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Theorem cdlemefrs29bpre0 39731
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 3061 . . 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 468 . . . . . . 7 (((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) ↔ (𝑠𝐴 ∧ ((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅)))
32imbi1i 349 . . . . . 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 450 . . . . . 6 ((((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ ((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) → ((𝑠 (𝑅 𝑊)) = 𝑅𝑧 = (𝑁 (𝑅 𝑊)))))
5 impexp 450 . . . . . 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 302 . . . . 5 ((𝑠𝐴 → (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))) ↔ ((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) → ((𝑠 (𝑅 𝑊)) = 𝑅𝑧 = (𝑁 (𝑅 𝑊)))))
7 simpl11 1247 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
8 simpl2r 1226 . . . . . . . . . . . 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 2731 . . . . . . . . . . . . 13 (0.‘𝐾) = (0.‘𝐾)
12 cdlemefrs27.a . . . . . . . . . . . . 13 𝐴 = (Atoms‘𝐾)
13 cdlemefrs27.h . . . . . . . . . . . . 13 𝐻 = (LHyp‘𝐾)
149, 10, 11, 12, 13lhpmat 39365 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 𝑊) = (0.‘𝐾))
157, 8, 14syl2anc 583 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝑅 𝑊) = (0.‘𝐾))
1615oveq2d 7428 . . . . . . . . . 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 1283 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝐾 ∈ HL)
18 hlol 38695 . . . . . . . . . . . . 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 768 . . . . . . . . . . . 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 38623 . . . . . . . . . . . 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 38559 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ 𝑠𝐵) → (𝑠 (0.‘𝐾)) = 𝑠)
2720, 24, 26syl2anc 583 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝑠 (0.‘𝐾)) = 𝑠)
2816, 27eqtrd 2771 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝑠 (𝑅 𝑊)) = 𝑠)
2928eqeq1d 2733 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → ((𝑠 (𝑅 𝑊)) = 𝑅𝑠 = 𝑅))
3015oveq2d 7428 . . . . . . . . . 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 1190 . . . . . . . . . . . 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 1225 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → 𝑃𝑄)
33 simprr 770 . . . . . . . . . . . 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 1373 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → 𝑁𝐵)
3622, 25, 11olj01 38559 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ 𝑁𝐵) → (𝑁 (0.‘𝐾)) = 𝑁)
3720, 35, 36syl2anc 583 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝑁 (0.‘𝐾)) = 𝑁)
3830, 37eqtrd 2771 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝑁 (𝑅 𝑊)) = 𝑁)
3938eqeq2d 2742 . . . . . . . 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 344 . . . . . . 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 801 . . . . . 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 450 . . . . . . 7 ((((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) ∧ 𝑠 = 𝑅) → 𝑧 = 𝑁) ↔ ((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) → (𝑠 = 𝑅𝑧 = 𝑁)))
43 simp2rl 1241 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝑅𝐴)
44 simp2rr 1242 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → ¬ 𝑅 𝑊)
45 simp3 1137 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝜓)
46 eleq1 2820 . . . . . . . . . . . . 13 (𝑠 = 𝑅 → (𝑠𝐴𝑅𝐴))
47 breq1 5151 . . . . . . . . . . . . . . 15 (𝑠 = 𝑅 → (𝑠 𝑊𝑅 𝑊))
4847notbid 318 . . . . . . . . . . . . . 14 (𝑠 = 𝑅 → (¬ 𝑠 𝑊 ↔ ¬ 𝑅 𝑊))
49 cdlemefrs27.eq . . . . . . . . . . . . . 14 (𝑠 = 𝑅 → (𝜑𝜓))
5048, 49anbi12d 630 . . . . . . . . . . . . 13 (𝑠 = 𝑅 → ((¬ 𝑠 𝑊𝜑) ↔ (¬ 𝑅 𝑊𝜓)))
5146, 50anbi12d 630 . . . . . . . . . . . 12 (𝑠 = 𝑅 → ((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) ↔ (𝑅𝐴 ∧ (¬ 𝑅 𝑊𝜓))))
5251biimprcd 249 . . . . . . . . . . 11 ((𝑅𝐴 ∧ (¬ 𝑅 𝑊𝜓)) → (𝑠 = 𝑅 → (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))))
5343, 44, 45, 52syl12anc 834 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (𝑠 = 𝑅 → (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))))
5453pm4.71rd 562 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (𝑠 = 𝑅 ↔ ((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) ∧ 𝑠 = 𝑅)))
5554imbi1d 341 . . . . . . . 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)))
56 eqcom 2738 . . . . . . . . 9 (𝑧 = 𝑁𝑁 = 𝑧)
5756imbi2i 336 . . . . . . . 8 ((𝑠 = 𝑅𝑧 = 𝑁) ↔ (𝑠 = 𝑅𝑁 = 𝑧))
5855, 57bitr3di 286 . . . . . . 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, 58bitr3id 285 . . . . . 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 279 . . . . 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, 60bitrid 283 . . . 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 1922 . . 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, 62bitrid 283 . 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 2902 . . . . 5 𝑠𝑧
6564csbiebg 3926 . . . 4 (𝑅𝐴 → (∀𝑠(𝑠 = 𝑅𝑁 = 𝑧) ↔ 𝑅 / 𝑠𝑁 = 𝑧))
6643, 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 2738 . . 3 (𝑅 / 𝑠𝑁 = 𝑧𝑧 = 𝑅 / 𝑠𝑁)
6866, 67bitrdi 287 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (∀𝑠(𝑠 = 𝑅𝑁 = 𝑧) ↔ 𝑧 = 𝑅 / 𝑠𝑁))
6963, 68bitrd 279 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 205  wa 395  w3a 1086  wal 1538   = wceq 1540  wcel 2105  wne 2939  wral 3060  csb 3893   class class class wbr 5148  cfv 6543  (class class class)co 7412  Basecbs 17151  lecple 17211  joincjn 18274  meetcmee 18275  0.cp0 18386  OLcol 38508  Atomscatm 38597  HLchlt 38684  LHypclh 39319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-proset 18258  df-poset 18276  df-plt 18293  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-p0 18388  df-lat 18395  df-oposet 38510  df-ol 38512  df-oml 38513  df-covers 38600  df-ats 38601  df-atl 38632  df-cvlat 38656  df-hlat 38685  df-lhyp 39323
This theorem is referenced by:  cdlemefrs29bpre1  39732  cdlemefrs32fva  39735  cdlemefr29bpre0N  39741  cdlemefs29bpre0N  39751
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