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Theorem tgasa1 25452
Description: Second congruence theorem: ASA. (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. Theorem 11.50 of [Schwabhauser] p. 108. (Contributed by Thierry Arnoux, 15-Aug-2020.)
Hypotheses
Ref Expression
tgsas.p 𝑃 = (Base‘𝐺)
tgsas.m = (dist‘𝐺)
tgsas.i 𝐼 = (Itv‘𝐺)
tgsas.g (𝜑𝐺 ∈ TarskiG)
tgsas.a (𝜑𝐴𝑃)
tgsas.b (𝜑𝐵𝑃)
tgsas.c (𝜑𝐶𝑃)
tgsas.d (𝜑𝐷𝑃)
tgsas.e (𝜑𝐸𝑃)
tgsas.f (𝜑𝐹𝑃)
tgasa.l 𝐿 = (LineG‘𝐺)
tgasa.1 (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
tgasa.2 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
tgasa.3 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
tgasa.4 (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)
Assertion
Ref Expression
tgasa1 (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))

Proof of Theorem tgasa1
Dummy variables 𝑎 𝑏 𝑓 𝑤 𝑡 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprr 791 . . 3 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐸 𝑓) = (𝐵 𝐶))
2 tgsas.p . . . . 5 𝑃 = (Base‘𝐺)
3 tgsas.i . . . . 5 𝐼 = (Itv‘𝐺)
4 tgasa.l . . . . 5 𝐿 = (LineG‘𝐺)
5 tgsas.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
65ad2antrr 757 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐺 ∈ TarskiG)
7 tgsas.f . . . . . 6 (𝜑𝐹𝑃)
87ad2antrr 757 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐹𝑃)
9 tgsas.d . . . . . 6 (𝜑𝐷𝑃)
109ad2antrr 757 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐷𝑃)
11 tgsas.e . . . . . 6 (𝜑𝐸𝑃)
1211ad2antrr 757 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐸𝑃)
13 tgsas.m . . . . . . 7 = (dist‘𝐺)
14 tgsas.a . . . . . . 7 (𝜑𝐴𝑃)
15 tgsas.b . . . . . . 7 (𝜑𝐵𝑃)
16 tgsas.c . . . . . . 7 (𝜑𝐶𝑃)
17 tgasa.3 . . . . . . 7 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
18 tgasa.1 . . . . . . 7 (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
192, 3, 13, 5, 14, 15, 16, 9, 11, 7, 17, 4, 18cgrancol 25433 . . . . . 6 (𝜑 → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
2019ad2antrr 757 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
21 eqid 2604 . . . . . 6 (hlG‘𝐺) = (hlG‘𝐺)
22 simplr 787 . . . . . 6 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓𝑃)
2316ad2antrr 757 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐶𝑃)
2414ad2antrr 757 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐴𝑃)
2515ad2antrr 757 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐵𝑃)
2618ad2antrr 757 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
276adantr 479 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐺 ∈ TarskiG)
2810adantr 479 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐷𝑃)
2912adantr 479 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐸𝑃)
308adantr 479 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐹𝑃)
3124adantr 479 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐴𝑃)
3225adantr 479 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐵𝑃)
3323adantr 479 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐶𝑃)
342, 3, 5, 21, 14, 15, 16, 9, 11, 7, 17cgracom 25427 . . . . . . . . . 10 (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
3534ad3antrrr 761 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
36 simpr 475 . . . . . . . . . . 11 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹))
372, 4, 3, 27, 28, 30, 29, 36colcom 25166 . . . . . . . . . 10 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → (𝐸 ∈ (𝐹𝐿𝐷) ∨ 𝐹 = 𝐷))
382, 4, 3, 27, 30, 28, 29, 37colrot1 25167 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
