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Theorem tgbtwnconn1lem3 25690
Description: Lemma for tgbtwnconn1 25691. (Contributed by Thierry Arnoux, 30-Apr-2019.)
Hypotheses
Ref Expression
tgbtwnconn1.p 𝑃 = (Base‘𝐺)
tgbtwnconn1.i 𝐼 = (Itv‘𝐺)
tgbtwnconn1.g (𝜑𝐺 ∈ TarskiG)
tgbtwnconn1.a (𝜑𝐴𝑃)
tgbtwnconn1.b (𝜑𝐵𝑃)
tgbtwnconn1.c (𝜑𝐶𝑃)
tgbtwnconn1.d (𝜑𝐷𝑃)
tgbtwnconn1.1 (𝜑𝐴𝐵)
tgbtwnconn1.2 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
tgbtwnconn1.3 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
tgbtwnconn1.m = (dist‘𝐺)
tgbtwnconn1.e (𝜑𝐸𝑃)
tgbtwnconn1.f (𝜑𝐹𝑃)
tgbtwnconn1.h (𝜑𝐻𝑃)
tgbtwnconn1.j (𝜑𝐽𝑃)
tgbtwnconn1.4 (𝜑𝐷 ∈ (𝐴𝐼𝐸))
tgbtwnconn1.5 (𝜑𝐶 ∈ (𝐴𝐼𝐹))
tgbtwnconn1.6 (𝜑𝐸 ∈ (𝐴𝐼𝐻))
tgbtwnconn1.7 (𝜑𝐹 ∈ (𝐴𝐼𝐽))
tgbtwnconn1.8 (𝜑 → (𝐸 𝐷) = (𝐶 𝐷))
tgbtwnconn1.9 (𝜑 → (𝐶 𝐹) = (𝐶 𝐷))
tgbtwnconn1.10 (𝜑 → (𝐸 𝐻) = (𝐵 𝐶))
tgbtwnconn1.11 (𝜑 → (𝐹 𝐽) = (𝐵 𝐷))
tgbtwnconn1.x (𝜑𝑋𝑃)
tgbtwnconn1.12 (𝜑𝑋 ∈ (𝐶𝐼𝐸))
tgbtwnconn1.13 (𝜑𝑋 ∈ (𝐷𝐼𝐹))
tgbtwnconn1.14 (𝜑𝐶𝐸)
Assertion
Ref Expression
tgbtwnconn1lem3 (𝜑𝐷 = 𝐹)

Proof of Theorem tgbtwnconn1lem3
Dummy variables 𝑞 𝑝 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tgbtwnconn1.p . . . . . 6 𝑃 = (Base‘𝐺)
2 tgbtwnconn1.m . . . . . 6 = (dist‘𝐺)
3 tgbtwnconn1.i . . . . . 6 𝐼 = (Itv‘𝐺)
4 tgbtwnconn1.g . . . . . . 7 (𝜑𝐺 ∈ TarskiG)
54ad6antr 720 . . . . . 6 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐺 ∈ TarskiG)
6 tgbtwnconn1.f . . . . . . 7 (𝜑𝐹𝑃)
76ad6antr 720 . . . . . 6 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐹𝑃)
8 tgbtwnconn1.d . . . . . . 7 (𝜑𝐷𝑃)
98ad6antr 720 . . . . . 6 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐷𝑃)
10 simplr 752 . . . . . 6 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝑞𝑃)
115adantr 466 . . . . . . . . . 10 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → 𝐺 ∈ TarskiG)
129adantr 466 . . . . . . . . . 10 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → 𝐷𝑃)
13 simpllr 760 . . . . . . . . . 10 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → 𝑞𝑃)
141, 2, 3, 11, 12, 13tgcgrtriv 25600 . . . . . . . . 9 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → (𝐷 𝐷) = (𝑞 𝑞))
15 simpr 471 . . . . . . . . . . 11 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → 𝐹 = 𝑋)
16 tgbtwnconn1.x . . . . . . . . . . . . . 14 (𝜑𝑋𝑃)
1716ad6antr 720 . . . . . . . . . . . . 13 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝑋𝑃)
1817adantr 466 . . . . . . . . . . . 12 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → 𝑋𝑃)
19 tgbtwnconn1.c . . . . . . . . . . . . . . . 16 (𝜑𝐶𝑃)
20 tgbtwnconn1.e . . . . . . . . . . . . . . . 16 (𝜑𝐸𝑃)
21 tgbtwnconn1.12 . . . . . . . . . . . . . . . 16 (𝜑𝑋 ∈ (𝐶𝐼𝐸))
22 eqidd 2772 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐶 𝐸) = (𝐶 𝐸))
23 eqidd 2772 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑋 𝐸) = (𝑋 𝐸))
24 tgbtwnconn1.9 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐶 𝐹) = (𝐶 𝐷))
2524eqcomd 2777 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐶 𝐷) = (𝐶 𝐹))
26 tgbtwnconn1.8 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐸 𝐷) = (𝐶 𝐷))
27 tgbtwnconn1.a . . . . . . . . . . . . . . . . . 18 (𝜑𝐴𝑃)
28 tgbtwnconn1.b . . . . . . . . . . . . . . . . . 18 (𝜑𝐵𝑃)
29 tgbtwnconn1.1 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴𝐵)
30 tgbtwnconn1.2 . . . . . . . . . . . . . . . . . 18 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
