝐵,𝑎,𝑏,𝑐,𝑢,𝑣   𝑣,𝐷   𝐺,𝑠,𝑡,𝑥,𝑦,𝑧   𝐻,𝑎,𝑏,𝑐,𝑢,𝑣   ∨ ,𝑎,𝑏,𝑐,𝑢,𝑣   𝐾,𝑎,𝑏,𝑐,𝑢,𝑣   ≤ ,𝑎,𝑏,𝑐,𝑢,𝑣   ∧ ,𝑎,𝑏,𝑐,𝑢,𝑣   𝑁,𝑎,𝑏,𝑐   𝑂,𝑎,𝑏,𝑐   𝑃,𝑎,𝑏,𝑐,𝑢,𝑣   𝑄,𝑎,𝑏,𝑐,𝑢,𝑣   𝑆,𝑎,𝑏,𝑐,𝑢,𝑣   𝑉,𝑎,𝑏,𝑐   𝑊,𝑎,𝑏,𝑐,𝑢,𝑣   𝑥,𝑢,𝑦,𝑧,𝑁   𝑥,𝑂,𝑦,𝑧   𝑣,𝑡   𝑢,𝑉   𝑥,𝑣,𝑦,𝑧,𝑉

Allowed substitution hints:   𝐷(𝑢,𝑡,𝑎,𝑏,𝑐)   𝑈(𝑣,𝑢,𝑎,𝑏,𝑐)   𝐸(𝑣,𝑢,𝑡,𝑠,𝑎,𝑏,𝑐)   𝐹(𝑥,𝑦,𝑧,𝑣,𝑢,𝑡,𝑠,𝑎,𝑏,𝑐)   𝐺(𝑣,𝑢,𝑎,𝑏,𝑐)   𝑁(𝑣,𝑡,𝑠)   𝑂(𝑣,𝑢,𝑡,𝑠)   𝑉(𝑡,𝑠)

