Theorem cdleme42ke 40488

Description: Part of proof of Lemma E in [Crawley] p. 113. Remove 𝑅 ≠ 𝑆 condition. TODO: FIX COMMENT. (Contributed by NM, 2-Apr-2013.)

Hypotheses

Ref	Expression
cdleme41.b	⊢ 𝐵 = (Base‘𝐾)
cdleme41.l	⊢ ≤ = (le‘𝐾)
cdleme41.j	⊢ ∨ = (join‘𝐾)
cdleme41.m	⊢ ∧ = (meet‘𝐾)
cdleme41.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme41.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme41.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme41.d	⊢ 𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
cdleme41.e	⊢ 𝐸 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdleme41.g	⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
cdleme41.i	⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺))
cdleme41.n	⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
cdleme41.o	⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))))
cdleme41.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥))
cdleme34e.v	⊢ 𝑉 = ((𝑅 ∨ 𝑆) ∧ 𝑊)

Assertion

Ref	Expression
cdleme42ke	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → ((𝐹‘𝑅) ∨ (𝐹‘𝑆)) = ((𝐹‘𝑅) ∨ 𝑉))

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

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

Proof of Theorem cdleme42ke

Step	Hyp	Ref	Expression
1		simpl1l 1224	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → 𝐾 ∈ HL)
2		simpr2 1195	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
3		cdleme41.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
4		cdleme41.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
5		cdleme41.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
6		cdleme41.m	. . . . . . . 8 ⊢ ∧ = (meet‘𝐾)
7		cdleme41.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
8		cdleme41.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
9		cdleme41.u	. . . . . . . 8 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
10		cdleme41.d	. . . . . . . 8 ⊢ 𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
11		cdleme41.e	. . . . . . . 8 ⊢ 𝐸 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
12		cdleme41.g	. . . . . . . 8 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
13		cdleme41.i	. . . . . . . 8 ⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺))
14		cdleme41.n	. . . . . . . 8 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
15		cdleme41.o	. . . . . . . 8 ⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))))
16		cdleme41.f	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥))
17	3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16	cdleme32fvaw 40442	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ((𝐹‘𝑅) ∈ 𝐴 ∧ ¬ (𝐹‘𝑅) ≤ 𝑊))
18	2, 17	syldan 591	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → ((𝐹‘𝑅) ∈ 𝐴 ∧ ¬ (𝐹‘𝑅) ≤ 𝑊))
19	18	simpld 494	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → (𝐹‘𝑅) ∈ 𝐴)
20	5, 7	hlatjidm 39371	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑅) ∈ 𝐴) → ((𝐹‘𝑅) ∨ (𝐹‘𝑅)) = (𝐹‘𝑅))
21	1, 19, 20	syl2anc 584	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → ((𝐹‘𝑅) ∨ (𝐹‘𝑅)) = (𝐹‘𝑅))
22		fveq2 6905	. . . . 5 ⊢ (𝑅 = 𝑆 → (𝐹‘𝑅) = (𝐹‘𝑆))
23	22	oveq2d 7448	. . . 4 ⊢ (𝑅 = 𝑆 → ((𝐹‘𝑅) ∨ (𝐹‘𝑅)) = ((𝐹‘𝑅) ∨ (𝐹‘𝑆)))
24	21, 23	sylan9req 2797	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) ∧ 𝑅 = 𝑆) → (𝐹‘𝑅) = ((𝐹‘𝑅) ∨ (𝐹‘𝑆)))
25		simpr2l 1232	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → 𝑅 ∈ 𝐴)
