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Theorem tgasa1 25960
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 756 . . 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 705 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐺 ∈ TarskiG)
7 tgsas.f . . . . . 6 (𝜑𝐹𝑃)
87ad2antrr 705 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐹𝑃)
9 tgsas.d . . . . . 6 (𝜑𝐷𝑃)
109ad2antrr 705 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐷𝑃)
11 tgsas.e . . . . . 6 (𝜑𝐸𝑃)
1211ad2antrr 705 . . . . 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 25941 . . . . . 6 (𝜑 → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
2019ad2antrr 705 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
21 eqid 2771 . . . . . 6 (hlG‘𝐺) = (hlG‘𝐺)
22 simplr 752 . . . . . 6 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓𝑃)
2316ad2antrr 705 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐶𝑃)
2414ad2antrr 705 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐴𝑃)
2515ad2antrr 705 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐵𝑃)
2618ad2antrr 705 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
276adantr 466 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐺 ∈ TarskiG)
2810adantr 466 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐷𝑃)
2912adantr 466 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐸𝑃)
308adantr 466 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐹𝑃)
3124adantr 466 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐴𝑃)
3225adantr 466 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐵𝑃)
3323adantr 466 . . . . . . . . 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 25935 . . . . . . . . . 10 (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
3534ad3antrrr 709 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
36 simpr 471 . . . . . . . . . . 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 25674 . . . . . . . . . 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 25675 . . . . . . . . 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 25940 . . . . . . . 8 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
4026adantr 466 . . . . . . . 8 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
4139, 40pm2.65da 818 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ¬ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹))
42 eqid 2771 . . . . . . . . . 10 (cgrG‘𝐺) = (cgrG‘𝐺)
4317ad2antrr 705 . . . . . . . . . . . . 13 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
44 simprl 754 . . . . . . . . . . . . 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 25930 . . . . . . . . . . . 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 25925 . . . . . . . . . . . . . 14 (𝜑𝐴𝐵)
472, 3, 21, 14, 14, 15, 5, 46hlid 25725 . . . . . . . . . . . . 13 (𝜑𝐴((hlG‘𝐺)‘𝐵)𝐴)
4847ad2antrr 705 . . . . . . . . . . . 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 25926 . . . . . . . . . . . . . . 15 (𝜑𝐵𝐶)
5049necomd 2998 . . . . . . . . . . . . . 14 (𝜑𝐶𝐵)
512, 3, 21, 16, 14, 15, 5, 50hlid 25725 . . . . . . . . . . . . 13 (𝜑𝐶((hlG‘𝐺)‘𝐵)𝐶)
5251ad2antrr 705 . . . . . . . . . . . 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 25596 . . . . . . . . . . . . 13 (𝜑 → (𝐵 𝐴) = (𝐸 𝐷))
5554ad2antrr 705 . . . . . . . . . . . 12 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐵 𝐴) = (𝐸 𝐷))
561eqcomd 2777 . . . . . . . . . . . 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 25931 . . . . . . . . . . 11 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐴 𝐶) = (𝐷 𝑓))
582, 13, 3, 6, 24, 23, 10, 22, 57tgcgrcomlr 25596 . . . . . . . . . 10 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐶 𝐴) = (𝑓 𝐷))
5953ad2antrr 705 . . . . . . . . . 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 25632 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrG‘𝐺)⟨“𝑓𝐷𝐸”⟩)
612, 3, 4, 5, 16, 14, 15, 18ncolne1 25741 . . . . . . . . . . . 12 (𝜑𝐶𝐴)
6261ad2antrr 705 . . . . . . . . . . 11 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐶𝐴)
632, 13, 3, 6, 23, 24, 22, 10, 58, 62tgcgrneq 25599 . . . . . . . . . 10 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓𝐷)
642, 3, 21, 22, 8, 10, 6, 63hlid 25725 . . . . . . . . 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 25925 . . . . . . . . . . . 12 (𝜑𝐷𝐸)
6665necomd 2998 . . . . . . . . . . 11 (𝜑𝐸𝐷)
672, 3, 21, 11, 14, 9, 5, 66hlid 25725 . . . . . . . . . 10 (𝜑𝐸((hlG‘𝐺)‘𝐷)𝐸)
6867ad2antrr 705 . . . . . . . . 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 25924 . . . . . . . 8 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝑓𝐷𝐸”⟩)
7065ad2antrr 705 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐷𝐸)
712, 3, 6, 21, 22, 10, 12, 63, 70cgraswap 25933 . . . . . . . 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 25936 . . . . . . 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 25927 . . . . . . . . . . 11 (𝜑𝐷𝐹)
7574necomd 2998 . . . . . . . . . 10 (𝜑𝐹𝐷)
762, 3, 5, 21, 7, 9, 11, 75, 65cgraswap 25933 . . . . . . . . 9 (𝜑 → ⟨“𝐹𝐷𝐸”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