31 tgbtwnconn1.3 . . . . . . . . . . . . . . . . . 18 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
32 tgbtwnconn1.h . . . . . . . . . . . . . . . . . 18 (𝜑𝐻𝑃)
33 tgbtwnconn1.j . . . . . . . . . . . . . . . . . 18 (𝜑𝐽𝑃)
34 tgbtwnconn1.4 . . . . . . . . . . . . . . . . . 18 (𝜑𝐷 ∈ (𝐴𝐼𝐸))
35 tgbtwnconn1.5 . . . . . . . . . . . . . . . . . 18 (𝜑𝐶 ∈ (𝐴𝐼𝐹))
36 tgbtwnconn1.6 . . . . . . . . . . . . . . . . . 18 (𝜑𝐸 ∈ (𝐴𝐼𝐻))
37 tgbtwnconn1.7 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 ∈ (𝐴𝐼𝐽))
38 tgbtwnconn1.10 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐸 𝐻) = (𝐵 𝐶))
39 tgbtwnconn1.11 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐹 𝐽) = (𝐵 𝐷))
401, 3, 4, 27, 28, 19, 8, 29, 30, 31, 2, 20, 6, 32, 33, 34, 35, 36, 37, 26, 24, 38, 39tgbtwnconn1lem2 25689 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐸 𝐹) = (𝐶 𝐷))
4126, 40eqtr4d 2808 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐸 𝐷) = (𝐸 𝐹))
421, 2, 3, 4, 19, 16, 20, 8, 19, 16, 20, 6, 21, 21, 22, 23, 25, 41tgifscgr 25624 . . . . . . . . . . . . . . 15 (𝜑 → (𝑋 𝐷) = (𝑋 𝐹))
4342ad6antr 720 . . . . . . . . . . . . . 14 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝑋 𝐷) = (𝑋 𝐹))
4443adantr 466 . . . . . . . . . . . . 13 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → (𝑋 𝐷) = (𝑋 𝐹))
4515oveq2d 6809 . . . . . . . . . . . . 13 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → (𝑋 𝐹) = (𝑋 𝑋))
4644, 45eqtrd 2805 . . . . . . . . . . . 12 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → (𝑋 𝐷) = (𝑋 𝑋))
471, 2, 3, 11, 18, 12, 18, 46axtgcgrid 25583 . . . . . . . . . . 11 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → 𝑋 = 𝐷)
4815, 47eqtrd 2805 . . . . . . . . . 10 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → 𝐹 = 𝐷)
4948oveq1d 6808 . . . . . . . . 9 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → (𝐹 𝐷) = (𝐷 𝐷))
507adantr 466 . . . . . . . . . . . 12 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → 𝐹𝑃)
51 simp-4r 770 . . . . . . . . . . . . . 14 (((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) → 𝑝𝑃)
5251ad2antrr 705 . . . . . . . . . . . . 13 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝑝𝑃)
5352adantr 466 . . . . . . . . . . . 12 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → 𝑝𝑃)
54 simp-4r 770 . . . . . . . . . . . . 13 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝑟𝑃)
5554adantr 466 . . . . . . . . . . . 12 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → 𝑟𝑃)
5619ad6antr 720 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐶𝑃)
57 simpr 471 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝐶 = 𝐹) → 𝐶 = 𝐹)
584adantr 466 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝐶 = 𝐹) → 𝐺 ∈ TarskiG)
5919adantr 466 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝐶 = 𝐹) → 𝐶𝑃)
606adantr 466 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝐶 = 𝐹) → 𝐹𝑃)
6120adantr 466 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝐶 = 𝐹) → 𝐸𝑃)
6224, 40eqtr4d 2808 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝐶 𝐹) = (𝐸 𝐹))
6362adantr 466 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝐶 = 𝐹) → (𝐶 𝐹) = (𝐸 𝐹))
641, 2, 3, 58, 59, 60, 61, 60, 63, 57tgcgreq 25598 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝐶 = 𝐹) → 𝐸 = 𝐹)
6557, 64eqtr4d 2808 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝐶 = 𝐹) → 𝐶 = 𝐸)
66 tgbtwnconn1.14 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐶𝐸)
