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Theorem cdlemefrs29bpre0 34496
Description: TODO fix comment. (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)
Assertion
Ref Expression
cdlemefrs29bpre0 ((((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))
Distinct variable groups:   𝑧,𝑠   𝐴,𝑠   𝐻,𝑠   ,𝑠   𝐾,𝑠   ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠   𝑊,𝑠   𝜓,𝑠
Allowed substitution hints:   𝜑(𝑧,𝑠)   𝜓(𝑧)   𝐴(𝑧)   𝐵(𝑧,𝑠)   𝑃(𝑧)   𝑄(𝑧)   𝑅(𝑧)   𝐻(𝑧)   (𝑧)   𝐾(𝑧)   (𝑧)   (𝑧,𝑠)   𝑁(𝑧,𝑠)   𝑊(𝑧)

Proof of Theorem cdlemefrs29bpre0
StepHypRef Expression
1 df-ral 2901 . . 3 (∀𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ ∀𝑠(𝑠𝐴 → (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))))
2 anass 679 . . . . . . 7 (((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) ↔ (𝑠𝐴 ∧ ((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅)))
32imbi1i 338 . . . . . 6 ((((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ ((𝑠𝐴 ∧ ((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅)) → 𝑧 = (𝑁 (𝑅 𝑊))))
4 impexp 461 . . . . . 6 ((((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ ((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) → ((𝑠 (𝑅 𝑊)) = 𝑅𝑧 = (𝑁 (𝑅 𝑊)))))
5 impexp 461 . . . . . 6 (((𝑠𝐴 ∧ ((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅)) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ (𝑠𝐴 → (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))))
63, 4, 53bitr3ri 290 . . . . 5 ((𝑠𝐴 → (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))) ↔ ((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) → ((𝑠 (𝑅 𝑊)) = 𝑅𝑧 = (𝑁 (𝑅 𝑊)))))
7 simpl11 1129 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
8 simpl2r 1108 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
9 cdlemefrs27.l . . . . . . . . . . . . 13 = (le‘𝐾)
10 cdlemefrs27.m . . . . . . . . . . . . 13 = (meet‘𝐾)
11 eqid 2610 . . . . . . . . . . . . 13 (0.‘𝐾) = (0.‘𝐾)
12 cdlemefrs27.a . . . . . . . . . . . . 13 𝐴 = (Atoms‘𝐾)
13 cdlemefrs27.h . . . . . . . . . . . . 13 𝐻 = (LHyp‘𝐾)
149, 10, 11, 12, 13lhpmat 34128 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 𝑊) = (0.‘𝐾))
157, 8, 14syl2anc 691 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝑅 𝑊) = (0.‘𝐾))
1615oveq2d 6543 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝑠 (𝑅 𝑊)) = (𝑠 (0.‘𝐾)))
17 simp11l 1165 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝐾 ∈ HL)
18 hlol 33460 . . . . . . . . . . . . 13 (𝐾 ∈ HL → 𝐾 ∈ OL)
1917, 18syl 17 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝐾 ∈ OL)
2019adantr 480 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → 𝐾 ∈ OL)
21 simprl 790 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → 𝑠𝐴)
22 cdlemefrs27.b . . . . . . . . . . . . 13 𝐵 = (Base‘𝐾)
2322, 12atbase 33388 . . . . . . . . . . . 12 (𝑠𝐴𝑠𝐵)
2421, 23syl 17 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → 𝑠𝐵)
25 cdlemefrs27.j . . . . . . . . . . . 12 = (join‘𝐾)
2622, 25, 11olj01 33324 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ 𝑠𝐵) → (𝑠 (0.‘𝐾)) = 𝑠)
2720, 24, 26syl2anc 691 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝑠 (0.‘𝐾)) = 𝑠)
2816, 27eqtrd 2644 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝑠 (𝑅 𝑊)) = 𝑠)
2928eqeq1d 2612 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → ((𝑠 (𝑅 𝑊)) = 𝑅𝑠 = 𝑅))
3015oveq2d 6543 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝑁 (𝑅 𝑊)) = (𝑁 (0.‘𝐾)))
31 simpl1 1057 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
32 simpl2l 1107 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → 𝑃𝑄)
33 simprr 792 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (¬ 𝑠 𝑊𝜑))
34 cdlemefrs27.nb . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → 𝑁𝐵)
3531, 32, 21, 33, 34syl112anc 1322 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → 𝑁𝐵)
3622, 25, 11olj01 33324 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ 𝑁𝐵) → (𝑁 (0.‘𝐾)) = 𝑁)
3720, 35, 36syl2anc 691 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝑁 (0.‘𝐾)) = 𝑁)
