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Theorem cdlemefrs29bpre0 36925
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 3087 . . 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 461 . . . . . . 7 (((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) ↔ (𝑠𝐴 ∧ ((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅)))
32imbi1i 342 . . . . . 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 443 . . . . . 6 ((((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ ((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) → ((𝑠 (𝑅 𝑊)) = 𝑅𝑧 = (𝑁 (𝑅 𝑊)))))
5 impexp 443 . . . . . 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 294 . . . . 5 ((𝑠𝐴 → (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))) ↔ ((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) → ((𝑠 (𝑅 𝑊)) = 𝑅𝑧 = (𝑁 (𝑅 𝑊)))))
7 simpl11 1228 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
8 simpl2r 1207 . . . . . . . . . . . 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 2772 . . . . . . . . . . . . 13 (0.‘𝐾) = (0.‘𝐾)
12 cdlemefrs27.a . . . . . . . . . . . . 13 𝐴 = (Atoms‘𝐾)
13 cdlemefrs27.h . . . . . . . . . . . . 13 𝐻 = (LHyp‘𝐾)
149, 10, 11, 12, 13lhpmat 36559 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 𝑊) = (0.‘𝐾))
157, 8, 14syl2anc 576 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝑅 𝑊) = (0.‘𝐾))
1615oveq2d 6986 . . . . . . . . . 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 1264 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝐾 ∈ HL)
18 hlol 35890 . . . . . . . . . . . . 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 473 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → 𝐾 ∈ OL)
21 simprl 758 . . . . . . . . . . . 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 35818 . . . . . . . . . . . 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 35754 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ 𝑠𝐵) → (𝑠 (0.‘𝐾)) = 𝑠)
2720, 24, 26syl2anc 576 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝑠 (0.‘𝐾)) = 𝑠)
2816, 27eqtrd 2808 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝑠 (𝑅 𝑊)) = 𝑠)
2928eqeq1d 2774 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → ((𝑠 (𝑅 𝑊)) = 𝑅𝑠 = 𝑅))
3015oveq2d 6986 . . . . . . . . . 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 1171 . . . . . . . . . . . 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 1206 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → 𝑃𝑄)
33 simprr 760 . . . . . . . . . . . 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 1354 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → 𝑁𝐵)
3622, 25, 11olj01 35754 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ 𝑁𝐵) → (𝑁 (0.‘𝐾)) = 𝑁)
3720, 35, 36syl2anc 576 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝑁 (0.‘𝐾)) = 𝑁)
3830, 37eqtrd 2808 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → (𝑁 (𝑅 𝑊)) = 𝑁)
3938eqeq2d 2782 . . . . . . . 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 337 . . . . . . 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 791 . . . . . 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 443 . . . . . . 7 ((((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) ∧ 𝑠 = 𝑅) → 𝑧 = 𝑁) ↔ ((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) → (𝑠 = 𝑅𝑧 = 𝑁)))
43 eqcom 2779 . . . . . . . . 9 (𝑧 = 𝑁𝑁 = 𝑧)
4443imbi2i 328 . . . . . . . 8 ((𝑠 = 𝑅𝑧 = 𝑁) ↔ (𝑠 = 𝑅𝑁 = 𝑧))
45 simp2rl 1222 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝑅𝐴)
46 simp2rr 1223 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → ¬ 𝑅 𝑊)
47 simp3 1118 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝜓)
48 eleq1 2847 . . . . . . . . . . . . 13 (𝑠 = 𝑅 → (𝑠𝐴𝑅𝐴))
49 breq1 4926 . . . . . . . . . . . . . . 15 (𝑠 = 𝑅 → (𝑠 𝑊𝑅 𝑊))
5049notbid 310 . . . . . . . . . . . . . 14 (𝑠 = 𝑅 → (¬ 𝑠 𝑊 ↔ ¬ 𝑅 𝑊))
51 cdlemefrs27.eq . . . . . . . . . . . . . 14 (𝑠 = 𝑅 → (𝜑𝜓))
5250, 51anbi12d 621 . . . . . . . . . . . . 13 (𝑠 = 𝑅 → ((¬ 𝑠 𝑊𝜑) ↔ (¬ 𝑅 𝑊𝜓)))
5348, 52anbi12d 621 . . . . . . . . . . . 12 (𝑠 = 𝑅 → ((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) ↔ (𝑅𝐴 ∧ (¬ 𝑅 𝑊𝜓))))
5453biimprcd 242 . . . . . . . . . . 11 ((𝑅𝐴 ∧ (¬ 𝑅 𝑊𝜓)) → (𝑠 = 𝑅 → (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))))
5545, 46, 47, 54syl12anc 824 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (𝑠 = 𝑅 → (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))))
5655pm4.71rd 555 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (𝑠 = 𝑅 ↔ ((𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑)) ∧ 𝑠 = 𝑅)))
5756imbi1d 334 . . . . . . . 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 278 . . . . . . 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 277 . . . . . 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 271 . . . . 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 275 . . . 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 1879 . . 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 275 . 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 2926 . . . . 5 𝑠𝑧
6564csbiebg 3807 . . . 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 2779 . . 3 (𝑅 / 𝑠𝑁 = 𝑧𝑧 = 𝑅 / 𝑠𝑁)
6866, 67syl6bb 279 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (∀𝑠(𝑠 = 𝑅𝑁 = 𝑧) ↔ 𝑧 = 𝑅 / 𝑠𝑁))
6963, 68bitrd 271 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 198  wa 387  w3a 1068  wal 1505   = wceq 1507  wcel 2048  wne 2961  wral 3082  csb 3782   class class class wbr 4923  cfv 6182  (class class class)co 6970  Basecbs 16329  lecple 16418  joincjn 17402  meetcmee 17403  0.cp0 17495  OLcol 35703  Atomscatm 35792  HLchlt 35879  LHypclh 36513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-proset 17386  df-poset 17404  df-plt 17416  df-lub 17432  df-glb 17433  df-join 17434  df-meet 17435  df-p0 17497  df-lat 17504  df-oposet 35705  df-ol 35707  df-oml 35708  df-covers 35795  df-ats 35796  df-atl 35827  df-cvlat 35851  df-hlat 35880  df-lhyp 36517
This theorem is referenced by:  cdlemefrs29bpre1  36926  cdlemefrs32fva  36929  cdlemefr29bpre0N  36935  cdlemefs29bpre0N  36945
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