6766adantr 466 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝐶 = 𝐹) → 𝐶𝐸)
6867neneqd 2948 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝐶 = 𝐹) → ¬ 𝐶 = 𝐸)
6965, 68pm2.65da 818 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ¬ 𝐶 = 𝐹)
7069neqned 2950 . . . . . . . . . . . . . . . . . 18 (𝜑𝐶𝐹)
7170necomd 2998 . . . . . . . . . . . . . . . . 17 (𝜑𝐹𝐶)
7271ad6antr 720 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐹𝐶)
73 simpllr 760 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋)))
7473simpld 482 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐶 ∈ (𝐹𝐼𝑟))
7520ad6antr 720 . . . . . . . . . . . . . . . . . 18 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐸𝑃)
761, 2, 3, 4, 19, 16, 20, 21tgbtwncom 25604 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑋 ∈ (𝐸𝐼𝐶))
7776ad6antr 720 . . . . . . . . . . . . . . . . . 18 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝑋 ∈ (𝐸𝐼𝐶))
78 simp-5r 774 . . . . . . . . . . . . . . . . . . 19 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹)))
7978simpld 482 . . . . . . . . . . . . . . . . . 18 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐶 ∈ (𝐸𝐼𝑝))
801, 2, 3, 5, 75, 17, 56, 52, 77, 79tgbtwnexch3 25610 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐶 ∈ (𝑋𝐼𝑝))
811, 2, 3, 5, 17, 56, 52, 80tgbtwncom 25604 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐶 ∈ (𝑝𝐼𝑋))
8278simprd 483 . . . . . . . . . . . . . . . . . 18 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐶 𝑝) = (𝐶 𝐹))
8382eqcomd 2777 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐶 𝐹) = (𝐶 𝑝))
841, 2, 3, 5, 56, 7, 56, 52, 83tgcgrcomlr 25596 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐹 𝐶) = (𝑝 𝐶))
8573simprd 483 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐶 𝑟) = (𝐶 𝑋))
861, 2, 3, 5, 7, 52axtgcgrrflx 25582 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐹 𝑝) = (𝑝 𝐹))
871, 2, 3, 5, 7, 56, 54, 52, 56, 17, 52, 7, 72, 74, 81, 84, 85, 86, 82axtg5seg 25585 . . . . . . . . . . . . . . 15 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝑟 𝑝) = (𝑋 𝐹))
8887eqcomd 2777 . . . . . . . . . . . . . 14 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝑋 𝐹) = (𝑟 𝑝))
891, 2, 3, 5, 17, 7, 54, 52, 88tgcgrcomlr 25596 . . . . . . . . . . . . 13 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐹 𝑋) = (𝑝 𝑟))
9089adantr 466 . . . . . . . . . . . 12 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → (𝐹 𝑋) = (𝑝 𝑟))
911, 2, 3, 11, 50, 18, 53, 55, 90, 15tgcgreq 25598 . . . . . . . . . . 11 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → 𝑝 = 𝑟)
92 simprr 756 . . . . . . . . . . . . . 14 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝑟 𝑞) = (𝑟 𝑝))
9392adantr 466 . . . . . . . . . . . . 13 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → (𝑟 𝑞) = (𝑟 𝑝))
9491oveq2d 6809 . . . . . . . . . . . . 13 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → (𝑟 𝑝) = (𝑟 𝑟))
9593, 94eqtrd 2805 . . . . . . . . . . . 12 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → (𝑟 𝑞) = (𝑟 𝑟))
961, 2, 3, 11, 55, 13, 55, 95axtgcgrid 25583 . . . . . . . . . . 11 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → 𝑟 = 𝑞)
9791, 96eqtrd 2805 . . . . . . . . . 10 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → 𝑝 = 𝑞)
9897oveq1d 6808 . . . . . . . . 9 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → (𝑝 𝑞) = (𝑞 𝑞))