3830, 37eqtrd 2644 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝑁 (𝑅 𝑊)) = 𝑁)
3938eqeq2d 2620 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝑧 = (𝑁 (𝑅 𝑊)) ↔ 𝑧 = 𝑁))
4029, 39imbi12d 333 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (((𝑠 (𝑅 𝑊)) = 𝑅𝑧 = (𝑁 (𝑅 𝑊))) ↔ (𝑠 = 𝑅𝑧 = 𝑁)))
4140pm5.74da 719 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) → ((𝑠 (𝑅 𝑊)) = 𝑅𝑧 = (𝑁 (𝑅 𝑊)))) ↔ ((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) → (𝑠 = 𝑅𝑧 = 𝑁))))
42 impexp 461 . . . . . . 7 ((((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) ∧ 𝑠 = 𝑅) → 𝑧 = 𝑁) ↔ ((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) → (𝑠 = 𝑅𝑧 = 𝑁)))
43 eqcom 2617 . . . . . . . . 9 (𝑧 = 𝑁𝑁 = 𝑧)
4443imbi2i 325 . . . . . . . 8 ((𝑠 = 𝑅𝑧 = 𝑁) ↔ (𝑠 = 𝑅𝑁 = 𝑧))
45 simp2rl 1123 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝑅𝐴)
46 simp2rr 1124 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → ¬ 𝑅 𝑊)
47 simp3 1056 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝜓)
48 eleq1 2676 . . . . . . . . . . . . 13 (𝑠 = 𝑅 → (𝑠𝐴𝑅𝐴))
49 breq1 4581 . . . . . . . . . . . . . . 15 (𝑠 = 𝑅 → (𝑠 𝑊𝑅 𝑊))
5049notbid 307 . . . . . . . . . . . . . 14 (𝑠 = 𝑅 → (¬ 𝑠 𝑊 ↔ ¬ 𝑅 𝑊))
51 cdlemefrs27.eq . . . . . . . . . . . . . 14 (𝑠 = 𝑅 → (𝜑𝜓))
5250, 51anbi12d 743 . . . . . . . . . . . . 13 (𝑠 = 𝑅 → ((¬ 𝑠 𝑊𝜑) ↔ (¬ 𝑅 𝑊𝜓)))
5348, 52anbi12d 743 . . . . . . . . . . . 12 (𝑠 = 𝑅 → ((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) ↔ (𝑅𝐴 ∧ (¬ 𝑅 𝑊𝜓))))
5453biimprcd 239 . . . . . . . . . . 11 ((𝑅𝐴 ∧ (¬ 𝑅 𝑊𝜓)) → (𝑠 = 𝑅 → (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))))
5545, 46, 47, 54syl12anc 1316 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (𝑠 = 𝑅 → (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))))
5655pm4.71rd 665 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (𝑠 = 𝑅 ↔ ((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) ∧ 𝑠 = 𝑅)))
5756imbi1d 330 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → ((𝑠 = 𝑅𝑧 = 𝑁) ↔ (((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) ∧ 𝑠 = 𝑅) → 𝑧 = 𝑁)))
5844, 57syl5rbbr 274 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → ((((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) ∧ 𝑠 = 𝑅) → 𝑧 = 𝑁) ↔ (𝑠 = 𝑅𝑁 = 𝑧)))
5942, 58syl5bbr 273 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) → (𝑠 = 𝑅𝑧 = 𝑁)) ↔ (𝑠 = 𝑅𝑁 = 𝑧)))
6041, 59bitrd 267 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) → ((𝑠 (𝑅 𝑊)) = 𝑅𝑧 = (𝑁 (𝑅 𝑊)))) ↔ (𝑠 = 𝑅𝑁 = 𝑧)))
616, 60syl5bb 271 . . . 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)))) ↔ (𝑠 = 𝑅𝑁 = 𝑧)))
6261albidv 1836 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (∀𝑠(𝑠𝐴 → (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))) ↔ ∀𝑠(𝑠 = 𝑅𝑁 = 𝑧)))
631, 62syl5bb 271 . 2 ((((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(𝑠 = 𝑅𝑁 = 𝑧)))
64 nfcv 2751 . . . . 5 𝑠𝑧
6564csbiebg 3522 . . . 4 (𝑅𝐴 → (∀𝑠(𝑠 = 𝑅𝑁 = 𝑧) ↔ 𝑅 / 𝑠𝑁 = 𝑧))
6645, 65syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (∀𝑠(𝑠 = 𝑅𝑁 = 𝑧) ↔ 𝑅 / 𝑠𝑁 = 𝑧))
67 eqcom 2617 . . 3 (𝑅 / 𝑠𝑁 = 𝑧𝑧 = 𝑅 / 𝑠𝑁)
6866, 67syl6bb 275 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (∀𝑠(𝑠 = 𝑅𝑁 = 𝑧) ↔ 𝑧 = 𝑅 / 𝑠𝑁))
6963, 68bitrd 267 1 ((((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))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031  wal 1473   = wceq 1475  wcel 1977  wne 2780  wral 2896  csb 3499   class class class wbr 4578  cfv 5790  (class class class)co 6527  Basecbs 15644  lecple 15724  joincjn 16716  meetcmee 16717  0.cp0 16809  OLcol 33273  Atomscatm 33362  HLchlt 33449  LHypclh 34082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-preset 16700  df-poset 16718  df-plt 16730  df-lub 16746  df-glb 16747  df-join 16748  df-meet 16749  df-p0 16811  df-lat 16818  df-oposet 33275  df-ol 33277  df-oml 33278  df-covers 33365  df-ats 33366  df-atl 33397  df-cvlat 33421  df-hlat 33450  df-lhyp 34086
This theorem is referenced by:  cdlemefrs29bpre1  34497  cdlemefrs32fva  34500  cdlemefr29bpre0N  34506  cdlemefs29bpre0N  34516
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