MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgasa1 Structured version   Visualization version   GIF version

Theorem tgasa1 27800
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 771 . . 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 724 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐺 ∈ TarskiG)
7 tgsas.f . . . . . 6 (𝜑𝐹𝑃)
87ad2antrr 724 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐹𝑃)
9 tgsas.d . . . . . 6 (𝜑𝐷𝑃)
109ad2antrr 724 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐷𝑃)
11 tgsas.e . . . . . 6 (𝜑𝐸𝑃)
1211ad2antrr 724 . . . . 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 27771 . . . . . 6 (𝜑 → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
2019ad2antrr 724 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
21 eqid 2736 . . . . . 6 (hlG‘𝐺) = (hlG‘𝐺)
22 simplr 767 . . . . . 6 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓𝑃)
2316ad2antrr 724 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐶𝑃)
2414ad2antrr 724 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐴𝑃)
2515ad2antrr 724 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐵𝑃)
2618ad2antrr 724 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
275ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐺 ∈ TarskiG)
289ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐷𝑃)
2911ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐸𝑃)
307ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐹𝑃)
3114ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐴𝑃)
3215ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐵𝑃)
3316ad3antrrr 728 . . . . . . . . 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 27764 . . . . . . . . . 10 (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
3534ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
36 simpr 485 . . . . . . . . . . 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 27500 . . . . . . . . . 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 27501 . . . . . . . . 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 27770 . . . . . . . 8 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
4018ad3antrrr 728 . . . . . . . 8 ((((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
4139, 40pm2.65da 815 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ¬ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹))
42 eqid 2736 . . . . . . . . . 10 (cgrG‘𝐺) = (cgrG‘𝐺)
4317ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
44 simprl 769 . . . . . . . . . . . . 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 27759 . . . . . . . . . . . 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 27754 . . . . . . . . . . . . . 14 (𝜑𝐴𝐵)
472, 3, 21, 14, 14, 15, 5, 46hlid 27551 . . . . . . . . . . . . 13 (𝜑𝐴((hlG‘𝐺)‘𝐵)𝐴)
4847ad2antrr 724 . . . . . . . . . . . 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 27755 . . . . . . . . . . . . . . 15 (𝜑𝐵𝐶)
5049necomd 2999 . . . . . . . . . . . . . 14 (𝜑𝐶𝐵)
512, 3, 21, 16, 14, 15, 5, 50hlid 27551 . . . . . . . . . . . . 13 (𝜑𝐶((hlG‘𝐺)‘𝐵)𝐶)
5251ad2antrr 724 . . . . . . . . . . . 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 27422 . . . . . . . . . . . . 13 (𝜑 → (𝐵 𝐴) = (𝐸 𝐷))
5554ad2antrr 724 . . . . . . . . . . . 12 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐵 𝐴) = (𝐸 𝐷))
561eqcomd 2742 . . . . . . . . . . . 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 27760 . . . . . . . . . . 11 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐴 𝐶) = (𝐷 𝑓))
582, 13, 3, 6, 24, 23, 10, 22, 57tgcgrcomlr 27422 . . . . . . . . . 10 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐶 𝐴) = (𝑓 𝐷))