392, 3, 13, 27, 28, 29, 30, 31, 32, 33, 35, 4, 38cgracol 25432 . . . . . . . 8 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
4026adantr 479 . . . . . . . 8 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
4139, 40pm2.65da 597 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ¬ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹))
42 eqid 2604 . . . . . . . . . 10 (cgrG‘𝐺) = (cgrG‘𝐺)
4317ad2antrr 757 . . . . . . . . . . . . 13 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
44 simprl 789 . . . . . . . . . . . . 13 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓((hlG‘𝐺)‘𝐸)𝐹)
452, 3, 21, 6, 24, 25, 23, 10, 12, 8, 43, 22, 44cgrahl2 25422 . . . . . . . . . . . 12 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑓”⟩)
462, 3, 21, 5, 14, 15, 16, 9, 11, 7, 17cgrane1 25417 . . . . . . . . . . . . . 14 (𝜑𝐴𝐵)
472, 3, 21, 14, 14, 15, 5, 46hlid 25217 . . . . . . . . . . . . 13 (𝜑𝐴((hlG‘𝐺)‘𝐵)𝐴)
4847ad2antrr 757 . . . . . . . . . . . 12 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐴((hlG‘𝐺)‘𝐵)𝐴)
492, 3, 21, 5, 14, 15, 16, 9, 11, 7, 17cgrane2 25418 . . . . . . . . . . . . . . 15 (𝜑𝐵𝐶)
5049necomd 2831 . . . . . . . . . . . . . 14 (𝜑𝐶𝐵)
512, 3, 21, 16, 14, 15, 5, 50hlid 25217 . . . . . . . . . . . . 13 (𝜑𝐶((hlG‘𝐺)‘𝐵)𝐶)
5251ad2antrr 757 . . . . . . . . . . . 12 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐶((hlG‘𝐺)‘𝐵)𝐶)
53 tgasa.2 . . . . . . . . . . . . . 14 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
542, 13, 3, 5, 14, 15, 9, 11, 53tgcgrcomlr 25087 . . . . . . . . . . . . 13 (𝜑 → (𝐵 𝐴) = (𝐸 𝐷))
5554ad2antrr 757 . . . . . . . . . . . 12 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐵 𝐴) = (𝐸 𝐷))
561eqcomd 2610 . . . . . . . . . . . 12 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐵 𝐶) = (𝐸 𝑓))
572, 3, 21, 6, 24, 25, 23, 10, 12, 22, 45, 24, 13, 23, 48, 52, 55, 56cgracgr 25423 . . . . . . . . . . 11 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐴 𝐶) = (𝐷 𝑓))
582, 13, 3, 6, 24, 23, 10, 22, 57tgcgrcomlr 25087 . . . . . . . . . 10 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐶 𝐴) = (𝑓 𝐷))
5953ad2antrr 757 . . . . . . . . . 10 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐴 𝐵) = (𝐷 𝐸))
602, 13, 42, 6, 23, 24, 25, 22, 10, 12, 58, 59, 56trgcgr 25124 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrG‘𝐺)⟨“𝑓𝐷𝐸”⟩)
612, 3, 4, 5, 16, 14, 15, 18ncolne1 25233 . . . . . . . . . . . 12 (𝜑𝐶𝐴)
6261ad2antrr 757 . . . . . . . . . . 11 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐶𝐴)
632, 13, 3, 6, 23, 24, 22, 10, 58, 62tgcgrneq 25090 . . . . . . . . . 10 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓𝐷)
642, 3, 21, 22, 8, 10, 6, 63hlid 25217 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓((hlG‘𝐺)‘𝐷)𝑓)
652, 3, 21, 5, 9, 11, 7, 14, 15, 16, 34cgrane1 25417 . . . . . . . . . . . 12 (𝜑𝐷𝐸)
6665necomd 2831 . . . . . . . . . . 11 (𝜑𝐸𝐷)
672, 3, 21, 11, 14, 9, 5, 66hlid 25217 . . . . . . . . . 10 (𝜑𝐸((hlG‘𝐺)‘𝐷)𝐸)
6867ad2antrr 757 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐸((hlG‘𝐺)‘𝐷)𝐸)
692, 3, 21, 6, 23, 24, 25, 22, 10, 12, 22, 12, 60, 64, 68iscgrad 25416 . . . . . . . 8 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝑓𝐷𝐸”⟩)
7065ad2antrr 757 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐷𝐸)
712, 3, 6, 21, 22, 10, 12, 63, 70cgraswap 25425 . . . . . . . 8 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝑓𝐷𝐸”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝑓”⟩)
722, 3, 6, 21, 23, 24, 25, 22, 10, 12, 69, 12, 10, 22, 71cgratr 25428 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝑓”⟩)
