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Theorem cdleme37m 38038
 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 1133 . . 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 1205 . . 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 1295 . . 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 1297 . . 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 1203 . . 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 1296 . . . 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 1298 . . . 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 1294 . . . 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 1125 . . 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 2758 . . . 4 ((𝑆 𝑡) 𝑊) = ((𝑆 𝑡) 𝑊)
19 eqid 2758 . . . 4 ((𝑆 𝑢) 𝑊) = ((𝑆 𝑢) 𝑊)
20 eqid 2758 . . . 4 ((𝑃 𝑄) (𝐸 ((𝑆 𝑡) 𝑊))) = ((𝑃 𝑄) (𝐸 ((𝑆 𝑡) 𝑊)))
21 eqid 2758 . . . 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 37914 . . 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 1385 . 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 cdleme37.c . . 3 𝐶 = ((𝑆 𝑉) (𝐸 ((𝑡 𝑆) 𝑊)))
25 simp11 1200 . . . . . 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))
26 simp12l 1283 . . . . . 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)
27 simp13l 1285 . . . . . 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)
2810, 11, 12, 13, 14, 15cdleme4 37814 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ 𝑆 (𝑃 𝑄)) → (𝑃 𝑄) = (𝑆 𝑈))
2925, 26, 27, 2, 8, 28syl131anc 1380 . . . . 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))
30 cdleme37.v . . . . . . 7 𝑉 = ((𝑡 𝐸) 𝑊)
3110, 11, 12, 13, 14, 15, 16cdleme2 37804 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊))) → ((𝑡 𝐸) 𝑊) = 𝑈)
3225, 26, 27, 3, 31syl13anc 1369 . . . . . . 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)
3330, 32syl5eq 2805 . . . . . 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)
3433oveq2d 7166 . . . . 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))
3529, 34eqtr4d 2796 . . . 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))
36 simp11l 1281 . . . . . . 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)
37 simp23l 1291 . . . . . . 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)
383simpld 498 . . . . . . 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)
3911, 13hlatjcom 36944 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑡𝐴) → (𝑆 𝑡) = (𝑡 𝑆))
4036, 37, 38, 39syl3anc 1368 . . . . . 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))
4140oveq1d 7165 . . . . 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))
4241oveq2d 7166 . . . 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)))
4335, 42oveq12d 7168 . . 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))))
4424, 43eqtr4id 2812 . 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.g . . 3 𝐺 = ((𝑆 𝑋) (𝐷 ((𝑢 𝑆) 𝑊)))
46 cdleme37.x . . . . . . 7 𝑋 = ((𝑢 𝐷) 𝑊)
4710, 11, 12, 13, 14, 15, 17cdleme2 37804 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑢𝐴 ∧ ¬ 𝑢 𝑊))) → ((𝑢 𝐷) 𝑊) = 𝑈)
4825, 26, 27, 4, 47syl13anc 1369 . . . . . . 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)
4946, 48syl5eq 2805 . . . . . 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)
5049oveq2d 7166 . . . . 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))
5129, 50eqtr4d 2796 . . . 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))
524simpld 498 . . . . . . 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)
5311, 13hlatjcom 36944 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑢𝐴) → (𝑆 𝑢) = (𝑢 𝑆))
5436, 37, 52, 53syl3anc 1368 . . . . . 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))
5554oveq1d 7165 . . . . 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))
5655oveq2d 7166 . . . 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)))
5751, 56oveq12d 7168 . . 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))))
5845, 57eqtr4id 2812 . 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 2803 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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951   class class class wbr 5032  ‘cfv 6335  (class class class)co 7150  lecple 16630  joincjn 17620  meetcmee 17621  Atomscatm 36839  HLchlt 36926  LHypclh 37560 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-iin 4886  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694  df-proset 17604  df-poset 17622  df-plt 17634  df-lub 17650  df-glb 17651  df-join 17652  df-meet 17653  df-p0 17715  df-p1 17716  df-lat 17722  df-clat 17784  df-oposet 36752  df-ol 36754  df-oml 36755  df-covers 36842  df-ats 36843  df-atl 36874  df-cvlat 36898  df-hlat 36927  df-llines 37074  df-lplanes 37075  df-lvols 37076  df-lines 37077  df-psubsp 37079  df-pmap 37080  df-padd 37372  df-lhyp 37564 This theorem is referenced by:  cdleme38m  38039
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