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Theorem cdleme37m 34768
Description: Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one on 𝑃 𝑄 line. TODO: FIX COMMENT. (Contributed by NM, 13-Mar-2013.)
Hypotheses
Ref Expression
cdleme37.l = (le‘𝐾)
cdleme37.j = (join‘𝐾)
cdleme37.m = (meet‘𝐾)
cdleme37.a 𝐴 = (Atoms‘𝐾)
cdleme37.h 𝐻 = (LHyp‘𝐾)
cdleme37.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme37.e 𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdleme37.d 𝐷 = ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊)))
cdleme37.v 𝑉 = ((𝑡 𝐸) 𝑊)
cdleme37.x 𝑋 = ((𝑢 𝐷) 𝑊)
cdleme37.c 𝐶 = ((𝑆 𝑉) (𝐸 ((𝑡 𝑆) 𝑊)))
cdleme37.g 𝐺 = ((𝑆 𝑋) (𝐷 ((𝑢 𝑆) 𝑊)))
Assertion
Ref Expression
cdleme37m ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝐶 = 𝐺)

Proof of Theorem cdleme37m
StepHypRef Expression
1 simp1 1054 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
2 simp23 1089 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
3 simp32l 1179 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → (𝑡𝐴 ∧ ¬ 𝑡 𝑊))
4 simp33l 1181 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → (𝑢𝐴 ∧ ¬ 𝑢 𝑊))
5 simp21 1087 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝑃𝑄)
6 simp32r 1180 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → ¬ 𝑡 (𝑃 𝑄))
7 simp33r 1182 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → ¬ 𝑢 (𝑃 𝑄))
8 simp31r 1178 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝑆 (𝑃 𝑄))
96, 7, 83jca 1235 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → (¬ 𝑡 (𝑃 𝑄) ∧ ¬ 𝑢 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)))
10 cdleme37.l . . . 4 = (le‘𝐾)
11 cdleme37.j . . . 4 = (join‘𝐾)
12 cdleme37.m . . . 4 = (meet‘𝐾)
13 cdleme37.a . . . 4 𝐴 = (Atoms‘𝐾)
14 cdleme37.h . . . 4 𝐻 = (LHyp‘𝐾)
15 cdleme37.u . . . 4 𝑈 = ((𝑃 𝑄) 𝑊)
16 cdleme37.e . . . 4 𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
17 cdleme37.d . . . 4 𝐷 = ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊)))
18 eqid 2610 . . . 4 ((𝑆 𝑡) 𝑊) = ((𝑆 𝑡) 𝑊)
19 eqid 2610 . . . 4 ((𝑆 𝑢) 𝑊) = ((𝑆 𝑢) 𝑊)
20 eqid 2610 . . . 4 ((𝑃 𝑄) (𝐸 ((𝑆 𝑡) 𝑊))) = ((𝑃 𝑄) (𝐸 ((𝑆 𝑡) 𝑊)))
21 eqid 2610 . . . 4 ((𝑃 𝑄) (𝐷 ((𝑆 𝑢) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑆 𝑢) 𝑊)))
2210, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21cdleme21k 34644 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ (𝑢𝐴 ∧ ¬ 𝑢 𝑊)) ∧ (𝑃𝑄 ∧ (¬ 𝑡 (𝑃 𝑄) ∧ ¬ 𝑢 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)))) → ((𝑃 𝑄) (𝐸 ((𝑆 𝑡) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑆 𝑢) 𝑊))))
231, 2, 3, 4, 5, 9, 22syl132anc 1336 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → ((𝑃 𝑄) (𝐸 ((𝑆 𝑡) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑆 𝑢) 𝑊))))
24 simp11 1084 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
25 simp12l 1167 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝑃𝐴)
26 simp13l 1169 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝑄𝐴)
2710, 11, 12, 13, 14, 15cdleme4 34543 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ 𝑆 (𝑃 𝑄)) → (𝑃 𝑄) = (𝑆 𝑈))
2824, 25, 26, 2, 8, 27syl131anc 1331 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → (𝑃 𝑄) = (𝑆 𝑈))
29 cdleme37.v . . . . . . 7 𝑉 = ((𝑡 𝐸) 𝑊)
3010, 11, 12, 13, 14, 15, 16cdleme2 34533 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊))) → ((𝑡 𝐸) 𝑊) = 𝑈)
