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Theorem cgrg3col4 27795
Description: Lemma 11.28 of [Schwabhauser] p. 102. Extend a congruence of three points with a fourth colinear point. (Contributed by Thierry Arnoux, 8-Oct-2020.)
Hypotheses
Ref Expression
isleag.p 𝑃 = (Base‘𝐺)
isleag.g (𝜑𝐺 ∈ TarskiG)
isleag.a (𝜑𝐴𝑃)
isleag.b (𝜑𝐵𝑃)
isleag.c (𝜑𝐶𝑃)
isleag.d (𝜑𝐷𝑃)
isleag.e (𝜑𝐸𝑃)
isleag.f (𝜑𝐹𝑃)
cgrg3col4.l 𝐿 = (LineG‘𝐺)
cgrg3col4.x (𝜑𝑋𝑃)
cgrg3col4.1 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
cgrg3col4.2 (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
Assertion
Ref Expression
cgrg3col4 (𝜑 → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝐶   𝑦,𝐷   𝑦,𝐸   𝑦,𝐹   𝑦,𝐺   𝑦,𝐿   𝑦,𝑃   𝑦,𝑋   𝜑,𝑦

Proof of Theorem cgrg3col4
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 isleag.p . . . . 5 𝑃 = (Base‘𝐺)
2 cgrg3col4.l . . . . 5 𝐿 = (LineG‘𝐺)
3 eqid 2736 . . . . 5 (Itv‘𝐺) = (Itv‘𝐺)
4 isleag.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54ad2antrr 724 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺 ∈ TarskiG)
6 isleag.a . . . . . 6 (𝜑𝐴𝑃)
76ad2antrr 724 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴𝑃)
8 isleag.b . . . . . 6 (𝜑𝐵𝑃)
98ad2antrr 724 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵𝑃)
10 cgrg3col4.x . . . . . 6 (𝜑𝑋𝑃)
1110ad2antrr 724 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝑋𝑃)
12 eqid 2736 . . . . 5 (cgrG‘𝐺) = (cgrG‘𝐺)
13 isleag.d . . . . . 6 (𝜑𝐷𝑃)
1413ad2antrr 724 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷𝑃)
15 isleag.e . . . . . 6 (𝜑𝐸𝑃)
1615ad2antrr 724 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐸𝑃)
17 eqid 2736 . . . . 5 (dist‘𝐺) = (dist‘𝐺)
18 simpr 485 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
19 isleag.c . . . . . . 7 (𝜑𝐶𝑃)
20 isleag.f . . . . . . 7 (𝜑𝐹𝑃)
21 cgrg3col4.1 . . . . . . 7 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
221, 17, 3, 12, 4, 6, 8, 19, 13, 15, 20, 21cgr3simp1 27462 . . . . . 6 (𝜑 → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
2322ad2antrr 724 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
241, 2, 3, 5, 7, 9, 11, 12, 14, 16, 17, 18, 23lnext 27509 . . . 4 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦𝑃 ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩)
2521ad4antr 730 . . . . . . 7 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
265ad2antrr 724 . . . . . . . . 9 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐺 ∈ TarskiG)
2711ad2antrr 724 . . . . . . . . 9 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝑋𝑃)
287ad2antrr 724 . . . . . . . . 9 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐴𝑃)
29 simplr 767 . . . . . . . . 9 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝑦𝑃)
3014ad2antrr 724 . . . . . . . . 9 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐷𝑃)
319ad2antrr 724 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐵𝑃)
3216ad2antrr 724 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐸𝑃)
33 simpr 485 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩)
341, 17, 3, 12, 26, 28, 31, 27, 30, 32, 29, 33cgr3simp3 27464 . . . . . . . . 9 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝑋(dist‘𝐺)𝐴) = (𝑦(dist‘𝐺)𝐷))
351, 17, 3, 26, 27, 28, 29, 30, 34tgcgrcomlr 27422 . . . . . . . 8 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦))
361, 17, 3, 12, 26, 28, 31, 27, 30, 32, 29, 33cgr3simp2 27463 . . . . . . . 8 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦))
3719ad4antr 730 . . . . . . . . 9 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐶𝑃)
3820ad4antr 730 . . . . . . . . 9 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐹𝑃)
39 simpr 485 . . . . . . . . . . . 12 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐶)
4039ad3antrrr 728 . . . . . . . . . . 11 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐴 = 𝐶)
4140oveq2d 7373 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝑋(dist‘𝐺)𝐴) = (𝑋(dist‘𝐺)𝐶))
424adantr 481 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐶) → 𝐺 ∈ TarskiG)
436adantr 481 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐶) → 𝐴𝑃)
4419adantr 481 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐶) → 𝐶𝑃)
4513adantr 481 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐶) → 𝐷𝑃)
4620adantr 481 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐶) → 𝐹𝑃)
471, 17, 3, 12, 4, 6, 8, 19, 13, 15, 20, 21cgr3simp3 27464 . . . . . . . . . . . . . . 15 (𝜑 → (𝐶(dist‘𝐺)𝐴) = (𝐹(dist‘𝐺)𝐷))
481, 17, 3, 4, 19, 6, 20, 13, 47tgcgrcomlr 27422 . . . . . . . . . . . . . 14 (𝜑 → (𝐴(dist‘𝐺)𝐶) = (𝐷(dist‘𝐺)𝐹))