26	5, 7	hlatjidm 39371	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴) → (𝑅 ∨ 𝑅) = 𝑅)
27	1, 25, 26	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → (𝑅 ∨ 𝑅) = 𝑅)
28	27	oveq1d 7447	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → ((𝑅 ∨ 𝑅) ∧ 𝑊) = (𝑅 ∧ 𝑊))
29		simpl1 1191	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
30		eqid 2736	. . . . . . . . 9 ⊢ (0.‘𝐾) = (0.‘𝐾)
31	4, 6, 30, 7, 8	lhpmat 40033	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑅 ∧ 𝑊) = (0.‘𝐾))
32	29, 2, 31	syl2anc 584	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → (𝑅 ∧ 𝑊) = (0.‘𝐾))
33	28, 32	eqtrd 2776	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → ((𝑅 ∨ 𝑅) ∧ 𝑊) = (0.‘𝐾))
34	33	oveq2d 7448	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → ((𝐹‘𝑅) ∨ ((𝑅 ∨ 𝑅) ∧ 𝑊)) = ((𝐹‘𝑅) ∨ (0.‘𝐾)))
35		hlol 39363	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
36	1, 35	syl 17	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → 𝐾 ∈ OL)
37	3, 7	atbase 39291	. . . . . . 7 ⊢ ((𝐹‘𝑅) ∈ 𝐴 → (𝐹‘𝑅) ∈ 𝐵)
38	19, 37	syl 17	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → (𝐹‘𝑅) ∈ 𝐵)
39	3, 5, 30	olj01 39227	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (𝐹‘𝑅) ∈ 𝐵) → ((𝐹‘𝑅) ∨ (0.‘𝐾)) = (𝐹‘𝑅))
40	36, 38, 39	syl2anc 584	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → ((𝐹‘𝑅) ∨ (0.‘𝐾)) = (𝐹‘𝑅))
41	34, 40	eqtrd 2776	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → ((𝐹‘𝑅) ∨ ((𝑅 ∨ 𝑅) ∧ 𝑊)) = (𝐹‘𝑅))
42		oveq2 7440	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑅 ∨ 𝑅) = (𝑅 ∨ 𝑆))
43	42	oveq1d 7447	. . . . . 6 ⊢ (𝑅 = 𝑆 → ((𝑅 ∨ 𝑅) ∧ 𝑊) = ((𝑅 ∨ 𝑆) ∧ 𝑊))
44		cdleme34e.v	. . . . . 6 ⊢ 𝑉 = ((𝑅 ∨ 𝑆) ∧ 𝑊)
45	43, 44	eqtr4di 2794	. . . . 5 ⊢ (𝑅 = 𝑆 → ((𝑅 ∨ 𝑅) ∧ 𝑊) = 𝑉)
46	45	oveq2d 7448	. . . 4 ⊢ (𝑅 = 𝑆 → ((𝐹‘𝑅) ∨ ((𝑅 ∨ 𝑅) ∧ 𝑊)) = ((𝐹‘𝑅) ∨ 𝑉))
47	41, 46	sylan9req 2797	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) ∧ 𝑅 = 𝑆) → (𝐹‘𝑅) = ((𝐹‘𝑅) ∨ 𝑉))
48	24, 47	eqtr3d 2778	. 2 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) ∧ 𝑅 = 𝑆) → ((𝐹‘𝑅) ∨ (𝐹‘𝑆)) = ((𝐹‘𝑅) ∨ 𝑉))
49	3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 44	cdleme42k 40487	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ 𝑅 ≠ 𝑆) → ((𝐹‘𝑅) ∨ (𝐹‘𝑆)) = ((𝐹‘𝑅) ∨ 𝑉))
50	49	3expa 1118	. 2 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) ∧ 𝑅 ≠ 𝑆) → ((𝐹‘𝑅) ∨ (𝐹‘𝑆)) = ((𝐹‘𝑅) ∨ 𝑉))
51	48, 50	pm2.61dane 3028	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))) → ((𝐹‘𝑅) ∨ (𝐹‘𝑆)) = ((𝐹‘𝑅) ∨ 𝑉))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  ifcif 4524   class class class wbr 5142   ↦ cmpt 5224  ‘cfv 6560  ℩crio 7388  (class class class)co 7432  Basecbs 17248  lecple 17305  joincjn 18358  meetcmee 18359  0.cp0 18469  OLcol 39176  Atomscatm 39265  HLchlt 39352  LHypclh 39987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-riotaBAD 38955

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-undef 8299  df-proset 18341  df-poset 18360  df-plt 18376  df-lub 18392  df-glb 18393  df-join 18394  df-meet 18395  df-p0 18471  df-p1 18472  df-lat 18478  df-clat 18545  df-oposet 39178  df-ol 39180  df-oml 39181  df-covers 39268  df-ats 39269  df-atl 39300  df-cvlat 39324  df-hlat 39353  df-llines 39501  df-lplanes 39502  df-lvols 39503  df-lines 39504  df-psubsp 39506  df-pmap 39507  df-padd 39799  df-lhyp 39991

This theorem is referenced by:  cdleme42keg  40489  cdleme42mN  40490  cdlemeg46fjv  40526