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Theorem hypcgrlem2 26686
 Description: Lemma for hypcgr 26687, case where triangles share one vertex 𝐵. (Contributed by Thierry Arnoux, 16-Dec-2019.)
Hypotheses
Ref Expression
hypcgr.p 𝑃 = (Base‘𝐺)
hypcgr.m = (dist‘𝐺)
hypcgr.i 𝐼 = (Itv‘𝐺)
hypcgr.g (𝜑𝐺 ∈ TarskiG)
hypcgr.h (𝜑𝐺DimTarskiG≥2)
hypcgr.a (𝜑𝐴𝑃)
hypcgr.b (𝜑𝐵𝑃)
hypcgr.c (𝜑𝐶𝑃)
hypcgr.d (𝜑𝐷𝑃)
hypcgr.e (𝜑𝐸𝑃)
hypcgr.f (𝜑𝐹𝑃)
hypcgr.1 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
hypcgr.2 (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
hypcgr.3 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
hypcgr.4 (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))
hypcgrlem2.b (𝜑𝐵 = 𝐸)
hypcgrlem2.s 𝑆 = ((lInvG‘𝐺)‘((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
Assertion
Ref Expression
hypcgrlem2 (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))

Proof of Theorem hypcgrlem2
StepHypRef Expression
1 hypcgr.p . . . 4 𝑃 = (Base‘𝐺)
2 hypcgr.m . . . 4 = (dist‘𝐺)
3 hypcgr.i . . . 4 𝐼 = (Itv‘𝐺)
4 hypcgr.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
54adantr 485 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐺 ∈ TarskiG)
6 hypcgr.h . . . . 5 (𝜑𝐺DimTarskiG≥2)
76adantr 485 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐺DimTarskiG≥2)
8 hypcgr.a . . . . 5 (𝜑𝐴𝑃)
98adantr 485 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐴𝑃)
10 hypcgr.b . . . . 5 (𝜑𝐵𝑃)
1110adantr 485 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐵𝑃)
12 hypcgr.c . . . . 5 (𝜑𝐶𝑃)
1312adantr 485 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐶𝑃)
14 eqid 2759 . . . . 5 (LineG‘𝐺) = (LineG‘𝐺)
15 eqid 2759 . . . . 5 (pInvG‘𝐺) = (pInvG‘𝐺)
16 eqid 2759 . . . . 5 ((pInvG‘𝐺)‘𝐵) = ((pInvG‘𝐺)‘𝐵)
17 hypcgr.d . . . . . 6 (𝜑𝐷𝑃)
1817adantr 485 . . . . 5 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐷𝑃)
191, 2, 3, 14, 15, 5, 11, 16, 18mircl 26547 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (((pInvG‘𝐺)‘𝐵)‘𝐷) ∈ 𝑃)
20 hypcgr.e . . . . 5 (𝜑𝐸𝑃)
2120adantr 485 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐸𝑃)
22 hypcgr.1 . . . . 5 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
2322adantr 485 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
24 eqidd 2760 . . . . . 6 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (((pInvG‘𝐺)‘𝐵)‘𝐷) = (((pInvG‘𝐺)‘𝐵)‘𝐷))
25 hypcgrlem2.b . . . . . . . . 9 (𝜑𝐵 = 𝐸)
2625adantr 485 . . . . . . . 8 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐵 = 𝐸)
271, 2, 3, 14, 15, 5, 11, 16, 21mirinv 26552 . . . . . . . 8 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐸) = 𝐸𝐵 = 𝐸))
2826, 27mpbird 260 . . . . . . 7 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (((pInvG‘𝐺)‘𝐵)‘𝐸) = 𝐸)
2928eqcomd 2765 . . . . . 6 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐸 = (((pInvG‘𝐺)‘𝐵)‘𝐸))
30 hypcgr.f . . . . . . . . . 10 (𝜑𝐹𝑃)
3130adantr 485 . . . . . . . . 9 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐹𝑃)
321, 2, 3, 5, 7, 13, 31midcom 26668 . . . . . . . 8 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐶(midG‘𝐺)𝐹) = (𝐹(midG‘𝐺)𝐶))
33 simpr 489 . . . . . . . 8 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐶(midG‘𝐺)𝐹) = 𝐵)
3432, 33eqtr3d 2796 . . . . . . 7 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐹(midG‘𝐺)𝐶) = 𝐵)
351, 2, 3, 5, 7, 31, 13, 15, 11ismidb 26664 . . . . . . 7 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐶 = (((pInvG‘𝐺)‘𝐵)‘𝐹) ↔ (𝐹(midG‘𝐺)𝐶) = 𝐵))
3634, 35mpbird 260 . . . . . 6 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐶 = (((pInvG‘𝐺)‘𝐵)‘𝐹))
3724, 29, 36s3eqd 14266 . . . . 5 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝐷)𝐸𝐶”⟩ = ⟨“(((pInvG‘𝐺)‘𝐵)‘𝐷)(((pInvG‘𝐺)‘𝐵)‘𝐸)(((pInvG‘𝐺)‘𝐵)‘𝐹)”⟩)
38 hypcgr.2 . . . . . . 7 (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
3938adantr 485 . . . . . 6 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
401, 2, 3, 14, 15, 5, 18, 21, 31, 39, 16, 11mirrag 26587 . . . . 5 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝐷)(((pInvG‘𝐺)‘𝐵)‘𝐸)(((pInvG‘𝐺)‘𝐵)‘𝐹)”⟩ ∈ (∟G‘𝐺))
4137, 40eqeltrd 2853 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝐷)𝐸𝐶”⟩ ∈ (∟G‘𝐺))
42 hypcgr.3 . . . . . 6 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
4342adantr 485 . . . . 5 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 𝐵) = (𝐷 𝐸))
