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Theorem cgrg3col4 25448
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 2605 . . . . 5 (Itv‘𝐺) = (Itv‘𝐺)
4 isleag.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54ad2antrr 757 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺 ∈ TarskiG)
6 isleag.a . . . . . 6 (𝜑𝐴𝑃)
76ad2antrr 757 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴𝑃)
8 isleag.b . . . . . 6 (𝜑𝐵𝑃)
98ad2antrr 757 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵𝑃)
10 cgrg3col4.x . . . . . 6 (𝜑𝑋𝑃)
1110ad2antrr 757 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝑋𝑃)
12 eqid 2605 . . . . 5 (cgrG‘𝐺) = (cgrG‘𝐺)
13 isleag.d . . . . . 6 (𝜑𝐷𝑃)
1413ad2antrr 757 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷𝑃)
15 isleag.e . . . . . 6 (𝜑𝐸𝑃)
1615ad2antrr 757 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐸𝑃)
17 eqid 2605 . . . . 5 (dist‘𝐺) = (dist‘𝐺)
18 simpr 475 . . . . 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 25129 . . . . . 6 (𝜑 → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
2322ad2antrr 757 . . . . 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 25176 . . . 4 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦𝑃 ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩)
2521ad4antr 763 . . . . . . . 8 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
265ad2antrr 757 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐺 ∈ TarskiG)
2711ad2antrr 757 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝑋𝑃)
287ad2antrr 757 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐴𝑃)
29 simplr 787 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝑦𝑃)
3014ad2antrr 757 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐷𝑃)
319ad2antrr 757 . . . . . . . . . . 11 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐵𝑃)
3216ad2antrr 757 . . . . . . . . . . 11 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐸𝑃)
33 simpr 475 . . . . . . . . . . 11 (((((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 25131 . . . . . . . . . 10 (((((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 25088 . . . . . . . . 9 (((((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 25130 . . . . . . . . 9 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦))
3719ad4antr 763 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐶𝑃)
3820ad4antr 763 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐹𝑃)
39 simpr 475 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐶)
4039ad3antrrr 761 . . . . . . . . . . . 12 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐴 = 𝐶)
4140oveq2d 6539 . . . . . . . . . . 11 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝑋(dist‘𝐺)𝐴) = (𝑋(dist‘𝐺)𝐶))
424adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐶) → 𝐺 ∈ TarskiG)
436adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐶) → 𝐴𝑃)
4419adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐶) → 𝐶𝑃)
4513adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐶) → 𝐷𝑃)
4620adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐶) → 𝐹𝑃)
471, 17, 3, 12, 4, 6, 8, 19, 13, 15, 20, 21cgr3simp3 25131 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐶(dist‘𝐺)𝐴) = (𝐹(dist‘𝐺)𝐷))
481, 17, 3, 4, 19, 6, 20, 13, 47tgcgrcomlr 25088 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴(dist‘𝐺)𝐶) = (𝐷(dist‘𝐺)𝐹))
4948adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐶) → (𝐴(dist‘𝐺)𝐶) = (𝐷(dist‘𝐺)𝐹))
501, 17, 3, 42, 43, 44, 45, 46, 49, 39tgcgreq 25090 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐶) → 𝐷 = 𝐹)
5150ad3antrrr 761 . . . . . . . . . . . 12 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐷 = 𝐹)
5251oveq2d 6539 . . . . . . . . . . 11 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝑦(dist‘𝐺)𝐷) = (𝑦(dist‘𝐺)𝐹))
5334, 41, 523eqtr3d 2647 . . . . . . . . . 10 (((((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 25088 . . . . . . . . 9 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦))
5535, 36, 543jca 1234 . . . . . . . 8 (((((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)))
