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Theorem cgrg3col4 25668
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 2621 . . . . 5 (Itv‘𝐺) = (Itv‘𝐺)
4 isleag.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54ad2antrr 761 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺 ∈ TarskiG)
6 isleag.a . . . . . 6 (𝜑𝐴𝑃)
76ad2antrr 761 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴𝑃)
8 isleag.b . . . . . 6 (𝜑𝐵𝑃)
98ad2antrr 761 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵𝑃)
10 cgrg3col4.x . . . . . 6 (𝜑𝑋𝑃)
1110ad2antrr 761 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝑋𝑃)
12 eqid 2621 . . . . 5 (cgrG‘𝐺) = (cgrG‘𝐺)
13 isleag.d . . . . . 6 (𝜑𝐷𝑃)
1413ad2antrr 761 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷𝑃)
15 isleag.e . . . . . 6 (𝜑𝐸𝑃)
1615ad2antrr 761 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐸𝑃)
17 eqid 2621 . . . . 5 (dist‘𝐺) = (dist‘𝐺)
18 simpr 477 . . . . 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 25349 . . . . . 6 (𝜑 → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
2322ad2antrr 761 . . . . 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 25396 . . . 4 (((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦𝑃 ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩)
2521ad4antr 767 . . . . . . . 8 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
265ad2antrr 761 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐺 ∈ TarskiG)
2711ad2antrr 761 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝑋𝑃)
287ad2antrr 761 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐴𝑃)
29 simplr 791 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝑦𝑃)
3014ad2antrr 761 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐷𝑃)
319ad2antrr 761 . . . . . . . . . . 11 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐵𝑃)
3216ad2antrr 761 . . . . . . . . . . 11 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐸𝑃)
33 simpr 477 . . . . . . . . . . 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 25351 . . . . . . . . . 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 25309 . . . . . . . . 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 25350 . . . . . . . . 9 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦))
3719ad4antr 767 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐶𝑃)
3820ad4antr 767 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐹𝑃)
39 simpr 477 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐶)
4039ad3antrrr 765 . . . . . . . . . . . 12 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐴 = 𝐶)
4140oveq2d 6631 . . . . . . . . . . 11 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝑋(dist‘𝐺)𝐴) = (𝑋(dist‘𝐺)𝐶))
424adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐶) → 𝐺 ∈ TarskiG)
436adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐶) → 𝐴𝑃)
4419adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐶) → 𝐶𝑃)
4513adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐶) → 𝐷𝑃)
4620adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐶) → 𝐹𝑃)
471, 17, 3, 12, 4, 6, 8, 19, 13, 15, 20, 21cgr3simp3 25351 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐶(dist‘𝐺)𝐴) = (𝐹(dist‘𝐺)𝐷))
481, 17, 3, 4, 19, 6, 20, 13, 47tgcgrcomlr 25309 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴(dist‘𝐺)𝐶) = (𝐷(dist‘𝐺)𝐹))
4948adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐶) → (𝐴(dist‘𝐺)𝐶) = (𝐷(dist‘𝐺)𝐹))
501, 17, 3, 42, 43, 44, 45, 46, 49, 39tgcgreq 25311 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐶) → 𝐷 = 𝐹)
5150ad3antrrr 765 . . . . . . . . . . . 12 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → 𝐷 = 𝐹)
5251oveq2d 6631 . . . . . . . . . . 11 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝑦(dist‘𝐺)𝐷) = (𝑦(dist‘𝐺)𝐹))
5334, 41, 523eqtr3d 2663 . . . . . . . . . 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 25309 . . . . . . . . 9 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦))
5535, 36, 543jca 1240 . . . . . . . 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 554 . . . . . . 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 25360 . . . . . . 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 247 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
5958ex 450 . . . . 5 ((((𝜑𝐴 = 𝐶) ∧ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑦𝑃) → (⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑦”⟩ → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
6059reximdva 3013 . . . 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 2621 . . . . . 6 (hlG‘𝐺) = (hlG‘𝐺)
6342adantr 481 . . . . . . 7 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺 ∈ TarskiG)
6463ad2antrr 761 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐺 ∈ TarskiG)
658ad2antrr 761 . . . . . . 7 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵𝑃)
6665ad2antrr 761 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐵𝑃)
