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Theorem tgasa1 26211
 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 763 . . 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 716 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐺 ∈ TarskiG)
7 tgsas.f . . . . . 6 (𝜑𝐹𝑃)
87ad2antrr 716 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐹𝑃)
9 tgsas.d . . . . . 6 (𝜑𝐷𝑃)
109ad2antrr 716 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐷𝑃)
11 tgsas.e . . . . . 6 (𝜑𝐸𝑃)
1211ad2antrr 716 . . . . 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 26181 . . . . . 6 (𝜑 → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
2019ad2antrr 716 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
21 eqid 2778 . . . . . 6 (hlG‘𝐺) = (hlG‘𝐺)
22 simplr 759 . . . . . 6 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓𝑃)
2316ad2antrr 716 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐶𝑃)
2414ad2antrr 716 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐴𝑃)
2515ad2antrr 716 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐵𝑃)
2618ad2antrr 716 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
276adantr 474 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐺 ∈ TarskiG)
2810adantr 474 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐷𝑃)
2912adantr 474 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐸𝑃)
308adantr 474 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐹𝑃)
3124adantr 474 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐴𝑃)
3225adantr 474 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐵𝑃)
3323adantr 474 . . . . . . . . 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 26174 . . . . . . . . . 10 (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
3534ad3antrrr 720 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
36 simpr 479 . . . . . . . . . . 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 25913 . . . . . . . . . 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 25914 . . . . . . . . 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 26180 . . . . . . . 8 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
4026adantr 474 . . . . . . . 8 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
4139, 40pm2.65da 807 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ¬ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹))
42 eqid 2778 . . . . . . . . . 10 (cgrG‘𝐺) = (cgrG‘𝐺)
4317ad2antrr 716 . . . . . . . . . . . . 13 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
44 simprl 761 . . . . . . . . . . . . 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 26169 . . . . . . . . . . . 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 26164 . . . . . . . . . . . . . 14 (𝜑𝐴𝐵)
472, 3, 21, 14, 14, 15, 5, 46hlid 25964 . . . . . . . . . . . . 13 (𝜑𝐴((hlG‘𝐺)‘𝐵)𝐴)
4847ad2antrr 716 . . . . . . . . . . . 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 26165 . . . . . . . . . . . . . . 15 (𝜑𝐵𝐶)
5049necomd 3024 . . . . . . . . . . . . . 14 (𝜑𝐶𝐵)
512, 3, 21, 16, 14, 15, 5, 50hlid 25964 . . . . . . . . . . . . 13 (𝜑𝐶((hlG‘𝐺)‘𝐵)𝐶)
5251ad2antrr 716 . . . . . . . . . . . 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 25835 . . . . . . . . . . . . 13 (𝜑 → (𝐵 𝐴) = (𝐸 𝐷))
5554ad2antrr 716 . . . . . . . . . . . 12 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐵 𝐴) = (𝐸 𝐷))
561eqcomd 2784 . . . . . . . . . . . 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 26170 . . . . . . . . . . 11 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐴 𝐶) = (𝐷 𝑓))
582, 13, 3, 6, 24, 23, 10, 22, 57tgcgrcomlr 25835 . . . . . . . . . 10 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐶 𝐴) = (𝑓 𝐷))
5953ad2antrr 716 . . . . . . . . . 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 25871 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrG‘𝐺)⟨“𝑓𝐷𝐸”⟩)
612, 3, 4, 5, 16, 14, 15, 18ncolne1 25980 . . . . . . . . . . . 12 (𝜑𝐶𝐴)
6261ad2antrr 716 . . . . . . . . . . 11 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐶𝐴)
632, 13, 3, 6, 23, 24, 22, 10, 58, 62tgcgrneq 25838 . . . . . . . . . 10 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓𝐷)
642, 3, 21, 22, 8, 10, 6, 63hlid 25964 . . . . . . . . 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 26164 . . . . . . . . . . . 12 (𝜑𝐷𝐸)
6665necomd 3024 . . . . . . . . . . 11 (𝜑𝐸𝐷)
672, 3, 21, 11, 14, 9, 5, 66hlid 25964 . . . . . . . . . 10 (𝜑𝐸((hlG‘𝐺)‘𝐷)𝐸)
6867ad2antrr 716 . . . . . . . . 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 26163 . . . . . . . 8 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝑓𝐷𝐸”⟩)
7065ad2antrr 716 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐷𝐸)
712, 3, 6, 21, 22, 10, 12, 63, 70cgraswap 26172 . . . . . . . 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 26175 . . . . . . 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 26166 . . . . . . . . . . 11 (𝜑𝐷𝐹)
7574necomd 3024 . . . . . . . . . 10 (𝜑𝐹𝐷)
762, 3, 5, 21, 7, 9, 11, 75, 65cgraswap 26172 . . . . . . . . 9 (𝜑 → ⟨“𝐹𝐷𝐸”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