5625, 55jca 552 . . . . . . 7 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (⟨“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))))
571, 17, 3, 12, 26, 28, 31, 37, 27, 30, 32, 38, 29tgcgr4 25140 . . . . . . 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)))))
5856, 57mpbird 245 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
5958ex 448 . . . . 5 ((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) → (⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩ → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
6059reximdva 2995 . . . 4 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → (∃𝑦𝑃 ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩ → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
6124, 60mpd 15 . . 3 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
62 eqid 2605 . . . . . 6 (hlG‘𝐺) = (hlG‘𝐺)
6342adantr 479 . . . . . . 7 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺 ∈ TarskiG)
6463ad2antrr 757 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐺 ∈ TarskiG)
658ad2antrr 757 . . . . . . 7 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵𝑃)
6665ad2antrr 757 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐵𝑃)
6743adantr 479 . . . . . . 7 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴𝑃)
6867ad2antrr 757 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐴𝑃)
6910ad2antrr 757 . . . . . . 7 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝑋𝑃)
7069ad2antrr 757 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝑋𝑃)
7115ad2antrr 757 . . . . . . 7 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐸𝑃)
7271ad2antrr 757 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐸𝑃)
7345adantr 479 . . . . . . 7 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷𝑃)
7473ad2antrr 757 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐷𝑃)
75 simplr 787 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝑥𝑃)
76 simpr 475 . . . . . . 7 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
7776ad2antrr 757 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
78 simpr 475 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ 𝑥 ∈ (𝐷𝐿𝐸))
7922ad2antrr 757 . . . . . . . . . . . . 13 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
801, 3, 2, 63, 65, 67, 69, 76ncolne1 25234 . . . . . . . . . . . . . 14 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵𝐴)
8180necomd 2832 . . . . . . . . . . . . 13 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴𝐵)
821, 17, 3, 63, 67, 65, 73, 71, 79, 81tgcgrneq 25091 . . . . . . . . . . . 12 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷𝐸)
8382ad2antrr 757 . . . . . . . . . . 11 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐷𝐸)
8483neneqd 2782 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ 𝐷 = 𝐸)
8578, 84jca 552 . . . . . . . . 9 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → (¬ 𝑥 ∈ (𝐷𝐿𝐸) ∧ ¬ 𝐷 = 𝐸))
86 ioran 509 . . . . . . . . 9 (¬ (𝑥 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸) ↔ (¬ 𝑥 ∈ (𝐷𝐿𝐸) ∧ ¬ 𝐷 = 𝐸))
8785, 86sylibr 222 . . . . . . . 8 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝑥 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
881, 2, 3, 64, 74, 72, 75, 87ncolcom 25170 . . . . . . 7 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝑥 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
891, 2, 3, 64, 72, 74, 75, 88ncolrot1 25171 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝐸 ∈ (𝐷𝐿𝑥) ∨ 𝐷 = 𝑥))
901, 17, 3, 4, 6, 8, 13, 15, 22tgcgrcomlr 25088 . . . . . . 7 (𝜑 → (𝐵(dist‘𝐺)𝐴) = (𝐸(dist‘𝐺)𝐷))
9190ad4antr 763 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → (𝐵(dist‘𝐺)𝐴) = (𝐸(dist‘𝐺)𝐷))
921, 17, 3, 2, 62, 64, 66, 68, 70, 72, 74, 75, 77, 89, 91trgcopy 25410 . . . . 5 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ∃𝑦𝑃 (⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩ ∧ 𝑦((hpG‘𝐺)‘(𝐸𝐿𝐷))𝑥))
9321ad6antr 767 . . . . . . . . . 10 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
9464ad2antrr 757 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐺 ∈ TarskiG)
9566ad2antrr 757 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐵𝑃)