73 tgasa.4 . . . . . . . . 9 (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)
742, 3, 21, 5, 16, 14, 15, 7, 9, 11, 73cgrane3 25419 . . . . . . . . . . 11 (𝜑𝐷𝐹)
7574necomd 2831 . . . . . . . . . 10 (𝜑𝐹𝐷)
762, 3, 5, 21, 7, 9, 11, 75, 65cgraswap 25425 . . . . . . . . 9 (𝜑 → ⟨“𝐹𝐷𝐸”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
772, 3, 5, 21, 16, 14, 15, 7, 9, 11, 73, 11, 9, 7, 76cgratr 25428 . . . . . . . 8 (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
7877ad2antrr 757 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
792, 3, 4, 5, 11, 9, 66tgelrnln 25238 . . . . . . . . 9 (𝜑 → (𝐸𝐿𝐷) ∈ ran 𝐿)
8079ad2antrr 757 . . . . . . . 8 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐸𝐿𝐷) ∈ ran 𝐿)
81 simpl 471 . . . . . . . . . . . 12 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑎 = 𝑢)
82 eqidd 2605 . . . . . . . . . . . 12 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑃 ∖ (𝐸𝐿𝐷)) = (𝑃 ∖ (𝐸𝐿𝐷)))
8381, 82eleq12d 2676 . . . . . . . . . . 11 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ↔ 𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷))))
84 simpr 475 . . . . . . . . . . . 12 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑏 = 𝑣)
8584, 82eleq12d 2676 . . . . . . . . . . 11 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ↔ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷))))
8683, 85anbi12d 742 . . . . . . . . . 10 ((𝑎 = 𝑢𝑏 = 𝑣) → ((𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷))) ↔ (𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷)))))
87 simpr 475 . . . . . . . . . . . 12 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑡 = 𝑤)
88 simpll 785 . . . . . . . . . . . . 13 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑎 = 𝑢)
89 simplr 787 . . . . . . . . . . . . 13 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑏 = 𝑣)
9088, 89oveq12d 6540 . . . . . . . . . . . 12 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → (𝑎𝐼𝑏) = (𝑢𝐼𝑣))
9187, 90eleq12d 2676 . . . . . . . . . . 11 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑤 ∈ (𝑢𝐼𝑣)))
9291cbvrexdva 3148 . . . . . . . . . 10 ((𝑎 = 𝑢𝑏 = 𝑣) → (∃𝑡 ∈ (𝐸𝐿𝐷)𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑤 ∈ (𝐸𝐿𝐷)𝑤 ∈ (𝑢𝐼𝑣)))
9386, 92anbi12d 742 . . . . . . . . 9 ((𝑎 = 𝑢𝑏 = 𝑣) → (((𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷))) ∧ ∃𝑡 ∈ (𝐸𝐿𝐷)𝑡 ∈ (𝑎𝐼𝑏)) ↔ ((𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷))) ∧ ∃𝑤 ∈ (𝐸𝐿𝐷)𝑤 ∈ (𝑢𝐼𝑣))))
9493cbvopabv 4643 . . . . . . . 8 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷))) ∧ ∃𝑡 ∈ (𝐸𝐿𝐷)𝑡 ∈ (𝑎𝐼𝑏))} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷))) ∧ ∃𝑤 ∈ (𝐸𝐿𝐷)𝑤 ∈ (𝑢𝐼𝑣))}
952, 3, 4, 5, 11, 9, 66tglinerflx1 25241 . . . . . . . . . 10 (𝜑𝐸 ∈ (𝐸𝐿𝐷))
9695ad2antrr 757 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐸 ∈ (𝐸𝐿𝐷))
972, 4, 3, 5, 9, 11, 7, 19ncolcom 25169 . . . . . . . . . . 11 (𝜑 → ¬ (𝐹 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
98 pm2.45 410 . . . . . . . . . . 11 (¬ (𝐹 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷) → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
9997, 98syl 17 . . . . . . . . . 10 (𝜑 → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
10099ad2antrr 757 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
1012, 3, 21, 22, 8, 12, 6, 44hlcomd 25212 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐹((hlG‘𝐺)‘𝐸)𝑓)
1022, 3, 4, 6, 80, 12, 94, 21, 96, 8, 22, 100, 101hphl 25376 . . . . . . . 8 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐹((hpG‘𝐺)‘(𝐸𝐿𝐷))𝑓)
