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Theorem cdleme38n 37486
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. TODO shorter if proved directly from cdleme36m 37483 and cdleme37m 37484? (Contributed by NM, 14-Mar-2013.)
Hypotheses
Ref Expression
cdleme38.l = (le‘𝐾)
cdleme38.j = (join‘𝐾)
cdleme38.m = (meet‘𝐾)
cdleme38.a 𝐴 = (Atoms‘𝐾)
cdleme38.h 𝐻 = (LHyp‘𝐾)
cdleme38.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme38.e 𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdleme38.d 𝐷 = ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊)))
cdleme38.v 𝑉 = ((𝑡 𝐸) 𝑊)
cdleme38.x 𝑋 = ((𝑢 𝐷) 𝑊)
cdleme38.f 𝐹 = ((𝑅 𝑉) (𝐸 ((𝑡 𝑅) 𝑊)))
cdleme38.g 𝐺 = ((𝑆 𝑋) (𝐷 ((𝑢 𝑆) 𝑊)))
Assertion
Ref Expression
cdleme38n ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝐹𝐺)

Proof of Theorem cdleme38n
StepHypRef Expression
1 simp313 1316 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝑅𝑆)
2 simpl1 1185 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) ∧ 𝐹 = 𝐺) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
3 simpl21 1245 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) ∧ 𝐹 = 𝐺) → 𝑃𝑄)
4 simpl22 1246 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) ∧ 𝐹 = 𝐺) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
5 simpl23 1247 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) ∧ 𝐹 = 𝐺) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
6 simp311 1314 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝑅 (𝑃 𝑄))
76adantr 481 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) ∧ 𝐹 = 𝐺) → 𝑅 (𝑃 𝑄))
8 simp312 1315 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝑆 (𝑃 𝑄))
98adantr 481 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) ∧ 𝐹 = 𝐺) → 𝑆 (𝑃 𝑄))
10 simpr 485 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) ∧ 𝐹 = 𝐺) → 𝐹 = 𝐺)
117, 9, 103jca 1122 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) ∧ 𝐹 = 𝐺) → (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝐹 = 𝐺))
12 simpl32 1249 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) ∧ 𝐹 = 𝐺) → ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)))
13 simpl33 1250 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) ∧ 𝐹 = 𝐺) → ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))
14 cdleme38.l . . . . . 6 = (le‘𝐾)
15 cdleme38.j . . . . . 6 = (join‘𝐾)
16 cdleme38.m . . . . . 6 = (meet‘𝐾)
17 cdleme38.a . . . . . 6 𝐴 = (Atoms‘𝐾)
18 cdleme38.h . . . . . 6 𝐻 = (LHyp‘𝐾)
19 cdleme38.u . . . . . 6 𝑈 = ((𝑃 𝑄) 𝑊)
20 cdleme38.e . . . . . 6 𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
21 cdleme38.d . . . . . 6 𝐷 = ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊)))
22 cdleme38.v . . . . . 6 𝑉 = ((𝑡 𝐸) 𝑊)
23 cdleme38.x . . . . . 6 𝑋 = ((𝑢 𝐷) 𝑊)
24 cdleme38.f . . . . . 6 𝐹 = ((𝑅 𝑉) (𝐸 ((𝑡 𝑅) 𝑊)))
25 cdleme38.g . . . . . 6 𝐺 = ((𝑆 𝑋) (𝐷 ((𝑢 𝑆) 𝑊)))
2614, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25cdleme38m 37485 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝐹 = 𝐺) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝑅 = 𝑆)
272, 3, 4, 5, 11, 12, 13, 26syl133anc 1387 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) ∧ 𝐹 = 𝐺) → 𝑅 = 𝑆)
2827ex 413 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → (𝐹 = 𝐺𝑅 = 𝑆))
2928necon3d 3042 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → (𝑅𝑆𝐹𝐺))
301, 29mpd 15 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝐹𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021   class class class wbr 5063  cfv 6354  (class class class)co 7150  lecple 16567  joincjn 17549  meetcmee 17550  Atomscatm 36285  HLchlt 36372  LHypclh 37006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686  df-proset 17533  df-poset 17551  df-plt 17563  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-p0 17644  df-p1 17645  df-lat 17651  df-clat 17713  df-oposet 36198  df-ol 36200  df-oml 36201  df-covers 36288  df-ats 36289  df-atl 36320  df-cvlat 36344  df-hlat 36373  df-llines 36520  df-lplanes 36521  df-lvols 36522  df-lines 36523  df-psubsp 36525  df-pmap 36526  df-padd 36818  df-lhyp 37010
This theorem is referenced by:  cdleme39n  37488
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