9668ad2antrr 757 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐴𝑃)
9770ad2antrr 757 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝑋𝑃)
9872ad2antrr 757 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐸𝑃)
9974ad2antrr 757 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐷𝑃)
100 simplr 787 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝑦𝑃)
101 simpr 475 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩)
1021, 17, 3, 12, 94, 95, 96, 97, 98, 99, 100, 101cgr3simp2 25130 . . . . . . . . . . 11 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦))
1031, 17, 3, 12, 94, 95, 96, 97, 98, 99, 100, 101cgr3simp3 25131 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝑋(dist‘𝐺)𝐵) = (𝑦(dist‘𝐺)𝐸))
1041, 17, 3, 94, 97, 95, 100, 98, 103tgcgrcomlr 25088 . . . . . . . . . . 11 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦))
10544ad5antr 765 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐶𝑃)
10646ad5antr 765 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐹𝑃)
1071, 17, 3, 94, 96, 97, 99, 100, 102tgcgrcomlr 25088 . . . . . . . . . . . . 13 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝑋(dist‘𝐺)𝐴) = (𝑦(dist‘𝐺)𝐷))
108 simp-6r 806 . . . . . . . . . . . . . 14 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐴 = 𝐶)
109108oveq2d 6539 . . . . . . . . . . . . 13 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝑋(dist‘𝐺)𝐴) = (𝑋(dist‘𝐺)𝐶))
11050ad5antr 765 . . . . . . . . . . . . . 14 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐷 = 𝐹)
111110oveq2d 6539 . . . . . . . . . . . . 13 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝑦(dist‘𝐺)𝐷) = (𝑦(dist‘𝐺)𝐹))
112107, 109, 1113eqtr3d 2647 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝑋(dist‘𝐺)𝐶) = (𝑦(dist‘𝐺)𝐹))
1131, 17, 3, 94, 97, 105, 100, 106, 112tgcgrcomlr 25088 . . . . . . . . . . 11 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦))
114102, 104, 1133jca 1234 . . . . . . . . . 10 (((((((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)))
11593, 114jca 552 . . . . . . . . 9 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (⟨“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))))
1161, 17, 3, 12, 94, 96, 95, 105, 97, 99, 98, 106, 100tgcgr4 25140 . . . . . . . . 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)))))
117115, 116mpbird 245 . . . . . . . 8 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
118117ex 448 . . . . . . 7 ((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) → (⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩ → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
119118adantrd 482 . . . . . 6 ((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) → ((⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩ ∧ 𝑦((hpG‘𝐺)‘(𝐸𝐿𝐷))𝑥) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
120119reximdva 2995 . . . . 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”⟩))
12192, 120mpd 15 . . . 4 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
1221, 2, 3, 63, 67, 69, 65, 76ncoltgdim2 25174 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺DimTarskiG≥2)
1231, 3, 2, 63, 122, 73, 71, 82tglowdim2ln 25260 . . . 4 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑥𝑃 ¬ 𝑥 ∈ (𝐷𝐿𝐸))
124121, 123r19.29a 3055 . . 3 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
12561, 124pm2.61dan 827 . 2 ((𝜑𝐴 = 𝐶) → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
126 cgrg3col4.2 . . . . . . 7 (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
1271, 2, 3, 4, 6, 19, 10, 126colcom 25167 . . . . . 6 (𝜑 → (𝑋 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
1281, 2, 3, 4, 19, 6, 10, 127colrot1 25168 . . . . 5 (𝜑 → (𝐶 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
1291, 2, 3, 4, 6, 19, 10, 12, 13, 20, 17, 128, 48lnext 25176 . . . 4 (𝜑 → ∃𝑦𝑃 ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩)
130129adantr 479 . . 3 ((𝜑𝐴𝐶) → ∃𝑦𝑃 ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩)
13121ad3antrrr 761 . . . . . . 7 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