772, 3, 5, 21, 16, 14, 15, 7, 9, 11, 73, 11, 9, 7, 76cgratr 26175 . . . . . . . 8 (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
7877ad2antrr 716 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
792, 3, 4, 5, 11, 9, 66tgelrnln 25985 . . . . . . . . 9 (𝜑 → (𝐸𝐿𝐷) ∈ ran 𝐿)
8079ad2antrr 716 . . . . . . . 8 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐸𝐿𝐷) ∈ ran 𝐿)
81 simpl 476 . . . . . . . . . . . 12 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑎 = 𝑢)
82 eqidd 2779 . . . . . . . . . . . 12 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑃 ∖ (𝐸𝐿𝐷)) = (𝑃 ∖ (𝐸𝐿𝐷)))
8381, 82eleq12d 2853 . . . . . . . . . . 11 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ↔ 𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷))))
84 simpr 479 . . . . . . . . . . . 12 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑏 = 𝑣)
8584, 82eleq12d 2853 . . . . . . . . . . 11 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ↔ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷))))
8683, 85anbi12d 624 . . . . . . . . . 10 ((𝑎 = 𝑢𝑏 = 𝑣) → ((𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷))) ↔ (𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷)))))
87 simpr 479 . . . . . . . . . . . 12 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑡 = 𝑤)
88 simpll 757 . . . . . . . . . . . . 13 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑎 = 𝑢)
89 simplr 759 . . . . . . . . . . . . 13 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑏 = 𝑣)
9088, 89oveq12d 6942 . . . . . . . . . . . 12 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → (𝑎𝐼𝑏) = (𝑢𝐼𝑣))
9187, 90eleq12d 2853 . . . . . . . . . . 11 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑤 ∈ (𝑢𝐼𝑣)))
9291cbvrexdva 3374 . . . . . . . . . 10 ((𝑎 = 𝑢𝑏 = 𝑣) → (∃𝑡 ∈ (𝐸𝐿𝐷)𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑤 ∈ (𝐸𝐿𝐷)𝑤 ∈ (𝑢𝐼𝑣)))
9386, 92anbi12d 624 . . . . . . . . 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 4960 . . . . . . . 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 25988 . . . . . . . . . 10 (𝜑𝐸 ∈ (𝐸𝐿𝐷))
9695ad2antrr 716 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐸 ∈ (𝐸𝐿𝐷))
972, 4, 3, 5, 9, 11, 7, 19ncolcom 25916 . . . . . . . . . . 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 716 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
1012, 3, 21, 22, 8, 12, 6, 44hlcomd 25959 . . . . . . . . 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 26123 . . . . . . . 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 26119 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓((hpG‘𝐺)‘(𝐸𝐿𝐷))𝐹)
1042, 3, 4, 5, 79, 7, 94, 99hpgid 26118 . . . . . . . 8 (𝜑𝐹((hpG‘𝐺)‘(𝐸𝐿𝐷))𝐹)
105104ad2antrr 716 . . . . . . 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 26187 . . . . . 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 25962 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓 ∈ (𝐹𝐿𝐷))
1082, 3, 4, 5, 7, 9, 75tglinerflx1 25988 . . . . . 6 (𝜑𝐹 ∈ (𝐹𝐿𝐷))
109108ad2antrr 716 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐹 ∈ (𝐹𝐿𝐷))
1102, 3, 21, 5, 14, 15, 16, 9, 11, 7, 17cgrane4 26167 . . . . . . 7 (𝜑𝐸𝐹)
111110ad2antrr 716 . . . . . 6 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐸𝐹)
1122, 3, 21, 22, 8, 12, 6, 4, 44hlln 25962 . . . . . 6 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓 ∈ (𝐹𝐿𝐸))
1132, 3, 4, 6, 12, 8, 22, 111, 112lncom 25977 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓 ∈ (𝐸𝐿𝐹))
1142, 3, 4, 6, 12, 8, 111tglinerflx2 25989 . . . . 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 26000 . . . 4 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓 = 𝐹)
116115oveq2d 6940 . . 3 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐸 𝑓) = (𝐸 𝐹))
1171, 116eqtr3d 2816 . 2 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐵 𝐶) = (𝐸 𝐹))
118110necomd 3024 . . 3 (𝜑𝐹𝐸)
1192, 3, 21, 11, 15, 16, 5, 7, 13, 118, 49hlcgrex 25971 . 2 (𝜑 → ∃𝑓𝑃 (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶)))
120117, 119r19.29a 3264 1 (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   ∨ wo 836   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ∃wrex 3091   ∖ cdif 3789   class class class wbr 4888  {copab 4950  ran crn 5358  ‘cfv 6137  (class class class)co 6924  ⟨“cs3 13997  Basecbs 16259  distcds 16351  TarskiGcstrkg 25785  Itvcitv 25791  LineGclng 25792  cgrGccgrg 25865  hlGchlg 25955  hpGchpg 26109  cgrAccgra 26159 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-er 8028  df-map 8144  df-pm 8145  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-card 9100  df-cda 9327  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-nn 11379  df-2 11442  df-3 11443  df-n0 11647  df-xnn0 11719  df-z 11733  df-uz 11997  df-fz 12648  df-fzo 12789  df-hash 13440  df-word 13604  df-concat 13665  df-s1 13690  df-s2 14003  df-s3 14004  df-trkgc 25803  df-trkgb 25804  df-trkgcb 25805  df-trkgld 25807  df-trkg 25808  df-cgrg 25866  df-leg 25938  df-hlg 25956  df-mir 26008  df-rag 26049  df-perpg 26051  df-hpg 26110  df-mid 26126  df-lmi 26127  df-cgra 26160 This theorem is referenced by:  tgasa  26212
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