4948adantr 481 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐶) → (𝐴(dist‘𝐺)𝐶) = (𝐷(dist‘𝐺)𝐹))
501, 17, 3, 42, 43, 44, 45, 46, 49, 39tgcgreq 27424 . . . . . . . . . . . 12 ((𝜑𝐴 = 𝐶) → 𝐷 = 𝐹)
5150ad3antrrr 728 . . . . . . . . . . 11 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐷 = 𝐹)
5251oveq2d 7373 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝑦(dist‘𝐺)𝐷) = (𝑦(dist‘𝐺)𝐹))
5334, 41, 523eqtr3d 2784 . . . . . . . . 9 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝑋(dist‘𝐺)𝐶) = (𝑦(dist‘𝐺)𝐹))
541, 17, 3, 26, 27, 37, 29, 38, 53tgcgrcomlr 27422 . . . . . . . 8 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦))
5535, 36, 543jca 1128 . . . . . . 7 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ((𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦) ∧ (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦) ∧ (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦)))
561, 17, 3, 12, 26, 28, 31, 37, 27, 30, 32, 38, 29tgcgr4 27473 . . . . . . 7 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩ ↔ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ((𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦) ∧ (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦) ∧ (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦)))))
5725, 55, 56mpbir2and 711 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
5857ex 413 . . . . 5 ((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) → (⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩ → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
5958reximdva 3165 . . . 4 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → (∃𝑦𝑃 ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩ → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
6024, 59mpd 15 . . 3 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
61 eqid 2736 . . . . . 6 (hlG‘𝐺) = (hlG‘𝐺)
624ad2antrr 724 . . . . . . 7 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺 ∈ TarskiG)
6362ad2antrr 724 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐺 ∈ TarskiG)
648ad2antrr 724 . . . . . . 7 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵𝑃)
6564ad2antrr 724 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐵𝑃)
666ad2antrr 724 . . . . . . 7 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴𝑃)
6766ad2antrr 724 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐴𝑃)
6810ad2antrr 724 . . . . . . 7 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝑋𝑃)
6968ad2antrr 724 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝑋𝑃)
7015ad2antrr 724 . . . . . . 7 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐸𝑃)
7170ad2antrr 724 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐸𝑃)
7213ad2antrr 724 . . . . . . 7 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷𝑃)
7372ad2antrr 724 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐷𝑃)
74 simplr 767 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝑥𝑃)
75 simpllr 774 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
76 simpr 485 . . . . . . . . 9 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ 𝑥 ∈ (𝐷𝐿𝐸))
7722ad2antrr 724 . . . . . . . . . . . 12 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
78 simpr 485 . . . . . . . . . . . . . 14 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
791, 3, 2, 62, 64, 66, 68, 78ncolne1 27567 . . . . . . . . . . . . 13 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵𝐴)
8079necomd 2999 . . . . . . . . . . . 12 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴𝐵)
811, 17, 3, 62, 66, 64, 72, 70, 77, 80tgcgrneq 27425 . . . . . . . . . . 11 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷𝐸)
8281ad2antrr 724 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐷𝐸)
8382neneqd 2948 . . . . . . . . 9 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ 𝐷 = 𝐸)
84 ioran 982 . . . . . . . . 9 (¬ (𝑥 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸) ↔ (¬ 𝑥 ∈ (𝐷𝐿𝐸) ∧ ¬ 𝐷 = 𝐸))
8576, 83, 84sylanbrc 583 . . . . . . . 8 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝑥 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
861, 2, 3, 63, 73, 71, 74, 85ncolcom 27503 . . . . . . 7 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝑥 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
871, 2, 3, 63, 71, 73, 74, 86ncolrot1 27504 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝐸 ∈ (𝐷𝐿𝑥) ∨ 𝐷 = 𝑥))
881, 17, 3, 4, 6, 8, 13, 15, 22tgcgrcomlr 27422 . . . . . . 7 (𝜑 → (𝐵(dist‘𝐺)𝐴) = (𝐸(dist‘𝐺)𝐷))
8988ad4antr 730 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → (𝐵(dist‘𝐺)𝐴) = (𝐸(dist‘𝐺)𝐷))