9914, 49, 983eqtr4d 2815 . . . . . . . 8 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → (𝐹 𝐷) = (𝑝 𝑞))
1005adantr 466 . . . . . . . . 9 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹𝑋) → 𝐺 ∈ TarskiG)
1017adantr 466 . . . . . . . . 9 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹𝑋) → 𝐹𝑃)
10217adantr 466 . . . . . . . . 9 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹𝑋) → 𝑋𝑃)
1039adantr 466 . . . . . . . . 9 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹𝑋) → 𝐷𝑃)
10452adantr 466 . . . . . . . . 9 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹𝑋) → 𝑝𝑃)
10554adantr 466 . . . . . . . . 9 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹𝑋) → 𝑟𝑃)
106 simpllr 760 . . . . . . . . 9 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹𝑋) → 𝑞𝑃)
107 tgbtwnconn1.13 . . . . . . . . . . 11 (𝜑𝑋 ∈ (𝐷𝐼𝐹))
1081, 2, 3, 4, 8, 16, 6, 107tgbtwncom 25604 . . . . . . . . . 10 (𝜑𝑋 ∈ (𝐹𝐼𝐷))
109108ad7antr 724 . . . . . . . . 9 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹𝑋) → 𝑋 ∈ (𝐹𝐼𝐷))
110 simplrl 762 . . . . . . . . 9 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹𝑋) → 𝑟 ∈ (𝑝𝐼𝑞))
11189adantr 466 . . . . . . . . 9 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹𝑋) → (𝐹 𝑋) = (𝑝 𝑟))
11287, 92, 433eqtr4rd 2816 . . . . . . . . . 10 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝑋 𝐷) = (𝑟 𝑞))
113112adantr 466 . . . . . . . . 9 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹𝑋) → (𝑋 𝐷) = (𝑟 𝑞))
1141, 2, 3, 100, 101, 102, 103, 104, 105, 106, 109, 110, 111, 113tgcgrextend 25601 . . . . . . . 8 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹𝑋) → (𝐹 𝐷) = (𝑝 𝑞))
11599, 114pm2.61dane 3030 . . . . . . 7 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐹 𝐷) = (𝑝 𝑞))
116 eqid 2771 . . . . . . . . 9 (LineG‘𝐺) = (LineG‘𝐺)
117 eqid 2771 . . . . . . . . 9 (cgrG‘𝐺) = (cgrG‘𝐺)
11866ad6antr 720 . . . . . . . . 9 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐶𝐸)
1191, 116, 3, 5, 56, 52, 75, 79btwncolg2 25672 . . . . . . . . 9 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐸 ∈ (𝐶(LineG‘𝐺)𝑝) ∨ 𝐶 = 𝑝))
12024ad6antr 720 . . . . . . . . . 10 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐶 𝐹) = (𝐶 𝐷))
12184adantr 466 . . . . . . . . . . . . 13 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → (𝐹 𝐶) = (𝑝 𝐶))
12248oveq1d 6808 . . . . . . . . . . . . 13 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → (𝐹 𝐶) = (𝐷 𝐶))
12397oveq1d 6808 . . . . . . . . . . . . 13 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → (𝑝 𝐶) = (𝑞 𝐶))
124121, 122, 1233eqtr3d 2813 . . . . . . . . . . . 12 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹 = 𝑋) → (𝐷 𝐶) = (𝑞 𝐶))
12556adantr 466 . . . . . . . . . . . . 13 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹𝑋) → 𝐶𝑃)
126 simpr 471 . . . . . . . . . . . . 13 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹𝑋) → 𝐹𝑋)
12784adantr 466 . . . . . . . . . . . . 13 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹𝑋) → (𝐹 𝐶) = (𝑝 𝐶))
12885eqcomd 2777 . . . . . . . . . . . . . . 15 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐶 𝑋) = (𝐶 𝑟))
1291, 2, 3, 5, 56, 17, 56, 54, 128tgcgrcomlr 25596 . . . . . . . . . . . . . 14 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝑋 𝐶) = (𝑟 𝐶))
130129adantr 466 . . . . . . . . . . . . 13 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹𝑋) → (𝑋 𝐶) = (𝑟 𝐶))
