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Theorem tgasa1 25720
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 795 . . 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 761 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐺 ∈ TarskiG)
7 tgsas.f . . . . . 6 (𝜑𝐹𝑃)
87ad2antrr 761 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐹𝑃)
9 tgsas.d . . . . . 6 (𝜑𝐷𝑃)
109ad2antrr 761 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐷𝑃)
11 tgsas.e . . . . . 6 (𝜑𝐸𝑃)
1211ad2antrr 761 . . . . 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 25701 . . . . . 6 (𝜑 → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
2019ad2antrr 761 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
21 eqid 2620 . . . . . 6 (hlG‘𝐺) = (hlG‘𝐺)
22 simplr 791 . . . . . 6 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓𝑃)
2316ad2antrr 761 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐶𝑃)
2414ad2antrr 761 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐴𝑃)
2515ad2antrr 761 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐵𝑃)
2618ad2antrr 761 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
276adantr 481 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐺 ∈ TarskiG)
2810adantr 481 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐷𝑃)
2912adantr 481 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐸𝑃)
308adantr 481 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐹𝑃)
3124adantr 481 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐴𝑃)
3225adantr 481 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐵𝑃)
3323adantr 481 . . . . . . . . 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 25695 . . . . . . . . . 10 (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
3534ad3antrrr 765 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
36 simpr 477 . . . . . . . . . . 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 25434 . . . . . . . . . 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 25435 . . . . . . . . 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 25700 . . . . . . . 8 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
4026adantr 481 . . . . . . . 8 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
4139, 40pm2.65da 599 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ¬ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹))
42 eqid 2620 . . . . . . . . . 10 (cgrG‘𝐺) = (cgrG‘𝐺)
4317ad2antrr 761 . . . . . . . . . . . . 13 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
44 simprl 793 . . . . . . . . . . . . 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 25690 . . . . . . . . . . . 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 25685 . . . . . . . . . . . . . 14 (𝜑𝐴𝐵)
472, 3, 21, 14, 14, 15, 5, 46hlid 25485 . . . . . . . . . . . . 13 (𝜑𝐴((hlG‘𝐺)‘𝐵)𝐴)
4847ad2antrr 761 . . . . . . . . . . . 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 25686 . . . . . . . . . . . . . . 15 (𝜑𝐵𝐶)
5049necomd 2846 . . . . . . . . . . . . . 14 (𝜑𝐶𝐵)
512, 3, 21, 16, 14, 15, 5, 50hlid 25485 . . . . . . . . . . . . 13 (𝜑𝐶((hlG‘𝐺)‘𝐵)𝐶)
5251ad2antrr 761 . . . . . . . . . . . 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 25356 . . . . . . . . . . . . 13 (𝜑 → (𝐵 𝐴) = (𝐸 𝐷))
5554ad2antrr 761 . . . . . . . . . . . 12 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐵 𝐴) = (𝐸 𝐷))
561eqcomd 2626 . . . . . . . . . . . 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 25691 . . . . . . . . . . 11 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐴 𝐶) = (𝐷 𝑓))
582, 13, 3, 6, 24, 23, 10, 22, 57tgcgrcomlr 25356 . . . . . . . . . 10 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐶 𝐴) = (𝑓 𝐷))
5953ad2antrr 761 . . . . . . . . . 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 25392 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrG‘𝐺)⟨“𝑓𝐷𝐸”⟩)
612, 3, 4, 5, 16, 14, 15, 18ncolne1 25501 . . . . . . . . . . . 12 (𝜑𝐶𝐴)
6261ad2antrr 761 . . . . . . . . . . 11 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐶𝐴)
632, 13, 3, 6, 23, 24, 22, 10, 58, 62tgcgrneq 25359 . . . . . . . . . 10 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓𝐷)
642, 3, 21, 22, 8, 10, 6, 63hlid 25485 . . . . . . . . 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 25685 . . . . . . . . . . . 12 (𝜑𝐷𝐸)
6665necomd 2846 . . . . . . . . . . 11 (𝜑𝐸𝐷)
672, 3, 21, 11, 14, 9, 5, 66hlid 25485 . . . . . . . . . 10 (𝜑𝐸((hlG‘𝐺)‘𝐷)𝐸)
6867ad2antrr 761 . . . . . . . . 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 25684 . . . . . . . 8 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝑓𝐷𝐸”⟩)
7065ad2antrr 761 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐷𝐸)
712, 3, 6, 21, 22, 10, 12, 63, 70cgraswap 25693 . . . . . . . 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 25696 . . . . . . 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 25687 . . . . . . . . . . 11 (𝜑𝐷𝐹)
7574necomd 2846 . . . . . . . . . 10 (𝜑𝐹𝐷)
762, 3, 5, 21, 7, 9, 11, 75, 65cgraswap 25693 . . . . . . . . 9 (𝜑 → ⟨“𝐹𝐷𝐸”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
772, 3, 5, 21, 16, 14, 15, 7, 9, 11, 73, 11, 9, 7, 76cgratr 25696 . . . . . . . 8 (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