6743adantr 481 . . . . . . 7 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴𝑃)
6867ad2antrr 761 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐴𝑃)
6910ad2antrr 761 . . . . . . 7 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝑋𝑃)
7069ad2antrr 761 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝑋𝑃)
7115ad2antrr 761 . . . . . . 7 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐸𝑃)
7271ad2antrr 761 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐸𝑃)
7345adantr 481 . . . . . . 7 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷𝑃)
7473ad2antrr 761 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐷𝑃)
75 simplr 791 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝑥𝑃)
76 simpr 477 . . . . . . 7 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
7776ad2antrr 761 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
78 simpr 477 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ 𝑥 ∈ (𝐷𝐿𝐸))
7922ad2antrr 761 . . . . . . . . . . . . 13 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → (𝐴(dist‘𝐺)𝐵) = (𝐷(dist‘𝐺)𝐸))
801, 3, 2, 63, 65, 67, 69, 76ncolne1 25454 . . . . . . . . . . . . . 14 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐵𝐴)
8180necomd 2845 . . . . . . . . . . . . 13 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐴𝐵)
821, 17, 3, 63, 67, 65, 73, 71, 79, 81tgcgrneq 25312 . . . . . . . . . . . 12 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐷𝐸)
8382ad2antrr 761 . . . . . . . . . . 11 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → 𝐷𝐸)
8483neneqd 2795 . . . . . . . . . 10 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ 𝐷 = 𝐸)
8578, 84jca 554 . . . . . . . . 9 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → (¬ 𝑥 ∈ (𝐷𝐿𝐸) ∧ ¬ 𝐷 = 𝐸))
86 ioran 511 . . . . . . . . 9 (¬ (𝑥 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸) ↔ (¬ 𝑥 ∈ (𝐷𝐿𝐸) ∧ ¬ 𝐷 = 𝐸))
8785, 86sylibr 224 . . . . . . . 8 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝑥 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
881, 2, 3, 64, 74, 72, 75, 87ncolcom 25390 . . . . . . 7 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝑥 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
891, 2, 3, 64, 72, 74, 75, 88ncolrot1 25391 . . . . . 6 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ¬ (𝐸 ∈ (𝐷𝐿𝑥) ∨ 𝐷 = 𝑥))
901, 17, 3, 4, 6, 8, 13, 15, 22tgcgrcomlr 25309 . . . . . . 7 (𝜑 → (𝐵(dist‘𝐺)𝐴) = (𝐸(dist‘𝐺)𝐷))
9190ad4antr 767 . . . . . 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 25630 . . . . 5 (((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) → ∃𝑦𝑃 (⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩ ∧ 𝑦((hpG‘𝐺)‘(𝐸𝐿𝐷))𝑥))
9321ad6antr 771 . . . . . . . . . 10 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
9464ad2antrr 761 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐺 ∈ TarskiG)
9566ad2antrr 761 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐵𝑃)
9668ad2antrr 761 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐴𝑃)
9770ad2antrr 761 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝑋𝑃)
9872ad2antrr 761 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐸𝑃)
9974ad2antrr 761 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐷𝑃)
100 simplr 791 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝑦𝑃)
101 simpr 477 . . . . . . . . . . . 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 25350 . . . . . . . . . . 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 25351 . . . . . . . . . . . 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 25309 . . . . . . . . . . 11 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝐵(dist‘𝐺)𝑋) = (𝐸(dist‘𝐺)𝑦))
10544ad5antr 769 . . . . . . . . . . . 12 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐶𝑃)
10646ad5antr 769 . . . . . . . . . . . 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 25309 . . . . . . . . . . . . 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 810 . . . . . . . . . . . . . 14 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐴 = 𝐶)
109108oveq2d 6631 . . . . . . . . . . . . 13 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → (𝑋(dist‘𝐺)𝐴) = (𝑋(dist‘𝐺)𝐶))
11050ad5antr 769 . . . . . . . . . . . . . 14 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → 𝐷 = 𝐹)
111110oveq2d 6631 . . . . . . . . . . . . 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 2663 . . . . . . . . . . . 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 25309 . . . . . . . . . . 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 1240 . . . . . . . . . 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 554 . . . . . . . . 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 25360 . . . . . . . . 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 247 . . . . . . . 8 (((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) ∧ ⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
