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Theorem cdlemg17h 35436
Description: TODO: fix comment. (Contributed by NM, 10-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l = (le‘𝐾)
cdlemg12.j = (join‘𝐾)
cdlemg12.m = (meet‘𝐾)
cdlemg12.a 𝐴 = (Atoms‘𝐾)
cdlemg12.h 𝐻 = (LHyp‘𝐾)
cdlemg12.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg12b.r 𝑅 = ((trL‘𝐾)‘𝑊)
Assertion
Ref Expression
cdlemg17h ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑆 = (𝐹𝑃) ∨ 𝑆 = (𝐹𝑄)))
Distinct variable groups:   𝐴,𝑟   𝐺,𝑟   ,𝑟   ,𝑟   𝑃,𝑟   𝑄,𝑟   𝑊,𝑟   𝐹,𝑟   𝑆,𝑟
Allowed substitution hints:   𝑅(𝑟)   𝑇(𝑟)   𝐻(𝑟)   𝐾(𝑟)   (𝑟)

Proof of Theorem cdlemg17h
StepHypRef Expression
1 simp11l 1170 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐾 ∈ HL)
2 simp23r 1181 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑆 ((𝐹𝑃) (𝐹𝑄)))
3 simp11 1089 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
4 simp22l 1178 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐹𝑇)
5 simp21l 1176 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑆𝐴)
6 cdlemg12.l . . . . . . . . 9 = (le‘𝐾)
7 cdlemg12.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
8 cdlemg12.h . . . . . . . . 9 𝐻 = (LHyp‘𝐾)
9 cdlemg12.t . . . . . . . . 9 𝑇 = ((LTrn‘𝐾)‘𝑊)
106, 7, 8, 9ltrncnvat 34907 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑆𝐴) → (𝐹𝑆) ∈ 𝐴)
113, 4, 5, 10syl3anc 1323 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐹𝑆) ∈ 𝐴)
12 eqid 2621 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
1312, 7atbase 34056 . . . . . . 7 ((𝐹𝑆) ∈ 𝐴 → (𝐹𝑆) ∈ (Base‘𝐾))
1411, 13syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐹𝑆) ∈ (Base‘𝐾))
15 simp12l 1172 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑃𝐴)
16 simp13l 1174 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑄𝐴)
17 cdlemg12.j . . . . . . . 8 = (join‘𝐾)
1812, 17, 7hlatjcl 34133 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
191, 15, 16, 18syl3anc 1323 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑃 𝑄) ∈ (Base‘𝐾))
2012, 6, 8, 9ltrnle 34895 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝐹𝑆) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((𝐹𝑆) (𝑃 𝑄) ↔ (𝐹‘(𝐹𝑆)) (𝐹‘(𝑃 𝑄))))
213, 4, 14, 19, 20syl112anc 1327 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝐹𝑆) (𝑃 𝑄) ↔ (𝐹‘(𝐹𝑆)) (𝐹‘(𝑃 𝑄))))
2212, 8, 9ltrn1o 34890 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
233, 4, 22syl2anc 692 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
2412, 7atbase 34056 . . . . . . . 8 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
255, 24syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑆 ∈ (Base‘𝐾))
26 f1ocnvfv2 6487 . . . . . . 7 ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝐹‘(𝐹𝑆)) = 𝑆)
2723, 25, 26syl2anc 692 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐹‘(𝐹𝑆)) = 𝑆)
2812, 7atbase 34056 . . . . . . . 8 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
2915, 28syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑃 ∈ (Base‘𝐾))
3012, 7atbase 34056 . . . . . . . 8 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
3116, 30syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑄 ∈ (Base‘𝐾))
3212, 17, 8, 9ltrnj 34898 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾))) → (𝐹‘(𝑃 𝑄)) = ((𝐹𝑃) (𝐹𝑄)))
333, 4, 29, 31, 32syl112anc 1327 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐹‘(𝑃 𝑄)) = ((𝐹𝑃) (𝐹𝑄)))
3427, 33breq12d 4626 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝐹‘(𝐹𝑆)) (𝐹‘(𝑃 𝑄)) ↔ 𝑆 ((𝐹𝑃) (𝐹𝑄))))
3521, 34bitr2d 269 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑆 ((𝐹𝑃) (𝐹𝑄)) ↔ (𝐹𝑆) (𝑃 𝑄)))
362, 35mpbid 222 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐹𝑆) (𝑃 𝑄))
37 simp33 1097 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
38 simp23l 1180 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑃𝑄)
39 simp21 1092 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
406, 7, 8, 9ltrncnvel 34908 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) → ((𝐹𝑆) ∈ 𝐴 ∧ ¬ (𝐹𝑆) 𝑊))
413, 4, 39, 40syl3anc 1323 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝐹𝑆) ∈ 𝐴 ∧ ¬ (𝐹𝑆) 𝑊))
426, 17, 7cdleme0nex 35057 . . 3 (((𝐾 ∈ HL ∧ (𝐹𝑆) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ ((𝐹𝑆) ∈ 𝐴 ∧ ¬ (𝐹𝑆) 𝑊)) → ((𝐹𝑆) = 𝑃 ∨ (𝐹𝑆) = 𝑄))
431, 36, 37, 15, 16, 38, 41, 42syl331anc 1348 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝐹𝑆) = 𝑃 ∨ (𝐹𝑆) = 𝑄))
44 eqcom 2628 . . . 4 ((𝐹𝑃) = 𝑆𝑆 = (𝐹𝑃))
45 f1ocnvfvb 6489 . . . . 5 ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((𝐹𝑃) = 𝑆 ↔ (𝐹𝑆) = 𝑃))
4623, 29, 25, 45syl3anc 1323 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝐹𝑃) = 𝑆 ↔ (𝐹𝑆) = 𝑃))
4744, 46syl5rbbr 275 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝐹𝑆) = 𝑃𝑆 = (𝐹𝑃)))
48 eqcom 2628 . . . 4 ((𝐹𝑄) = 𝑆𝑆 = (𝐹𝑄))
49 f1ocnvfvb 6489 . . . . 5 ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((𝐹𝑄) = 𝑆 ↔ (𝐹𝑆) = 𝑄))
5023, 31, 25, 49syl3anc 1323 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝐹𝑄) = 𝑆 ↔ (𝐹𝑆) = 𝑄))
5148, 50syl5rbbr 275 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝐹𝑆) = 𝑄𝑆 = (𝐹𝑄)))
5247, 51orbi12d 745 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (((𝐹𝑆) = 𝑃 ∨ (𝐹𝑆) = 𝑄) ↔ (𝑆 = (𝐹𝑃) ∨ 𝑆 = (𝐹𝑄))))
5343, 52mpbid 222 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑆 = (𝐹𝑃) ∨ 𝑆 = (𝐹𝑄)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2908   class class class wbr 4613  ccnv 5073  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  joincjn 16865  meetcmee 16866  Atomscatm 34030  HLchlt 34117  LHypclh 34750  LTrncltrn 34867  trLctrl 34925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-preset 16849  df-poset 16867  df-plt 16879  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-p0 16960  df-lat 16967  df-oposet 33943  df-ol 33945  df-oml 33946  df-covers 34033  df-ats 34034  df-atl 34065  df-cvlat 34089  df-hlat 34118  df-lhyp 34754  df-laut 34755  df-ldil 34870  df-ltrn 34871
This theorem is referenced by:  cdlemg17i  35437
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