441, 2, 3, 14, 15, 5, 11, 16, 18, 21miriso 26556 . . . . 5 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) (((pInvG‘𝐺)‘𝐵)‘𝐸)) = (𝐷 𝐸))
4528oveq2d 7167 . . . . 5 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) (((pInvG‘𝐺)‘𝐵)‘𝐸)) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) 𝐸))
4643, 44, 453eqtr2d 2800 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 𝐵) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) 𝐸))
4726oveq1d 7166 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐵 𝐶) = (𝐸 𝐶))
48 eqid 2759 . . . 4 ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)(((pInvG‘𝐺)‘𝐵)‘𝐷))(LineG‘𝐺)𝐵)) = ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)(((pInvG‘𝐺)‘𝐵)‘𝐷))(LineG‘𝐺)𝐵))
49 eqidd 2760 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐶 = 𝐶)
501, 2, 3, 5, 7, 9, 11, 13, 19, 21, 13, 23, 41, 46, 47, 26, 48, 49hypcgrlem1 26685 . . 3 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 𝐶) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) 𝐶))
5136oveq2d 7167 . . 3 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) 𝐶) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) (((pInvG‘𝐺)‘𝐵)‘𝐹)))
521, 2, 3, 14, 15, 5, 11, 16, 18, 31miriso 26556 . . 3 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) (((pInvG‘𝐺)‘𝐵)‘𝐹)) = (𝐷 𝐹))
5350, 51, 523eqtrd 2798 . 2 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 𝐶) = (𝐷 𝐹))
544ad2antrr 726 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐺 ∈ TarskiG)
556ad2antrr 726 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐺DimTarskiG≥2)
568ad2antrr 726 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐴𝑃)
5710ad2antrr 726 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐵𝑃)
5812ad2antrr 726 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐶𝑃)
5917ad2antrr 726 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐷𝑃)
6020ad2antrr 726 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐸𝑃)
6130ad2antrr 726 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐹𝑃)
6222ad2antrr 726 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
6338ad2antrr 726 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
6442ad2antrr 726 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → (𝐴 𝐵) = (𝐷 𝐸))
65 hypcgr.4 . . . . 5 (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))
6665ad2antrr 726 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → (𝐵 𝐶) = (𝐸 𝐹))
6725ad2antrr 726 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐵 = 𝐸)
68 eqid 2759 . . . 4 ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵)) = ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵))
69 simpr 489 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐶 = 𝐹)
701, 2, 3, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69hypcgrlem1 26685 . . 3 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → (𝐴 𝐶) = (𝐷 𝐹))
714ad2antrr 726 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐺 ∈ TarskiG)
726ad2antrr 726 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐺DimTarskiG≥2)
738ad2antrr 726 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐴𝑃)
7410ad2antrr 726 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐵𝑃)
7512ad2antrr 726 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐶𝑃)
76 hypcgrlem2.s . . . . . 6 𝑆 = ((lInvG‘𝐺)‘((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
7730ad2antrr 726 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐹𝑃)
781, 2, 3, 71, 72, 75, 77midcl 26663 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ 𝑃)
79 simplr 769 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ≠ 𝐵)
801, 3, 14, 71, 78, 74, 79tgelrnln 26516 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵) ∈ ran (LineG‘𝐺))
8117ad2antrr 726 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐷𝑃)
821, 2, 3, 71, 72, 76, 14, 80, 81lmicl 26672 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝑆𝐷) ∈ 𝑃)
8320ad2antrr 726 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐸𝑃)
841, 2, 3, 71, 72, 76, 14, 80, 83lmicl 26672 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝑆𝐸) ∈ 𝑃)
851, 2, 3, 71, 72, 76, 14, 80, 77lmicl 26672 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝑆𝐹) ∈ 𝑃)
8622ad2antrr 726 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
871, 2, 3, 71, 72, 76, 14, 80lmimot 26684 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝑆 ∈ (𝐺Ismt𝐺))