901, 17, 3, 2, 61, 63, 65, 67, 69, 71, 73, 74, 75, 87, 89trgcopy 27746 . . . . 5 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ∃𝑦𝑃 (⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩ ∧ 𝑦((hpG‘𝐺)‘(𝐸𝐿𝐷))𝑥))
9121ad6antr 734 . . . . . . . . 9 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
9263ad2antrr 724 . . . . . . . . . . 11 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐺 ∈ TarskiG)
9365ad2antrr 724 . . . . . . . . . . 11 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐵𝑃)
9467ad2antrr 724 . . . . . . . . . . 11 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐴𝑃)
9569ad2antrr 724 . . . . . . . . . . 11 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝑋𝑃)
9671ad2antrr 724 . . . . . . . . . . 11 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐸𝑃)
9773ad2antrr 724 . . . . . . . . . . 11 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐷𝑃)
98 simplr 767 . . . . . . . . . . 11 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝑦𝑃)
99 simpr 485 . . . . . . . . . . 11 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩)
1001, 17, 3, 12, 92, 93, 94, 95, 96, 97, 98, 99cgr3simp2 27463 . . . . . . . . . 10 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦))
1011, 17, 3, 12, 92, 93, 94, 95, 96, 97, 98, 99cgr3simp3 27464 . . . . . . . . . . 11 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝑋(dist‘𝐺)𝐵) = (𝑦(dist‘𝐺)𝐸))
1021, 17, 3, 92, 95, 93, 98, 96, 101tgcgrcomlr 27422 . . . . . . . . . 10 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦))
10344ad5antr 732 . . . . . . . . . . 11 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐶𝑃)
10446ad5antr 732 . . . . . . . . . . 11 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐹𝑃)
1051, 17, 3, 92, 94, 95, 97, 98, 100tgcgrcomlr 27422 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝑋(dist‘𝐺)𝐴) = (𝑦(dist‘𝐺)𝐷))
106 simp-6r 786 . . . . . . . . . . . . 13 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐴 = 𝐶)
107106oveq2d 7373 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝑋(dist‘𝐺)𝐴) = (𝑋(dist‘𝐺)𝐶))
10850ad5antr 732 . . . . . . . . . . . . 13 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐷 = 𝐹)
109108oveq2d 7373 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝑦(dist‘𝐺)𝐷) = (𝑦(dist‘𝐺)𝐹))
110105, 107, 1093eqtr3d 2784 . . . . . . . . . . 11 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝑋(dist‘𝐺)𝐶) = (𝑦(dist‘𝐺)𝐹))
1111, 17, 3, 92, 95, 103, 98, 104, 110tgcgrcomlr 27422 . . . . . . . . . 10 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦))
112100, 102, 1113jca 1128 . . . . . . . . 9 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → ((𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦) ∧ (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦) ∧ (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦)))
1131, 17, 3, 12, 92, 94, 93, 103, 95, 97, 96, 104, 98tgcgr4 27473 . . . . . . . . 9 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩ ↔ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ((𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦) ∧ (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦) ∧ (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦)))))
11491, 112, 113mpbir2and 711 . . . . . . . 8 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
115114ex 413 . . . . . . 7 ((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) → (⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩ → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
116115adantrd 492 . . . . . 6 ((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) → ((⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩ ∧ 𝑦((hpG‘𝐺)‘(𝐸𝐿𝐷))𝑥) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
117116reximdva 3165 . . . . 5 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → (∃𝑦𝑃 (⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩ ∧ 𝑦((hpG‘𝐺)‘(𝐸𝐿𝐷))𝑥) → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
11890, 117mpd 15 . . . 4 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
1191, 2, 3, 62, 66, 68, 64, 78ncoltgdim2 27507 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺DimTarskiG≥2)
1201, 3, 2, 62, 119, 72, 70, 81tglowdim2ln 27593 . . . 4 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑥𝑃 ¬ 𝑥 ∈ (𝐷𝐿𝐸))
121118, 120r19.29a 3159 . . 3 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
12260, 121pm2.61dan 811 . 2 ((𝜑𝐴 = 𝐶) → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
123 cgrg3col4.2 . . . . . . 7 (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
