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Theorem cdlemefrs32fva 36190
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 35819 here and elsewhere, and presence/absence of 𝑠 (𝑃 𝑄) term. Also, why can proof be shortened with cdleme29cl 36167? What is difference from cdlemefs27cl 36203? (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 1309 . . 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 35079 . . 3 (𝑅𝐴𝑅𝐵)
5 cdleme29frs.o . . . 4 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
6 eqid 2760 . . . 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 36169 . . 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 ssid 3765 . . . 4 𝐵𝐵
109a1i 11 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝐵𝐵)
11 simpll 807 . . . . . . . 8 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → ¬ 𝑠 𝑊)
12 simpr 479 . . . . . . . 8 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → (𝑠 (𝑅 𝑊)) = 𝑅)
1311, 12jca 555 . . . . . . 7 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → (¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅))
1413imim1i 63 . . . . . 6 (((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) → (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))))
1514ralimi 3090 . . . . 5 (∀𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) → ∀𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))))
1615rgenw 3062 . . . 4 𝑧𝐵 (∀𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) → ∀𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))))
1716a1i 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)))))
18 cdlemefrs27.l . . . . 5 = (le‘𝐾)
19 cdlemefrs27.j . . . . 5 = (join‘𝐾)
20 cdlemefrs27.m . . . . 5 = (meet‘𝐾)
21 cdlemefrs27.h . . . . 5 𝐻 = (LHyp‘𝐾)
22 cdlemefrs27.eq . . . . 5 (𝑠 = 𝑅 → (𝜑𝜓))
23 cdlemefrs27.nb . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → 𝑁𝐵)
24 cdlemefrs27.rnb . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝑅 / 𝑠𝑁𝐵)
252, 18, 19, 20, 3, 21, 22, 23, 24cdlemefrs29bpre1 36187 . . . 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))))
26 simpl11 1315 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → (𝐾 ∈ HL ∧ 𝑊𝐻))
27 simpl2r 1285 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
28 simpl3 1232 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → 𝜓)
29 simpr 479 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → 𝑠𝐴)
302, 18, 19, 20, 3, 21, 22cdlemefrs29pre00 36185 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝜓) ∧ 𝑠𝐴) → (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) ↔ (¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅)))
3126, 27, 28, 29, 30syl31anc 1480 . . . . . . 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)))
3231imbi1d 330 . . . . . 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)))))
3332ralbidva 3123 . . . . 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)))))
3433rexbidv 3190 . . . 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)))))
3525, 34mpbid 222 . . 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))))
362, 18, 19, 20, 3, 21, 22, 23, 24cdlemefrs29cpre1 36188 . . 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))))
37 riotass2 6801 . . 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)))))
3810, 17, 35, 36, 37syl22anc 1478 . 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)))))
392, 18, 19, 20, 3, 21, 22, 23cdlemefrs29bpre0 36186 . . . 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))
4039adantr 472 . . 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))
4124, 40riota5 6800 . 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)
428, 38, 413eqtrd 2798 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  ∃!wreu 3052  csb 3674  wss 3715   class class class wbr 4804  cfv 6049  crio 6773  (class class class)co 6813  Basecbs 16059  lecple 16150  joincjn 17145  meetcmee 17146  Atomscatm 35053  HLchlt 35140  LHypclh 35773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-preset 17129  df-poset 17147  df-plt 17159  df-lub 17175  df-glb 17176  df-join 17177  df-meet 17178  df-p0 17240  df-p1 17241  df-lat 17247  df-clat 17309  df-oposet 34966  df-ol 34968  df-oml 34969  df-covers 35056  df-ats 35057  df-atl 35088  df-cvlat 35112  df-hlat 35141  df-lhyp 35777
This theorem is referenced by:  cdlemefrs32fva1  36191  cdlemefr32fvaN  36199  cdlemefs32fvaN  36212
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