Proof of Theorem cdlemeg46c

Step	Hyp	Ref	Expression
1		cdlemef46g.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		cdlemef46g.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		cdlemef46g.j	. . . 4 ⊢ ∨ = (join‘𝐾)
4		cdlemef46g.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
5		cdlemef46g.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemef46g.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemef46.v	. . . 4 ⊢ 𝑉 = ((𝑄 ∨ 𝑃) ∧ 𝑊)
8		cdlemef46.n	. . . 4 ⊢ 𝑁 = ((𝑣 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊)))
9		cdlemefs46.o	. . . 4 ⊢ 𝑂 = ((𝑄 ∨ 𝑃) ∧ (𝑁 ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊)))
10		cdlemef46.g	. . . 4 ⊢ 𝐺 = (𝑎 ∈ 𝐵 ↦ if((𝑄 ≠ 𝑃 ∧ ¬ 𝑎 ≤ 𝑊), (℩𝑐 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ((¬ 𝑢 ≤ 𝑊 ∧ (𝑢 ∨ (𝑎 ∧ 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 ≤ (𝑄 ∨ 𝑃), (℩𝑏 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑃)) → 𝑏 = 𝑂)), ⦋𝑢 / 𝑣⦌𝑁) ∨ (𝑎 ∧ 𝑊)))), 𝑎))
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	cdlemeg47b 40627	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (𝐺‘𝑆) = ⦋𝑆 / 𝑣⦌𝑁)
12	11	csbeq1d 3850	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → ⦋(𝐺‘𝑆) / 𝑡⦌𝐷 = ⦋⦋𝑆 / 𝑣⦌𝑁 / 𝑡⦌𝐷)
13		simp1 1136	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
14		simp2l 1200	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → 𝑃 ≠ 𝑄)
15		simp11 1204	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
16		simp13 1206	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
17		simp12 1205	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
18		simp2r 1201	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	cdleme46fvaw 40620	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) → ((𝐺‘𝑆) ∈ 𝐴 ∧ ¬ (𝐺‘𝑆) ≤ 𝑊))
20	15, 16, 17, 18, 19	syl31anc 1375	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → ((𝐺‘𝑆) ∈ 𝐴 ∧ ¬ (𝐺‘𝑆) ≤ 𝑊))
21	3, 5	cdleme46f2g2 40612	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ≠ 𝑃 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑃)))
22	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	cdleme46frvlpq 40623	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ≠ 𝑃 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑃)) → ¬ (𝐺‘𝑆) ≤ (𝑄 ∨ 𝑃))
23	21, 22	syl 17	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → ¬ (𝐺‘𝑆) ≤ (𝑄 ∨ 𝑃))
24		simp11l 1285	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → 𝐾 ∈ HL)
25		simp12l 1287	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → 𝑃 ∈ 𝐴)
26		simp13l 1289	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → 𝑄 ∈ 𝐴)
27	3, 5	hlatjcom 39487	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
28	24, 25, 26, 27	syl3anc 1373	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
29	28	breq2d 5105	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → ((𝐺‘𝑆) ≤ (𝑃 ∨ 𝑄) ↔ (𝐺‘𝑆) ≤ (𝑄 ∨ 𝑃)))
30	23, 29	mtbird 325	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → ¬ (𝐺‘𝑆) ≤ (𝑃 ∨ 𝑄))
31		cdlemef46g.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
32		cdlemef46g.d	. . . 4 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
33		cdlemef46g.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃 ∨ 𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥))
34	1, 2, 3, 4, 5, 6, 31, 32, 33	cdlemefr45 40546	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ((𝐺‘𝑆) ∈ 𝐴 ∧ ¬ (𝐺‘𝑆) ≤ 𝑊)) ∧ ¬ (𝐺‘𝑆) ≤ (𝑃 ∨ 𝑄)) → (𝐹‘(𝐺‘𝑆)) = ⦋(𝐺‘𝑆) / 𝑡⦌𝐷)
35	13, 14, 20, 30, 34	syl121anc 1377	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (𝐹‘(𝐺‘𝑆)) = ⦋(𝐺‘𝑆) / 𝑡⦌𝐷)
36		simp2rl 1243	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → 𝑆 ∈ 𝐴)
37		csbnestgw 4373	. . 3 ⊢ (𝑆 ∈ 𝐴 → ⦋𝑆 / 𝑣⦌⦋𝑁 / 𝑡⦌𝐷 = ⦋⦋𝑆 / 𝑣⦌𝑁 / 𝑡⦌𝐷)
38	36, 37	syl 17	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑆 / 𝑣⦌⦋𝑁 / 𝑡⦌𝐷 = ⦋⦋𝑆 / 𝑣⦌𝑁 / 𝑡⦌𝐷)
39	12, 35, 38	3eqtr4d 2778	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (𝐹‘(𝐺‘𝑆)) = ⦋𝑆 / 𝑣⦌⦋𝑁 / 𝑡⦌𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ∀wral 3048  ⦋csb 3846  ifcif 4474   class class class wbr 5093   ↦ cmpt 5174  ‘cfv 6486  ℩crio 7308  (class class class)co 7352  Basecbs 17122  lecple 17170  joincjn 18219  meetcmee 18220  Atomscatm 39382  HLchlt 39469  LHypclh 40103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-riotaBAD 39072

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-undef 8209  df-proset 18202  df-poset 18221  df-plt 18236  df-lub 18252  df-glb 18253  df-join 18254  df-meet 18255  df-p0 18331  df-p1 18332  df-lat 18340  df-clat 18407  df-oposet 39295  df-ol 39297  df-oml 39298  df-covers 39385  df-ats 39386  df-atl 39417  df-cvlat 39441  df-hlat 39470  df-llines 39617  df-lplanes 39618  df-lvols 39619  df-lines 39620  df-psubsp 39622  df-pmap 39623  df-padd 39915  df-lhyp 40107

This theorem is referenced by:  cdlemeg46ngfr  40637