1311, 2, 3, 100, 101, 102, 103, 104, 105, 106, 125, 125, 126, 109, 110, 111, 113, 127, 130axtg5seg 25585 . . . . . . . . . . . 12 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐹𝑋) → (𝐷 𝐶) = (𝑞 𝐶))
132124, 131pm2.61dane 3030 . . . . . . . . . . 11 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐷 𝐶) = (𝑞 𝐶))
1331, 2, 3, 5, 9, 56, 10, 56, 132tgcgrcomlr 25596 . . . . . . . . . 10 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐶 𝐷) = (𝐶 𝑞))
13482, 120, 1333eqtrd 2809 . . . . . . . . 9 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐶 𝑝) = (𝐶 𝑞))
13528ad6antr 720 . . . . . . . . . 10 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐵𝑃)
13633ad6antr 720 . . . . . . . . . 10 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐽𝑃)
1375adantr 466 . . . . . . . . . . . . . 14 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐵 = 𝐽) → 𝐺 ∈ TarskiG)
138136adantr 466 . . . . . . . . . . . . . 14 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐵 = 𝐽) → 𝐽𝑃)
13956adantr 466 . . . . . . . . . . . . . 14 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐵 = 𝐽) → 𝐶𝑃)
1401, 2, 3, 4, 27, 19, 6, 33, 35, 37tgbtwnexch 25614 . . . . . . . . . . . . . . . . 17 (𝜑𝐶 ∈ (𝐴𝐼𝐽))
1411, 2, 3, 4, 27, 28, 19, 33, 30, 140tgbtwnexch3 25610 . . . . . . . . . . . . . . . 16 (𝜑𝐶 ∈ (𝐵𝐼𝐽))
142141ad7antr 724 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐵 = 𝐽) → 𝐶 ∈ (𝐵𝐼𝐽))
143 simpr 471 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐵 = 𝐽) → 𝐵 = 𝐽)
144143oveq1d 6808 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐵 = 𝐽) → (𝐵𝐼𝐽) = (𝐽𝐼𝐽))
145142, 144eleqtrd 2852 . . . . . . . . . . . . . 14 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐵 = 𝐽) → 𝐶 ∈ (𝐽𝐼𝐽))
1461, 2, 3, 137, 138, 139, 145axtgbtwnid 25586 . . . . . . . . . . . . 13 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐵 = 𝐽) → 𝐽 = 𝐶)
1477adantr 466 . . . . . . . . . . . . . 14 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐵 = 𝐽) → 𝐹𝑃)
1481, 2, 3, 4, 27, 19, 6, 33, 35, 37tgbtwnexch3 25610 . . . . . . . . . . . . . . . . 17 (𝜑𝐹 ∈ (𝐶𝐼𝐽))
1491, 2, 3, 4, 28, 19, 6, 33, 141, 148tgbtwnexch2 25612 . . . . . . . . . . . . . . . 16 (𝜑𝐹 ∈ (𝐵𝐼𝐽))
150149ad7antr 724 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐵 = 𝐽) → 𝐹 ∈ (𝐵𝐼𝐽))
151150, 144eleqtrd 2852 . . . . . . . . . . . . . 14 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐵 = 𝐽) → 𝐹 ∈ (𝐽𝐼𝐽))
1521, 2, 3, 137, 138, 147, 151axtgbtwnid 25586 . . . . . . . . . . . . 13 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐵 = 𝐽) → 𝐽 = 𝐹)
153146, 152eqtr3d 2807 . . . . . . . . . . . 12 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐵 = 𝐽) → 𝐶 = 𝐹)
15469ad7antr 724 . . . . . . . . . . . 12 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐵 = 𝐽) → ¬ 𝐶 = 𝐹)
155153, 154pm2.65da 818 . . . . . . . . . . 11 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → ¬ 𝐵 = 𝐽)
156155neqned 2950 . . . . . . . . . 10 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐵𝐽)
1571, 2, 3, 4, 27, 28, 8, 20, 31, 34tgbtwnexch 25614 . . . . . . . . . . . . 13 (𝜑𝐵 ∈ (𝐴𝐼𝐸))
1581, 3, 4, 27, 28, 19, 8, 29, 30, 31, 2, 20, 6, 32, 33, 34, 35, 36, 37, 26, 24, 38, 39tgbtwnconn1lem1 25688 . . . . . . . . . . . . . . 15 (𝜑𝐻 = 𝐽)
159158oveq2d 6809 . . . . . . . . . . . . . 14 (𝜑 → (𝐴𝐼𝐻) = (𝐴𝐼𝐽))
16036, 159eleqtrd 2852 . . . . . . . . . . . . 13 (𝜑𝐸 ∈ (𝐴𝐼𝐽))
1611, 2, 3, 4, 27, 28, 20, 33, 157, 160tgbtwnexch3 25610 . . . . . . . . . . . 12 (𝜑𝐸 ∈ (𝐵𝐼𝐽))
