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Theorem cdlemefrs32fva 38853
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 38483 here and elsewhere, and presence/absence of 𝑠 (𝑃 𝑄) term. Also, why can proof be shortened with cdleme29cl 38830? What is difference from cdlemefs27cl 38866? (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 → (𝜑𝜓))
cdlemefrs27.nb ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → 𝑁𝐵)
cdlemefrs27.rnb ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝑅 / 𝑠𝑁𝐵)
cdleme29frs.o 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
Assertion
Ref Expression
cdlemefrs32fva ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝑅 / 𝑥𝑂 = 𝑅 / 𝑠𝑁)
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
StepHypRef Expression
1 simp2rl 1242 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝑅𝐴)
2 cdlemefrs27.b . . . 4 𝐵 = (Base‘𝐾)
3 cdlemefrs27.a . . . 4 𝐴 = (Atoms‘𝐾)
42, 3atbase 37741 . . 3 (𝑅𝐴𝑅𝐵)
5 cdleme29frs.o . . . 4 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
6 eqid 2736 . . . 4 (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))) = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))))
75, 6cdleme31so 38832 . . 3 (𝑅𝐵𝑅 / 𝑥𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))))
81, 4, 73syl 18 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝑅 / 𝑥𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))))
9 ssidd 3967 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝐵𝐵)
10 simpll 765 . . . . . . . 8 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → ¬ 𝑠 𝑊)
11 simpr 485 . . . . . . . 8 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → (𝑠 (𝑅 𝑊)) = 𝑅)
1210, 11jca 512 . . . . . . 7 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → (¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅))
1312imim1i 63 . . . . . 6 (((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) → (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))))
1413ralimi 3086 . . . . 5 (∀𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) → ∀𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))))
1514rgenw 3068 . . . 4 𝑧𝐵 (∀𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) → ∀𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))))
1615a1i 11 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → ∀𝑧𝐵 (∀𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) → ∀𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (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 → (𝜑𝜓))
22 cdlemefrs27.nb . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → 𝑁𝐵)
23 cdlemefrs27.rnb . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝑅 / 𝑠𝑁𝐵)
242, 17, 18, 19, 3, 20, 21, 22, 23cdlemefrs29bpre1 38850 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → ∃𝑧𝐵𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))))
25 simpl11 1248 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → (𝐾 ∈ HL ∧ 𝑊𝐻))
26 simpl2r 1227 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
27 simpl3 1193 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → 𝜓)
28 simpr 485 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → 𝑠𝐴)
292, 17, 18, 19, 3, 20, 21cdlemefrs29pre00 38848 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝜓) ∧ 𝑠𝐴) → (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) ↔ (¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅)))
3025, 26, 27, 28, 29syl31anc 1373 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) ↔ (¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅)))
3130imbi1d 341 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → ((((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))))
3231ralbidva 3172 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (∀𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ ∀𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))))
3332rexbidv 3175 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (∃𝑧𝐵𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ ∃𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))))
3424, 33mpbid 231 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → ∃𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))))
352, 17, 18, 19, 3, 20, 21, 22, 23cdlemefrs29cpre1 38851 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → ∃!𝑧𝐵𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))))
36 riotass2 7343 . . 3 (((𝐵𝐵 ∧ ∀𝑧𝐵 (∀𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) → ∀𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))))) ∧ (∃𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ∧ ∃!𝑧𝐵𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))))) → (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))) = (𝑧𝐵𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))))
379, 16, 34, 35, 36syl22anc 837 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))) = (𝑧𝐵𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))))
382, 17, 18, 19, 3, 20, 21, 22cdlemefrs29bpre0 38849 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (∀𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ 𝑧 = 𝑅 / 𝑠𝑁))
3938adantr 481 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑧𝐵) → (∀𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ 𝑧 = 𝑅 / 𝑠𝑁))
4023, 39riota5 7342 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (𝑧𝐵𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))) = 𝑅 / 𝑠𝑁)
418, 37, 403eqtrd 2780 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝑅 / 𝑥𝑂 = 𝑅 / 𝑠𝑁)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  ∃!wreu 3351  csb 3855  wss 3910   class class class wbr 5105  cfv 6496  crio 7311  (class class class)co 7356  Basecbs 17082  lecple 17139  joincjn 18199  meetcmee 18200  Atomscatm 37715  HLchlt 37802  LHypclh 38437
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 7671
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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-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 7312  df-ov 7359  df-oprab 7360  df-proset 18183  df-poset 18201  df-plt 18218  df-lub 18234  df-glb 18235  df-join 18236  df-meet 18237  df-p0 18313  df-p1 18314  df-lat 18320  df-clat 18387  df-oposet 37628  df-ol 37630  df-oml 37631  df-covers 37718  df-ats 37719  df-atl 37750  df-cvlat 37774  df-hlat 37803  df-lhyp 38441
This theorem is referenced by:  cdlemefrs32fva1  38854  cdlemefr32fvaN  38862  cdlemefs32fvaN  38875
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