1324ad3antrrr 761 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐺 ∈ TarskiG)
13310ad3antrrr 761 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝑋𝑃)
1346ad3antrrr 761 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐴𝑃)
135 simplr 787 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝑦𝑃)
13613ad3antrrr 761 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐷𝑃)
13719ad3antrrr 761 . . . . . . . . . 10 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐶𝑃)
13820ad3antrrr 761 . . . . . . . . . 10 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐹𝑃)
139 simpr 475 . . . . . . . . . 10 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩)
1401, 17, 3, 12, 132, 134, 137, 133, 136, 138, 135, 139cgr3simp3 25131 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝑋(dist‘𝐺)𝐴) = (𝑦(dist‘𝐺)𝐷))
1411, 17, 3, 132, 133, 134, 135, 136, 140tgcgrcomlr 25088 . . . . . . . 8 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦))
1428ad3antrrr 761 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐵𝑃)
14315ad3antrrr 761 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐸𝑃)
144128ad3antrrr 761 . . . . . . . . . 10 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
14522ad3antrrr 761 . . . . . . . . . 10 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
1461, 17, 3, 12, 4, 6, 8, 19, 13, 15, 20, 21cgr3simp2 25130 . . . . . . . . . . . 12 (𝜑 → (𝐵(dist‘𝐺)𝐶) = (𝐸(dist‘𝐺)𝐹))
1471, 17, 3, 4, 8, 19, 15, 20, 146tgcgrcomlr 25088 . . . . . . . . . . 11 (𝜑 → (𝐶(dist‘𝐺)𝐵) = (𝐹(dist‘𝐺)𝐸))
148147ad3antrrr 761 . . . . . . . . . 10 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶(dist‘𝐺)𝐵) = (𝐹(dist‘𝐺)𝐸))
149 simpllr 794 . . . . . . . . . 10 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐴𝐶)
1501, 2, 3, 132, 134, 137, 133, 12, 136, 138, 17, 142, 135, 143, 144, 139, 145, 148, 149tgfscgr 25177 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝑋(dist‘𝐺)𝐵) = (𝑦(dist‘𝐺)𝐸))
1511, 17, 3, 132, 133, 142, 135, 143, 150tgcgrcomlr 25088 . . . . . . . 8 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦))
1521, 17, 3, 12, 132, 134, 137, 133, 136, 138, 135, 139cgr3simp2 25130 . . . . . . . 8 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦))
153141, 151, 1523jca 1234 . . . . . . 7 ((((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)))
154131, 153jca 552 . . . . . 6 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (⟨“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))))
1551, 17, 3, 12, 132, 134, 142, 137, 133, 136, 143, 138, 135tgcgr4 25140 . . . . . 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)))))
156154, 155mpbird 245 . . . . 5 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
157156ex 448 . . . 4 (((𝜑𝐴𝐶) ∧ 𝑦𝑃) → (⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩ → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
158157reximdva 2995 . . 3 ((𝜑𝐴𝐶) → (∃𝑦𝑃 ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩ → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
159130, 158mpd 15 . 2 ((𝜑𝐴𝐶) → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
160125, 159pm2.61dane 2864 1 (𝜑 → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2775  wrex 2892   class class class wbr 4573  cfv 5786  (class class class)co 6523  ⟨“cs3 13380  ⟨“cs4 13381  Basecbs 15637  distcds 15719  TarskiGcstrkg 25042  Itvcitv 25048  LineGclng 25049  cgrGccgrg 25119  hlGchlg 25209  hpGchpg 25363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-er 7602  df-map 7719  df-pm 7720  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-card 8621  df-cda 8846  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-2 10922  df-3 10923  df-4 10924  df-n0 11136  df-z 11207  df-uz 11516  df-fz 12149  df-fzo 12286  df-hash 12931  df-word 13096  df-concat 13098  df-s1 13099  df-s2 13386  df-s3 13387  df-s4 13388  df-trkgc 25060  df-trkgb 25061  df-trkgcb 25062  df-trkgld 25064  df-trkg 25065  df-cgrg 25120  df-ismt 25142  df-leg 25192  df-hlg 25210  df-mir 25262  df-rag 25303  df-perpg 25305  df-hpg 25364  df-mid 25380  df-lmi 25381
This theorem is referenced by: (None)
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