162161ad6antr 720 . . . . . . . . . . 11 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐸 ∈ (𝐵𝐼𝐽))
1631, 116, 3, 5, 135, 75, 136, 162btwncolg3 25673 . . . . . . . . . 10 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐽 ∈ (𝐵(LineG‘𝐺)𝐸) ∨ 𝐵 = 𝐸))
16470ad6antr 720 . . . . . . . . . . 11 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐶𝐹)
1651, 2, 3, 4, 6, 19, 28, 33, 148, 141tgbtwnintr 25609 . . . . . . . . . . . . 13 (𝜑𝐶 ∈ (𝐹𝐼𝐵))
166165ad6antr 720 . . . . . . . . . . . 12 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐶 ∈ (𝐹𝐼𝐵))
1671, 116, 3, 5, 56, 135, 7, 166btwncolg2 25672 . . . . . . . . . . 11 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐹 ∈ (𝐶(LineG‘𝐺)𝐵) ∨ 𝐶 = 𝐵))
1685adantr 466 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐶 = 𝑟) → 𝐺 ∈ TarskiG)
16956adantr 466 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐶 = 𝑟) → 𝐶𝑃)
17054adantr 466 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐶 = 𝑟) → 𝑟𝑃)
17117adantr 466 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐶 = 𝑟) → 𝑋𝑃)
17285adantr 466 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐶 = 𝑟) → (𝐶 𝑟) = (𝐶 𝑋))
173 simpr 471 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐶 = 𝑟) → 𝐶 = 𝑟)
1741, 2, 3, 168, 169, 170, 169, 171, 172, 173tgcgreq 25598 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐶 = 𝑟) → 𝐶 = 𝑋)
17575adantr 466 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐶 = 𝑟) → 𝐸𝑃)
176 eqidd 2772 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐷 𝐹) = (𝐷 𝐹))
177 eqidd 2772 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑋 𝐹) = (𝑋 𝐹))
17826eqcomd 2777 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐶 𝐷) = (𝐸 𝐷))
1791, 2, 3, 4, 19, 8, 20, 8, 178tgcgrcomlr 25596 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐷 𝐶) = (𝐷 𝐸))
1801, 2, 3, 4, 19, 6, 20, 6, 62tgcgrcomlr 25596 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐹 𝐶) = (𝐹 𝐸))
1811, 2, 3, 4, 8, 16, 6, 19, 8, 16, 6, 20, 107, 107, 176, 177, 179, 180tgifscgr 25624 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑋 𝐶) = (𝑋 𝐸))
182181ad7antr 724 . . . . . . . . . . . . . . . . 17 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐶 = 𝑟) → (𝑋 𝐶) = (𝑋 𝐸))
183174oveq2d 6809 . . . . . . . . . . . . . . . . 17 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐶 = 𝑟) → (𝑋 𝐶) = (𝑋 𝑋))
184182, 183eqtr3d 2807 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐶 = 𝑟) → (𝑋 𝐸) = (𝑋 𝑋))
1851, 2, 3, 168, 171, 175, 171, 184axtgcgrid 25583 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐶 = 𝑟) → 𝑋 = 𝐸)
186174, 185eqtrd 2805 . . . . . . . . . . . . . 14 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐶 = 𝑟) → 𝐶 = 𝐸)
18766neneqd 2948 . . . . . . . . . . . . . . 15 (𝜑 → ¬ 𝐶 = 𝐸)
188187ad7antr 724 . . . . . . . . . . . . . 14 ((((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) ∧ 𝐶 = 𝑟) → ¬ 𝐶 = 𝐸)
189186, 188pm2.65da 818 . . . . . . . . . . . . 13 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → ¬ 𝐶 = 𝑟)
190189neqned 2950 . . . . . . . . . . . 12 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐶𝑟)
1911, 2, 3, 5, 7, 56, 54, 74tgbtwncom 25604 . . . . . . . . . . . . 13 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐶 ∈ (𝑟𝐼𝐹))
1921, 116, 3, 5, 56, 7, 54, 191btwncolg2 25672 . . . . . . . . . . . 12 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝑟 ∈ (𝐶(LineG‘𝐺)𝐹) ∨ 𝐶 = 𝐹))