5953ad2antrr 724 . . . . . . . . . 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 27458 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrG‘𝐺)⟨“𝑓𝐷𝐸”⟩)
612, 3, 4, 5, 16, 14, 15, 18ncolne1 27567 . . . . . . . . . . . 12 (𝜑𝐶𝐴)
6261ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐶𝐴)
632, 13, 3, 6, 23, 24, 22, 10, 58, 62tgcgrneq 27425 . . . . . . . . . 10 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓𝐷)
642, 3, 21, 22, 8, 10, 6, 63hlid 27551 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓((hlG‘𝐺)‘𝐷)𝑓)
65 tgasa.4 . . . . . . . . . . . . 13 (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)
662, 3, 21, 5, 16, 14, 15, 7, 9, 11, 65cgrane4 27757 . . . . . . . . . . . 12 (𝜑𝐷𝐸)
6766necomd 2999 . . . . . . . . . . 11 (𝜑𝐸𝐷)
682, 3, 21, 11, 14, 9, 5, 67hlid 27551 . . . . . . . . . 10 (𝜑𝐸((hlG‘𝐺)‘𝐷)𝐸)
6968ad2antrr 724 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐸((hlG‘𝐺)‘𝐷)𝐸)
702, 3, 21, 6, 23, 24, 25, 22, 10, 12, 22, 12, 60, 64, 69iscgrad 27753 . . . . . . . 8 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝑓𝐷𝐸”⟩)
7166ad2antrr 724 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐷𝐸)
722, 3, 6, 21, 22, 10, 12, 63, 71cgraswap 27762 . . . . . . . 8 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝑓𝐷𝐸”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝑓”⟩)
732, 3, 6, 21, 23, 24, 25, 22, 10, 12, 70, 12, 10, 22, 72cgratr 27765 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝑓”⟩)
742, 3, 21, 5, 16, 14, 15, 7, 9, 11, 65cgrane3 27756 . . . . . . . . . . 11 (𝜑𝐷𝐹)
7574necomd 2999 . . . . . . . . . 10 (𝜑𝐹𝐷)
762, 3, 5, 21, 7, 9, 11, 75, 66cgraswap 27762 . . . . . . . . 9 (𝜑 → ⟨“𝐹𝐷𝐸”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
772, 3, 5, 21, 16, 14, 15, 7, 9, 11, 65, 11, 9, 7, 76cgratr 27765 . . . . . . . 8 (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
7877ad2antrr 724 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
792, 3, 4, 5, 11, 9, 67tgelrnln 27572 . . . . . . . . 9 (𝜑 → (𝐸𝐿𝐷) ∈ ran 𝐿)
8079ad2antrr 724 . . . . . . . 8 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐸𝐿𝐷) ∈ ran 𝐿)
81 simpl 483 . . . . . . . . . . . 12 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑎 = 𝑢)
8281eleq1d 2822 . . . . . . . . . . 11 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ↔ 𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷))))
83 simpr 485 . . . . . . . . . . . 12 ((𝑎 = 𝑢𝑏 = 𝑣) → 𝑏 = 𝑣)
8483eleq1d 2822 . . . . . . . . . . 11 ((𝑎 = 𝑢𝑏 = 𝑣) → (𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ↔ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷))))
8582, 84anbi12d 631 . . . . . . . . . 10 ((𝑎 = 𝑢𝑏 = 𝑣) → ((𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷))) ↔ (𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷)))))
86 simpr 485 . . . . . . . . . . . 12 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑡 = 𝑤)
87 simpll 765 . . . . . . . . . . . . 13 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑎 = 𝑢)
88 simplr 767 . . . . . . . . . . . . 13 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑏 = 𝑣)
8987, 88oveq12d 7375 . . . . . . . . . . . 12 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → (𝑎𝐼𝑏) = (𝑢𝐼𝑣))
9086, 89eleq12d 2832 . . . . . . . . . . 11 (((𝑎 = 𝑢𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑤 ∈ (𝑢𝐼𝑣)))
9190cbvrexdva 3327 . . . . . . . . . 10 ((𝑎 = 𝑢𝑏 = 𝑣) → (∃𝑡 ∈ (𝐸𝐿𝐷)𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑤 ∈ (𝐸𝐿𝐷)𝑤 ∈ (𝑢𝐼𝑣)))
9285, 91anbi12d 631 . . . . . . . . 9 ((𝑎 = 𝑢𝑏 = 𝑣) → (((𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷))) ∧ ∃𝑡 ∈ (𝐸𝐿𝐷)𝑡 ∈ (𝑎𝐼𝑏)) ↔ ((𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷))) ∧ ∃𝑤 ∈ (𝐸𝐿𝐷)𝑤 ∈ (𝑢𝐼𝑣))))