772, 3, 5, 21, 16, 14, 15, 7, 9, 11, 73, 11, 9, 7, 76cgratr 25936 . . . . . . . 8 (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
7877ad2antrr 705 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
792, 3, 4, 5, 11, 9, 66tgelrnln 25746 . . . . . . . . 9 (𝜑 → (𝐸𝐿𝐷) ∈ ran 𝐿)
8079ad2antrr 705 . . . . . . . 8 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐸𝐿𝐷) ∈ ran 𝐿)
81 simpl 468 . . . . . . . . . . . 12 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑎 = 𝑢)
82 eqidd 2772 . . . . . . . . . . . 12 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑃 ∖ (𝐸𝐿𝐷)) = (𝑃 ∖ (𝐸𝐿𝐷)))
8381, 82eleq12d 2844 . . . . . . . . . . 11 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ↔ 𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷))))
84 simpr 471 . . . . . . . . . . . 12 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑏 = 𝑣)
8584, 82eleq12d 2844 . . . . . . . . . . 11 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ↔ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷))))
8683, 85anbi12d 616 . . . . . . . . . 10 ((𝑎 = 𝑢𝑏 = 𝑣) → ((𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷))) ↔ (𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷)))))
87 simpr 471 . . . . . . . . . . . 12 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑡 = 𝑤)
88 simpll 750 . . . . . . . . . . . . 13 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑎 = 𝑢)
89 simplr 752 . . . . . . . . . . . . 13 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑏 = 𝑣)
9088, 89oveq12d 6811 . . . . . . . . . . . 12 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → (𝑎𝐼𝑏) = (𝑢𝐼𝑣))
9187, 90eleq12d 2844 . . . . . . . . . . 11 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑤 ∈ (𝑢𝐼𝑣)))
9291cbvrexdva 3327 . . . . . . . . . 10 ((𝑎 = 𝑢𝑏 = 𝑣) → (∃𝑡 ∈ (𝐸𝐿𝐷)𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑤 ∈ (𝐸𝐿𝐷)𝑤 ∈ (𝑢𝐼𝑣)))
9386, 92anbi12d 616 . . . . . . . . 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 4856 . . . . . . . 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 25749 . . . . . . . . . 10 (𝜑𝐸 ∈ (𝐸𝐿𝐷))
9695ad2antrr 705 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐸 ∈ (𝐸𝐿𝐷))
972, 4, 3, 5, 9, 11, 7, 19ncolcom 25677 . . . . . . . . . . 11 (𝜑 → ¬ (𝐹 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
98 pm2.45 868 . . . . . . . . . . 11 (¬ (𝐹 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷) → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
9997, 98syl 17 . . . . . . . . . 10 (𝜑 → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
10099ad2antrr 705 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
1012, 3, 21, 22, 8, 12, 6, 44hlcomd 25720 . . . . . . . . 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 25884 . . . . . . . 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 25880 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓((hpG‘𝐺)‘(𝐸𝐿𝐷))𝐹)
1042, 3, 4, 5, 79, 7, 94, 99hpgid 25879 . . . . . . . 8 (𝜑𝐹((hpG‘𝐺)‘(𝐸𝐿𝐷))𝐹)
105104ad2antrr 705 . . . . . . 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 25946 . . . . . 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 25723 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓 ∈ (𝐹𝐿𝐷))
1082, 3, 4, 5, 7, 9, 75tglinerflx1 25749 . . . . . 6 (𝜑𝐹 ∈ (𝐹𝐿𝐷))
109108ad2antrr 705 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐹 ∈ (𝐹𝐿𝐷))
1102, 3, 21, 5, 14, 15, 16, 9, 11, 7, 17cgrane4 25928 . . . . . . 7 (𝜑𝐸𝐹)
111110ad2antrr 705 . . . . . 6 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐸𝐹)
1122, 3, 21, 22, 8, 12, 6, 4, 44hlln 25723 . . . . . 6 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓 ∈ (𝐹𝐿𝐸))
1132, 3, 4, 6, 12, 8, 22, 111, 112lncom 25738 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓 ∈ (𝐸𝐿𝐹))
1142, 3, 4, 6, 12, 8, 111tglinerflx2 25750 . . . . 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 25761 . . . 4 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓 = 𝐹)
116115oveq2d 6809 . . 3 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐸 𝑓) = (𝐸 𝐹))
1171, 116eqtr3d 2807 . 2 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐵 𝐶) = (𝐸 𝐹))
118110necomd 2998 . . 3 (𝜑𝐹𝐸)
1192, 3, 21, 11, 15, 16, 5, 7, 13, 118, 49hlcgrex 25732 . 2 (𝜑 → ∃𝑓𝑃 (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶)))
120117, 119r19.29a 3226 1 (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wo 836   = wceq 1631  wcel 2145  wne 2943  wrex 3062  cdif 3720   class class class wbr 4786  {copab 4846  ran crn 5250  cfv 6031  (class class class)co 6793  ⟨“cs3 13796  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556  LineGclng 25557  cgrGccgrg 25626  hlGchlg 25716  hpGchpg 25870  cgrAccgra 25920
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-fal 1637  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-map 8011  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-trkgld 25572  df-trkg 25573  df-cgrg 25627  df-leg 25699  df-hlg 25717  df-mir 25769  df-rag 25810  df-perpg 25812  df-hpg 25871  df-mid 25887  df-lmi 25888  df-cgra 25921
This theorem is referenced by:  tgasa  25961
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