Theorem cdlemefrs32fva 41551

Description: Part of proof of Lemma E in [Crawley] p. 113. Value of 𝐹 at an atom not under 𝑊. TODO: FIX COMMENT. TODO: consolidate uses of lhpmat 41181 here and elsewhere, and presence/absence of 𝑠 ≤ (𝑃 ∨ 𝑄) term. Also, why can proof be shortened with cdleme29cl 41528? What is difference from cdlemefs27cl 41564? (Contributed by NM, 29-Mar-2013.)

Hypotheses

Ref	Expression
cdlemefrs27.b	⊢ 𝐵 = (Base‘𝐾)
cdlemefrs27.l	⊢ ≤ = (le‘𝐾)
cdlemefrs27.j	⊢ ∨ = (join‘𝐾)
cdlemefrs27.m	⊢ ∧ = (meet‘𝐾)
cdlemefrs27.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemefrs27.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemefrs27.eq	⊢ (𝑠 = 𝑅 → (𝜑 ↔ 𝜓))
cdlemefrs27.nb	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))) → 𝑁 ∈ 𝐵)
cdlemefrs27.rnb	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ⦋𝑅 / 𝑠⦌𝑁 ∈ 𝐵)
cdleme29frs.o	⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))))

Assertion

Ref	Expression
cdlemefrs32fva	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ⦋𝑅 / 𝑥⦌𝑂 = ⦋𝑅 / 𝑠⦌𝑁)

Distinct variable groups:   𝑧,𝑠,𝐴   𝐻,𝑠   ∨ ,𝑠   𝐾,𝑠   ≤ ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠   𝑊,𝑠   𝜓,𝑠   𝑧,𝐴   𝑧,𝐵   𝑧,𝐻   𝑧,𝐾   𝑧, ≤   𝑧,𝑁   𝑧,𝑃   𝑧,𝑄   𝑧,𝑅   𝑧,𝑊   𝜓,𝑧   𝐵,𝑠   𝑧, ∨   ∧ ,𝑠,𝑧   𝜑,𝑧   𝑥,𝑧,𝐴   𝑥,𝐵   𝑥, ∨   𝑥, ≤   𝑥, ∧   𝑥,𝑁   𝑥,𝑠,𝑅   𝑥,𝑊

Allowed substitution hints:   𝜑(𝑥, 𝑠)   𝜓(𝑥)   𝑃(𝑥)   𝑄(𝑥)   𝐻(𝑥)   𝐾(𝑥)   𝑁(𝑠)   𝑂(𝑥, 𝑧, 𝑠)