9392cbvopabv 5178 . . . . . . . 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))}
942, 3, 4, 5, 11, 9, 67tglinerflx1 27575 . . . . . . . . . 10 (𝜑𝐸 ∈ (𝐸𝐿𝐷))
9594ad2antrr 724 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐸 ∈ (𝐸𝐿𝐷))
962, 4, 3, 5, 9, 11, 7, 19ncolcom 27503 . . . . . . . . . . 11 (𝜑 → ¬ (𝐹 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
97 pm2.45 880 . . . . . . . . . . 11 (¬ (𝐹 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷) → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
9896, 97syl 17 . . . . . . . . . 10 (𝜑 → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
9998ad2antrr 724 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
1002, 3, 21, 22, 8, 12, 6, 44hlcomd 27546 . . . . . . . . 9 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐹((hlG‘𝐺)‘𝐸)𝑓)
1012, 3, 4, 6, 80, 12, 93, 21, 95, 8, 22, 99, 100hphl 27713 . . . . . . . 8 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐹((hpG‘𝐺)‘(𝐸𝐿𝐷))𝑓)
1022, 3, 4, 6, 80, 8, 93, 22, 101hpgcom 27709 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓((hpG‘𝐺)‘(𝐸𝐿𝐷))𝐹)
1032, 3, 4, 5, 79, 7, 93, 98hpgid 27708 . . . . . . . 8 (𝜑𝐹((hpG‘𝐺)‘(𝐸𝐿𝐷))𝐹)
104103ad2antrr 724 . . . . . . 7 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐹((hpG‘𝐺)‘(𝐸𝐿𝐷))𝐹)
1052, 3, 13, 6, 23, 24, 25, 12, 10, 8, 4, 26, 41, 22, 8, 21, 73, 78, 102, 104acopyeu 27776 . . . . . 6 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓((hlG‘𝐺)‘𝐷)𝐹)
1062, 3, 21, 22, 8, 10, 6, 4, 105hlln 27549 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓 ∈ (𝐹𝐿𝐷))
1072, 3, 4, 5, 7, 9, 75tglinerflx1 27575 . . . . . 6 (𝜑𝐹 ∈ (𝐹𝐿𝐷))
108107ad2antrr 724 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐹 ∈ (𝐹𝐿𝐷))
1092, 3, 21, 5, 14, 15, 16, 9, 11, 7, 17cgrane4 27757 . . . . . . 7 (𝜑𝐸𝐹)
110109ad2antrr 724 . . . . . 6 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐸𝐹)
1112, 3, 21, 22, 8, 12, 6, 4, 44hlln 27549 . . . . . 6 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓 ∈ (𝐹𝐿𝐸))
1122, 3, 4, 6, 12, 8, 22, 110, 111lncom 27564 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓 ∈ (𝐸𝐿𝐹))
1132, 3, 4, 6, 12, 8, 110tglinerflx2 27576 . . . . 5 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝐹 ∈ (𝐸𝐿𝐹))
1142, 3, 4, 6, 8, 10, 12, 8, 20, 106, 108, 112, 113tglineinteq 27587 . . . 4 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → 𝑓 = 𝐹)
115114oveq2d 7373 . . 3 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐸 𝑓) = (𝐸 𝐹))
1161, 115eqtr3d 2778 . 2 (((𝜑𝑓𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶))) → (𝐵 𝐶) = (𝐸 𝐹))
117109necomd 2999 . . 3 (𝜑𝐹𝐸)
1182, 3, 21, 11, 15, 16, 5, 7, 13, 117, 49hlcgrex 27558 . 2 (𝜑 → ∃𝑓𝑃 (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 𝑓) = (𝐵 𝐶)))
119116, 118r19.29a 3159 1 (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 845   = wceq 1541  wcel 2106  wne 2943  wrex 3073  cdif 3907   class class class wbr 5105  {copab 5167  ran crn 5634  cfv 6496  (class class class)co 7357  ⟨“cs3 14731  Basecbs 17083  distcds 17142  TarskiGcstrkg 27369  Itvcitv 27375  LineGclng 27376  cgrGccgrg 27452  hlGchlg 27542  hpGchpg 27699  cgrAccgra 27749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-concat 14459  df-s1 14484  df-s2 14737  df-s3 14738  df-trkgc 27390  df-trkgb 27391  df-trkgcb 27392  df-trkgld 27394  df-trkg 27395  df-cgrg 27453  df-leg 27525  df-hlg 27543  df-mir 27595  df-rag 27636  df-perpg 27638  df-hpg 27700  df-mid 27716  df-lmi 27717  df-cgra 27750
This theorem is referenced by:  tgasa  27801
  Copyright terms: Public domain W3C validator