1032, 3, 4, 6, 80, 8, 94, 22, 102hpgcom 25372 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓((hpG‘𝐺)‘(𝐸𝐿𝐷))𝐹)
1042, 3, 4, 5, 79, 7, 94, 99hpgid 25371 . . . . . . . 8 (𝜑𝐹((hpG‘𝐺)‘(𝐸𝐿𝐷))𝐹)
105104ad2antrr 757 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐹((hpG‘𝐺)‘(𝐸𝐿𝐷))𝐹)
1062, 3, 13, 6, 23, 24, 25, 12, 10, 8, 4, 26, 41, 22, 8, 21, 72, 78, 103, 105acopyeu 25438 . . . . . 6 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓((hlG‘𝐺)‘𝐷)𝐹)
1072, 3, 21, 22, 8, 10, 6, 4, 106hlln 25215 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓 ∈ (𝐹𝐿𝐷))
1082, 3, 4, 5, 7, 9, 75tglinerflx1 25241 . . . . . 6 (𝜑𝐹 ∈ (𝐹𝐿𝐷))
109108ad2antrr 757 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐹 ∈ (𝐹𝐿𝐷))
1102, 3, 21, 5, 14, 15, 16, 9, 11, 7, 17cgrane4 25420 . . . . . . 7 (𝜑𝐸𝐹)
111110ad2antrr 757 . . . . . 6 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐸𝐹)
1122, 3, 21, 22, 8, 12, 6, 4, 44hlln 25215 . . . . . 6 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓 ∈ (𝐹𝐿𝐸))
1132, 3, 4, 6, 12, 8, 22, 111, 112lncom 25230 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓 ∈ (𝐸𝐿𝐹))
1142, 3, 4, 6, 12, 8, 111tglinerflx2 25242 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐹 ∈ (𝐸𝐿𝐹))
1152, 3, 4, 6, 8, 10, 12, 8, 20, 107, 109, 113, 114tglineinteq 25253 . . . 4 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓 = 𝐹)
116115oveq2d 6538 . . 3 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐸 𝑓) = (𝐸 𝐹))
1171, 116eqtr3d 2640 . 2 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐵 𝐶) = (𝐸 𝐹))
118110necomd 2831 . . 3 (𝜑𝐹𝐸)
1192, 3, 21, 11, 15, 16, 5, 7, 13, 118, 49hlcgrex 25224 . 2 (𝜑 → ∃𝑓𝑃 (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶)))
120117, 119r19.29a 3054 1 (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 381  wa 382   = wceq 1474  wcel 1975  wne 2774  wrex 2891  cdif 3531   class class class wbr 4572  {copab 4631  ran crn 5024  cfv 5785  (class class class)co 6522  ⟨“cs3 13379  Basecbs 15636  distcds 15718  TarskiGcstrkg 25041  Itvcitv 25047  LineGclng 25048  cgrGccgrg 25118  hlGchlg 25208  hpGchpg 25362  cgrAccgra 25412
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 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-cnex 9843  ax-resscn 9844  ax-1cn 9845  ax-icn 9846  ax-addcl 9847  ax-addrcl 9848  ax-mulcl 9849  ax-mulrcl 9850  ax-mulcom 9851  ax-addass 9852  ax-mulass 9853  ax-distr 9854  ax-i2m1 9855  ax-1ne0 9856  ax-1rid 9857  ax-rnegex 9858  ax-rrecex 9859  ax-cnre 9860  ax-pre-lttri 9861  ax-pre-lttrn 9862  ax-pre-ltadd 9863  ax-pre-mulgt0 9864
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 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-nel 2777  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-om 6930  df-1st 7031  df-2nd 7032  df-wrecs 7266  df-recs 7327  df-rdg 7365  df-1o 7419  df-oadd 7423  df-er 7601  df-map 7718  df-pm 7719  df-en 7814  df-dom 7815  df-sdom 7816  df-fin 7817  df-card 8620  df-cda 8845  df-pnf 9927  df-mnf 9928  df-xr 9929  df-ltxr 9930  df-le 9931  df-sub 10114  df-neg 10115  df-nn 10863  df-2 10921  df-3 10922  df-n0 11135  df-z 11206  df-uz 11515  df-fz 12148  df-fzo 12285  df-hash 12930  df-word 13095  df-concat 13097  df-s1 13098  df-s2 13385  df-s3 13386  df-trkgc 25059  df-trkgb 25060  df-trkgcb 25061  df-trkgld 25063  df-trkg 25064  df-cgrg 25119  df-leg 25191  df-hlg 25209  df-mir 25261  df-rag 25302  df-perpg 25304  df-hpg 25363  df-mid 25379  df-lmi 25380  df-cgra 25413
This theorem is referenced by:  tgasa  25453
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