Proof of Theorem cdlemefrs32fva

Step	Hyp	Ref	Expression
1		simp2rl 1261	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → 𝑅 ∈ 𝐴)
2		cdlemefrs27.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
3		cdlemefrs27.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
4	2, 3	atbase 40440	. . 3 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ 𝐵)
5		cdleme29frs.o	. . . 4 ⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))))
6		eqid 2760	. . . 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))))
7	5, 6	cdleme31so 41530	. . 3 ⊢ (𝑅 ∈ 𝐵 → ⦋𝑅 / 𝑥⦌𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))))
8	1, 4, 7	3syl 19	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ⦋𝑅 / 𝑥⦌𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))))
9		ssidd 3931	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → 𝐵 ⊆ 𝐵)
10		simpll 779	. . . . . . . 8 ⊢ (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → ¬ 𝑠 ≤ 𝑊)
11		simpr 490	. . . . . . . 8 ⊢ (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅)
12	10, 11	jca 521	. . . . . . 7 ⊢ (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → (¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅))
13	12	imim1i 64	. . . . . 6 ⊢ (((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) → (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))))
14	13	ralimi 3099	. . . . 5 ⊢ (∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) → ∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))))
15	14	rgenw 3080	. . . 4 ⊢ ∀𝑧 ∈ 𝐵 (∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) → ∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))))
16	15	a1i 11	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ∀𝑧 ∈ 𝐵 (∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) → ∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))))
17		cdlemefrs27.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
18		cdlemefrs27.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
19		cdlemefrs27.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
20		cdlemefrs27.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
21		cdlemefrs27.eq	. . . . 5 ⊢ (𝑠 = 𝑅 → (𝜑 ↔ 𝜓))
22		cdlemefrs27.nb	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ 𝜑))) → 𝑁 ∈ 𝐵)
23		cdlemefrs27.rnb	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ⦋𝑅 / 𝑠⦌𝑁 ∈ 𝐵)
24	2, 17, 18, 19, 3, 20, 21, 22, 23	cdlemefrs29bpre1 41548	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ∃𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))))
25		simpl11 1267	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ 𝑠 ∈ 𝐴) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
26		simpl2r 1246	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ 𝑠 ∈ 𝐴) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
27		simpl3 1212	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ 𝑠 ∈ 𝐴) → 𝜓)
28		simpr 490	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ 𝐴)
29	2, 17, 18, 19, 3, 20, 21	cdlemefrs29pre00 41546	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝜓) ∧ 𝑠 ∈ 𝐴) → (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) ↔ (¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅)))
30	25, 26, 27, 28, 29	syl31anc 1400	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ 𝑠 ∈ 𝐴) → (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) ↔ (¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅)))
31	30	imbi1d 344	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ 𝑠 ∈ 𝐴) → ((((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))))
32	31	ralbidva 3183	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → (∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))))
33	32	rexbidv 3186	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → (∃𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ ∃𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))))
34	24, 33	mpbid 235	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ∃𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))))
35	2, 17, 18, 19, 3, 20, 21, 22, 23	cdlemefrs29cpre1 41549	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ∃!𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))))
36		riotass2 7408	. . 3 ⊢ (((𝐵 ⊆ 𝐵 ∧ ∀𝑧 ∈ 𝐵 (∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) → ∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))))) ∧ (∃𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ∧ ∃!𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))))) → (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))) = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))))
37	9, 16, 34, 35, 36	syl22anc 852	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))) = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))))
38	2, 17, 18, 19, 3, 20, 21, 22	cdlemefrs29bpre0 41547	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → (∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ 𝑧 = ⦋𝑅 / 𝑠⦌𝑁))
39	38	adantr 486	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) ∧ 𝑧 ∈ 𝐵) → (∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊))) ↔ 𝑧 = ⦋𝑅 / 𝑠⦌𝑁))
40	23, 39	riota5 7407	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ 𝜑) ∧ (𝑠 ∨ (𝑅 ∧ 𝑊)) = 𝑅) → 𝑧 = (𝑁 ∨ (𝑅 ∧ 𝑊)))) = ⦋𝑅 / 𝑠⦌𝑁)
41	8, 37, 40	3eqtrd 2799	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝜓) → ⦋𝑅 / 𝑥⦌𝑂 = ⦋𝑅 / 𝑠⦌𝑁)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  ∃!wreu 3363  ⦋csb 3846   ⊆ wss 3895   class class class wbr 5102  ‘cfv 6539  ℩crio 7377  (class class class)co 7421  Basecbs 17393  lecple 17441  joincjn 18491  meetcmee 18492  Atomscatm 40414  HLchlt 40501  LHypclh 41135

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-proset 18474  df-poset 18493  df-plt 18508  df-lub 18524  df-glb 18525  df-join 18526  df-meet 18527  df-p0 18603  df-p1 18604  df-lat 18612  df-clat 18679  df-oposet 40327  df-ol 40329  df-oml 40330  df-covers 40417  df-ats 40418  df-atl 40449  df-cvlat 40473  df-hlat 40502  df-lhyp 41139

This theorem is used by:  cdlemefrs32fva1  41552  cdlemefr32fvaN  41560  cdlemefs32fvaN  41573