19392eqcomd 2777 . . . . . . . . . . . 12 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝑟 𝑝) = (𝑟 𝑞))
1941, 116, 3, 5, 56, 54, 7, 117, 52, 10, 2, 190, 192, 134, 193lncgr 25685 . . . . . . . . . . 11 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐹 𝑝) = (𝐹 𝑞))
1951, 116, 3, 5, 56, 7, 135, 117, 52, 10, 2, 164, 167, 134, 194lncgr 25685 . . . . . . . . . 10 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐵 𝑝) = (𝐵 𝑞))
196148ad6antr 720 . . . . . . . . . . . 12 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐹 ∈ (𝐶𝐼𝐽))
1971, 116, 3, 5, 56, 136, 7, 196btwncolg1 25671 . . . . . . . . . . 11 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐹 ∈ (𝐶(LineG‘𝐺)𝐽) ∨ 𝐶 = 𝐽))
1981, 116, 3, 5, 56, 7, 136, 117, 52, 10, 2, 164, 197, 134, 194lncgr 25685 . . . . . . . . . 10 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐽 𝑝) = (𝐽 𝑞))
1991, 116, 3, 5, 135, 136, 75, 117, 52, 10, 2, 156, 163, 195, 198lncgr 25685 . . . . . . . . 9 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐸 𝑝) = (𝐸 𝑞))
2001, 116, 3, 5, 56, 75, 52, 117, 10, 56, 2, 118, 119, 134, 199lnid 25686 . . . . . . . 8 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝑝 = 𝑞)
201200oveq1d 6808 . . . . . . 7 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝑝 𝑞) = (𝑞 𝑞))
202115, 201eqtrd 2805 . . . . . 6 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → (𝐹 𝐷) = (𝑞 𝑞))
2031, 2, 3, 5, 7, 9, 10, 202axtgcgrid 25583 . . . . 5 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐹 = 𝐷)
204203eqcomd 2777 . . . 4 (((((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) ∧ 𝑞𝑃) ∧ (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝))) → 𝐷 = 𝐹)
2054ad2antrr 705 . . . . . 6 (((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) → 𝐺 ∈ TarskiG)
206205ad2antrr 705 . . . . 5 (((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) → 𝐺 ∈ TarskiG)
207 simplr 752 . . . . 5 (((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) → 𝑟𝑃)
2081, 2, 3, 206, 51, 207, 207, 51axtgsegcon 25584 . . . 4 (((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) → ∃𝑞𝑃 (𝑟 ∈ (𝑝𝐼𝑞) ∧ (𝑟 𝑞) = (𝑟 𝑝)))
209204, 208r19.29a 3226 . . 3 (((((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) ∧ 𝑟𝑃) ∧ (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋))) → 𝐷 = 𝐹)
2106ad2antrr 705 . . . 4 (((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) → 𝐹𝑃)
21119ad2antrr 705 . . . 4 (((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) → 𝐶𝑃)
21216ad2antrr 705 . . . 4 (((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) → 𝑋𝑃)
2131, 2, 3, 205, 210, 211, 211, 212axtgsegcon 25584 . . 3 (((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) → ∃𝑟𝑃 (𝐶 ∈ (𝐹𝐼𝑟) ∧ (𝐶 𝑟) = (𝐶 𝑋)))
214209, 213r19.29a 3226 . 2 (((𝜑𝑝𝑃) ∧ (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹))) → 𝐷 = 𝐹)
2151, 2, 3, 4, 20, 19, 19, 6axtgsegcon 25584 . 2 (𝜑 → ∃𝑝𝑃 (𝐶 ∈ (𝐸𝐼𝑝) ∧ (𝐶 𝑝) = (𝐶 𝐹)))
216214, 215r19.29a 3226 1 (𝜑𝐷 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  cfv 6031  (class class class)co 6793  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556  LineGclng 25557  cgrGccgrg 25626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-xnn0 11566  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-concat 13497  df-s1 13498  df-s2 13802  df-s3 13803  df-trkgc 25568  df-trkgb 25569  df-trkgcb 25570  df-trkg 25573  df-cgrg 25627
This theorem is referenced by:  tgbtwnconn1  25691
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