1241, 2, 3, 4, 6, 19, 10, 123colcom 27500 . . . . . 6 (𝜑 → (𝑋 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
1251, 2, 3, 4, 19, 6, 10, 124colrot1 27501 . . . . 5 (𝜑 → (𝐶 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
1261, 2, 3, 4, 6, 19, 10, 12, 13, 20, 17, 125, 48lnext 27509 . . . 4 (𝜑 → ∃𝑦𝑃 ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩)
127126adantr 481 . . 3 ((𝜑𝐴𝐶) → ∃𝑦𝑃 ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩)
12821ad3antrrr 728 . . . . . 6 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
1294ad3antrrr 728 . . . . . . . 8 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐺 ∈ TarskiG)
13010ad3antrrr 728 . . . . . . . 8 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝑋𝑃)
1316ad3antrrr 728 . . . . . . . 8 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐴𝑃)
132 simplr 767 . . . . . . . 8 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝑦𝑃)
13313ad3antrrr 728 . . . . . . . 8 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐷𝑃)
13419ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐶𝑃)
13520ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐹𝑃)
136 simpr 485 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩)
1371, 17, 3, 12, 129, 131, 134, 130, 133, 135, 132, 136cgr3simp3 27464 . . . . . . . 8 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝑋(dist‘𝐺)𝐴) = (𝑦(dist‘𝐺)𝐷))
1381, 17, 3, 129, 130, 131, 132, 133, 137tgcgrcomlr 27422 . . . . . . 7 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦))
1398ad3antrrr 728 . . . . . . . 8 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐵𝑃)
14015ad3antrrr 728 . . . . . . . 8 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐸𝑃)
141125ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
14222ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
1431, 17, 3, 12, 4, 6, 8, 19, 13, 15, 20, 21cgr3simp2 27463 . . . . . . . . . . 11 (𝜑 → (𝐵(dist‘𝐺)𝐶) = (𝐸(dist‘𝐺)𝐹))
1441, 17, 3, 4, 8, 19, 15, 20, 143tgcgrcomlr 27422 . . . . . . . . . 10 (𝜑 → (𝐶(dist‘𝐺)𝐵) = (𝐹(dist‘𝐺)𝐸))
145144ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶(dist‘𝐺)𝐵) = (𝐹(dist‘𝐺)𝐸))
146 simpllr 774 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐴𝐶)
1471, 2, 3, 129, 131, 134, 130, 12, 133, 135, 17, 139, 132, 140, 141, 136, 142, 145, 146tgfscgr 27510 . . . . . . . 8 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝑋(dist‘𝐺)𝐵) = (𝑦(dist‘𝐺)𝐸))
1481, 17, 3, 129, 130, 139, 132, 140, 147tgcgrcomlr 27422 . . . . . . 7 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦))
1491, 17, 3, 12, 129, 131, 134, 130, 133, 135, 132, 136cgr3simp2 27463 . . . . . . 7 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦))
150138, 148, 1493jca 1128 . . . . . 6 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ((𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦) ∧ (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦) ∧ (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦)))
1511, 17, 3, 12, 129, 131, 139, 134, 130, 133, 140, 135, 132tgcgr4 27473 . . . . . 6 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩ ↔ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ((𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦) ∧ (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦) ∧ (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦)))))
152128, 150, 151mpbir2and 711 . . . . 5 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
153152ex 413 . . . 4 (((𝜑𝐴𝐶) ∧ 𝑦𝑃) → (⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩ → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
154153reximdva 3165 . . 3 ((𝜑𝐴𝐶) → (∃𝑦𝑃 ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩ → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
155127, 154mpd 15 . 2 ((𝜑𝐴𝐶) → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
156122, 155pm2.61dane 3032 1 (𝜑 → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wrex 3073   class class class wbr 5105  cfv 6496  (class class class)co 7357  ⟨“cs3 14731  ⟨“cs4 14732  Basecbs 17083  distcds 17142  TarskiGcstrkg 27369  Itvcitv 27375  LineGclng 27376  cgrGccgrg 27452  hlGchlg 27542  hpGchpg 27699
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-concat 14459  df-s1 14484  df-s2 14737  df-s3 14738  df-s4 14739  df-trkgc 27390  df-trkgb 27391  df-trkgcb 27392  df-trkgld 27394  df-trkg 27395  df-cgrg 27453  df-ismt 27475  df-leg 27525  df-hlg 27543  df-mir 27595  df-rag 27636  df-perpg 27638  df-hpg 27700  df-mid 27716  df-lmi 27717
This theorem is referenced by: (None)
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