3124, 25, 26, 3, 30syl13anc 1320 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → ((𝑡 𝐸) 𝑊) = 𝑈)
3229, 31syl5eq 2656 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝑉 = 𝑈)
3332oveq2d 6565 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → (𝑆 𝑉) = (𝑆 𝑈))
3428, 33eqtr4d 2647 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → (𝑃 𝑄) = (𝑆 𝑉))
35 simp11l 1165 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝐾 ∈ HL)
36 simp23l 1175 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝑆𝐴)
373simpld 474 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝑡𝐴)
3811, 13hlatjcom 33672 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑡𝐴) → (𝑆 𝑡) = (𝑡 𝑆))
3935, 36, 37, 38syl3anc 1318 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → (𝑆 𝑡) = (𝑡 𝑆))
4039oveq1d 6564 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → ((𝑆 𝑡) 𝑊) = ((𝑡 𝑆) 𝑊))
4140oveq2d 6565 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → (𝐸 ((𝑆 𝑡) 𝑊)) = (𝐸 ((𝑡 𝑆) 𝑊)))
4234, 41oveq12d 6567 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → ((𝑃 𝑄) (𝐸 ((𝑆 𝑡) 𝑊))) = ((𝑆 𝑉) (𝐸 ((𝑡 𝑆) 𝑊))))
43 cdleme37.c . . 3 𝐶 = ((𝑆 𝑉) (𝐸 ((𝑡 𝑆) 𝑊)))
4442, 43syl6reqr 2663 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝐶 = ((𝑃 𝑄) (𝐸 ((𝑆 𝑡) 𝑊))))
45 cdleme37.x . . . . . . 7 𝑋 = ((𝑢 𝐷) 𝑊)
4610, 11, 12, 13, 14, 15, 17cdleme2 34533 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑢𝐴 ∧ ¬ 𝑢 𝑊))) → ((𝑢 𝐷) 𝑊) = 𝑈)
4724, 25, 26, 4, 46syl13anc 1320 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → ((𝑢 𝐷) 𝑊) = 𝑈)
4845, 47syl5eq 2656 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝑋 = 𝑈)
4948oveq2d 6565 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → (𝑆 𝑋) = (𝑆 𝑈))
5028, 49eqtr4d 2647 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → (𝑃 𝑄) = (𝑆 𝑋))
514simpld 474 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝑢𝐴)
5211, 13hlatjcom 33672 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑢𝐴) → (𝑆 𝑢) = (𝑢 𝑆))
5335, 36, 51, 52syl3anc 1318 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → (𝑆 𝑢) = (𝑢 𝑆))
5453oveq1d 6564 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → ((𝑆 𝑢) 𝑊) = ((𝑢 𝑆) 𝑊))
5554oveq2d 6565 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → (𝐷 ((𝑆 𝑢) 𝑊)) = (𝐷 ((𝑢 𝑆) 𝑊)))
5650, 55oveq12d 6567 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → ((𝑃 𝑄) (𝐷 ((𝑆 𝑢) 𝑊))) = ((𝑆 𝑋) (𝐷 ((𝑢 𝑆) 𝑊))))
57 cdleme37.g . . 3 𝐺 = ((𝑆 𝑋) (𝐷 ((𝑢 𝑆) 𝑊)))
5856, 57syl6reqr 2663 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝐺 = ((𝑃 𝑄) (𝐷 ((𝑆 𝑢) 𝑊))))
5923, 44, 583eqtr4d 2654 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝐶 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780   class class class wbr 4583  cfv 5804  (class class class)co 6549  lecple 15775  joincjn 16767  meetcmee 16768  Atomscatm 33568  HLchlt 33655  LHypclh 34288
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 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  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 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lvols 33804  df-lines 33805  df-psubsp 33807  df-pmap 33808  df-padd 34100  df-lhyp 34292
This theorem is referenced by:  cdleme38m  34769
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