7877ad2antrr 761 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
792, 3, 4, 5, 11, 9, 66tgelrnln 25506 . . . . . . . . 9 (𝜑 → (𝐸𝐿𝐷) ∈ ran 𝐿)
8079ad2antrr 761 . . . . . . . 8 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐸𝐿𝐷) ∈ ran 𝐿)
81 simpl 473 . . . . . . . . . . . 12 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑎 = 𝑢)
82 eqidd 2621 . . . . . . . . . . . 12 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑃 ∖ (𝐸𝐿𝐷)) = (𝑃 ∖ (𝐸𝐿𝐷)))
8381, 82eleq12d 2693 . . . . . . . . . . 11 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ↔ 𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷))))
84 simpr 477 . . . . . . . . . . . 12 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑏 = 𝑣)
8584, 82eleq12d 2693 . . . . . . . . . . 11 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ↔ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷))))
8683, 85anbi12d 746 . . . . . . . . . 10 ((𝑎 = 𝑢𝑏 = 𝑣) → ((𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷))) ↔ (𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷)))))
87 simpr 477 . . . . . . . . . . . 12 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑡 = 𝑤)
88 simpll 789 . . . . . . . . . . . . 13 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑎 = 𝑢)
89 simplr 791 . . . . . . . . . . . . 13 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑏 = 𝑣)
9088, 89oveq12d 6653 . . . . . . . . . . . 12 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → (𝑎𝐼𝑏) = (𝑢𝐼𝑣))
9187, 90eleq12d 2693 . . . . . . . . . . 11 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑤 ∈ (𝑢𝐼𝑣)))
9291cbvrexdva 3173 . . . . . . . . . 10 ((𝑎 = 𝑢𝑏 = 𝑣) → (∃𝑡 ∈ (𝐸𝐿𝐷)𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑤 ∈ (𝐸𝐿𝐷)𝑤 ∈ (𝑢𝐼𝑣)))
9386, 92anbi12d 746 . . . . . . . . 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 4713 . . . . . . . 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 25509 . . . . . . . . . 10 (𝜑𝐸 ∈ (𝐸𝐿𝐷))
9695ad2antrr 761 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐸 ∈ (𝐸𝐿𝐷))
972, 4, 3, 5, 9, 11, 7, 19ncolcom 25437 . . . . . . . . . . 11 (𝜑 → ¬ (𝐹 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
98 pm2.45 412 . . . . . . . . . . 11 (¬ (𝐹 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷) → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
9997, 98syl 17 . . . . . . . . . 10 (𝜑 → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
10099ad2antrr 761 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
1012, 3, 21, 22, 8, 12, 6, 44hlcomd 25480 . . . . . . . . 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 25644 . . . . . . . 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 25640 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓((hpG‘𝐺)‘(𝐸𝐿𝐷))𝐹)
1042, 3, 4, 5, 79, 7, 94, 99hpgid 25639 . . . . . . . 8 (𝜑𝐹((hpG‘𝐺)‘(𝐸𝐿𝐷))𝐹)
105104ad2antrr 761 . . . . . . 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 25706 . . . . . 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 25483 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓 ∈ (𝐹𝐿𝐷))
1082, 3, 4, 5, 7, 9, 75tglinerflx1 25509 . . . . . 6 (𝜑𝐹 ∈ (𝐹𝐿𝐷))
109108ad2antrr 761 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐹 ∈ (𝐹𝐿𝐷))
1102, 3, 21, 5, 14, 15, 16, 9, 11, 7, 17cgrane4 25688 . . . . . . 7 (𝜑𝐸𝐹)
111110ad2antrr 761 . . . . . 6 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐸𝐹)
1122, 3, 21, 22, 8, 12, 6, 4, 44hlln 25483 . . . . . 6 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓 ∈ (𝐹𝐿𝐸))
1132, 3, 4, 6, 12, 8, 22, 111, 112lncom 25498 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓 ∈ (𝐸𝐿𝐹))
1142, 3, 4, 6, 12, 8, 111tglinerflx2 25510 . . . . 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 25521 . . . 4 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓 = 𝐹)
116115oveq2d 6651 . . 3 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐸 𝑓) = (𝐸 𝐹))
1171, 116eqtr3d 2656 . 2 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐵 𝐶) = (𝐸 𝐹))
118110necomd 2846 . . 3 (𝜑𝐹𝐸)
1192, 3, 21, 11, 15, 16, 5, 7, 13, 118, 49hlcgrex 25492 . 2 (𝜑 → ∃𝑓𝑃 (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶)))
120117, 119r19.29a 3074 1 (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384   = wceq 1481  wcel 1988  wne 2791  wrex 2910  cdif 3564   class class class wbr 4644  {copab 4703  ran crn 5105  cfv 5876  (class class class)co 6635  ⟨“cs3 13568  Basecbs 15838  distcds 15931  TarskiGcstrkg 25310  Itvcitv 25316  LineGclng 25317  cgrGccgrg 25386  hlGchlg 25476  hpGchpg 25630  cgrAccgra 25680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-pm 7845  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-3 11065  df-n0 11278  df-xnn0 11349  df-z 11363  df-uz 11673  df-fz 12312  df-fzo 12450  df-hash 13101  df-word 13282  df-concat 13284  df-s1 13285  df-s2 13574  df-s3 13575  df-trkgc 25328  df-trkgb 25329  df-trkgcb 25330  df-trkgld 25332  df-trkg 25333  df-cgrg 25387  df-leg 25459  df-hlg 25477  df-mir 25529  df-rag 25570  df-perpg 25572  df-hpg 25631  df-mid 25647  df-lmi 25648  df-cgra 25681
This theorem is referenced by:  tgasa  25721
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