118117ex 450 . . . . . . 7 ((((((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) ∧ 𝑥𝑃) ∧ ¬ 𝑥 ∈ (𝐷𝐿𝐸)) ∧ 𝑦𝑃) → (⟨“𝐵𝐴𝑋”⟩(cgrG‘𝐺)⟨“𝐸𝐷𝑦”⟩ → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
119118adantrd 484 . . . . . 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 3013 . . . . 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 25394 . . . . 5 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → 𝐺DimTarskiG≥2)
1231, 3, 2, 63, 122, 73, 71, 82tglowdim2ln 25480 . . . 4 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑥𝑃 ¬ 𝑥 ∈ (𝐷𝐿𝐸))
124121, 123r19.29a 3073 . . 3 (((𝜑𝐴 = 𝐶) ∧ ¬ (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋)) → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
12561, 124pm2.61dan 831 . 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 25387 . . . . . 6 (𝜑 → (𝑋 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
1281, 2, 3, 4, 19, 6, 10, 127colrot1 25388 . . . . 5 (𝜑 → (𝐶 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
1291, 2, 3, 4, 6, 19, 10, 12, 13, 20, 17, 128, 48lnext 25396 . . . 4 (𝜑 → ∃𝑦𝑃 ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩)
130129adantr 481 . . 3 ((𝜑𝐴𝐶) → ∃𝑦𝑃 ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩)
13121ad3antrrr 765 . . . . . . 7 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
1324ad3antrrr 765 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐺 ∈ TarskiG)
13310ad3antrrr 765 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝑋𝑃)
1346ad3antrrr 765 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐴𝑃)
135 simplr 791 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝑦𝑃)
13613ad3antrrr 765 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐷𝑃)
13719ad3antrrr 765 . . . . . . . . . 10 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐶𝑃)
13820ad3antrrr 765 . . . . . . . . . 10 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐹𝑃)
139 simpr 477 . . . . . . . . . 10 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩)
1401, 17, 3, 12, 132, 134, 137, 133, 136, 138, 135, 139cgr3simp3 25351 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝑋(dist‘𝐺)𝐴) = (𝑦(dist‘𝐺)𝐷))
1411, 17, 3, 132, 133, 134, 135, 136, 140tgcgrcomlr 25309 . . . . . . . 8 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐴(dist‘𝐺)𝑋) = (𝐷(dist‘𝐺)𝑦))
1428ad3antrrr 765 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐵𝑃)
14315ad3antrrr 765 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → 𝐸𝑃)
144128ad3antrrr 765 . . . . . . . . . 10 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
14522ad3antrrr 765 . . . . . . . . . 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 25350 . . . . . . . . . . . 12 (𝜑 → (𝐵(dist‘𝐺)𝐶) = (𝐸(dist‘𝐺)𝐹))
1471, 17, 3, 4, 8, 19, 15, 20, 146tgcgrcomlr 25309 . . . . . . . . . . 11 (𝜑 → (𝐶(dist‘𝐺)𝐵) = (𝐹(dist‘𝐺)𝐸))
148147ad3antrrr 765 . . . . . . . . . 10 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶(dist‘𝐺)𝐵) = (𝐹(dist‘𝐺)𝐸))
149 simpllr 798 . . . . . . . . . 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 25397 . . . . . . . . 9 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝑋(dist‘𝐺)𝐵) = (𝑦(dist‘𝐺)𝐸))
1511, 17, 3, 132, 133, 142, 135, 143, 150tgcgrcomlr 25309 . . . . . . . 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 25350 . . . . . . . 8 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → (𝐶(dist‘𝐺)𝑋) = (𝐹(dist‘𝐺)𝑦))
153141, 151, 1523jca 1240 . . . . . . 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 554 . . . . . 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 25360 . . . . . 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 247 . . . . 5 ((((𝜑𝐴𝐶) ∧ 𝑦𝑃) ∧ ⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩) → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
157156ex 450 . . . 4 (((𝜑𝐴𝐶) ∧ 𝑦𝑃) → (⟨“𝐴𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐹𝑦”⟩ → ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩))
158157reximdva 3013 . . 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 2877 1 (𝜑 → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2909   class class class wbr 4623  cfv 5857  (class class class)co 6615  ⟨“cs3 13540  ⟨“cs4 13541  Basecbs 15800  distcds 15890  TarskiGcstrkg 25263  Itvcitv 25269  LineGclng 25270  cgrGccgrg 25339  hlGchlg 25429  hpGchpg 25583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-n0 11253  df-xnn0 11324  df-z 11338  df-uz 11648  df-fz 12285  df-fzo 12423  df-hash 13074  df-word 13254  df-concat 13256  df-s1 13257  df-s2 13546  df-s3 13547  df-s4 13548  df-trkgc 25281  df-trkgb 25282  df-trkgcb 25283  df-trkgld 25285  df-trkg 25286  df-cgrg 25340  df-ismt 25362  df-leg 25412  df-hlg 25430  df-mir 25482  df-rag 25523  df-perpg 25525  df-hpg 25584  df-mid 25600  df-lmi 25601
This theorem is referenced by: (None)
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