8838ad2antrr 726 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
891, 2, 3, 14, 15, 71, 81, 83, 77, 87, 88motrag 26594 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ⟨“(𝑆𝐷)(𝑆𝐸)(𝑆𝐹)”⟩ ∈ (∟G‘𝐺))
9042ad2antrr 726 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐴 𝐵) = (𝐷 𝐸))
911, 2, 3, 71, 72, 76, 14, 80, 81, 83lmiiso 26683 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝑆𝐷) (𝑆𝐸)) = (𝐷 𝐸))
9290, 91eqtr4d 2797 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐴 𝐵) = ((𝑆𝐷) (𝑆𝐸)))
9365ad2antrr 726 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐵 𝐶) = (𝐸 𝐹))
941, 2, 3, 71, 72, 76, 14, 80, 83, 77lmiiso 26683 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝑆𝐸) (𝑆𝐹)) = (𝐸 𝐹))
9593, 94eqtr4d 2797 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐵 𝐶) = ((𝑆𝐸) (𝑆𝐹)))
961, 3, 14, 71, 78, 74, 79tglinerflx2 26520 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐵 ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
971, 2, 3, 71, 72, 76, 14, 80, 74, 96lmicinv 26679 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝑆𝐵) = 𝐵)
9825ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐵 = 𝐸)
9998fveq2d 6663 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝑆𝐵) = (𝑆𝐸))
10097, 99eqtr3d 2796 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐵 = (𝑆𝐸))
101 eqid 2759 . . . . 5 ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)(𝑆𝐷))(LineG‘𝐺)𝐵)) = ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)(𝑆𝐷))(LineG‘𝐺)𝐵))
1021, 2, 3, 71, 72, 75, 77midcom 26668 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) = (𝐹(midG‘𝐺)𝐶))
1031, 3, 14, 71, 78, 74, 79tglinerflx1 26519 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
104102, 103eqeltrrd 2854 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐹(midG‘𝐺)𝐶) ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
105 simpr 489 . . . . . . . . . 10 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐶𝐹)
106105necomd 3007 . . . . . . . . 9 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐹𝐶)
1071, 3, 14, 71, 77, 75, 106tgelrnln 26516 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐹(LineG‘𝐺)𝐶) ∈ ran (LineG‘𝐺))
1081, 2, 3, 71, 72, 75, 77midbtwn 26665 . . . . . . . . . . 11 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (𝐶𝐼𝐹))
1091, 2, 3, 71, 75, 78, 77, 108tgbtwncom 26374 . . . . . . . . . 10 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (𝐹𝐼𝐶))
1101, 3, 14, 71, 77, 75, 78, 106, 109btwnlng1 26505 . . . . . . . . 9 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (𝐹(LineG‘𝐺)𝐶))
111103, 110elind 4100 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵) ∩ (𝐹(LineG‘𝐺)𝐶)))
1121, 3, 14, 71, 77, 75, 106tglinerflx2 26520 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐶 ∈ (𝐹(LineG‘𝐺)𝐶))
11379necomd 3007 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐵 ≠ (𝐶(midG‘𝐺)𝐹))
1144ad2antrr 726 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐺 ∈ TarskiG)
11512ad2antrr 726 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐶𝑃)
11630ad2antrr 726 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐹𝑃)
1176ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐺DimTarskiG≥2)
118 simpr 489 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐶 = (𝐶(midG‘𝐺)𝐹))
119118eqcomd 2765 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → (𝐶(midG‘𝐺)𝐹) = 𝐶)
1201, 2, 3, 114, 117, 115, 116, 119midcgr 26666 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → (𝐶 𝐶) = (𝐶 𝐹))
121120eqcomd 2765 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → (𝐶 𝐹) = (𝐶 𝐶))
1221, 2, 3, 114, 115, 116, 115, 121axtgcgrid 26349 . . . . . . . . . . 11 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐶 = 𝐹)
123122ex 417 . . . . . . . . . 10 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐶 = (𝐶(midG‘𝐺)𝐹) → 𝐶 = 𝐹))
124123necon3d 2973 . . . . . . . . 9 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐶𝐹𝐶 ≠ (𝐶(midG‘𝐺)𝐹)))
125124imp 411 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐶 ≠ (𝐶(midG‘𝐺)𝐹))
12698eqcomd 2765 . . . . . . . . . . 11 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐸 = 𝐵)
127 eqidd 2760 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) = (𝐶(midG‘𝐺)𝐹))
1281, 2, 3, 71, 72, 75, 77, 15, 78ismidb 26664 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐹 = (((pInvG‘𝐺)‘(𝐶(midG‘𝐺)𝐹))‘𝐶) ↔ (𝐶(midG‘𝐺)𝐹) = (𝐶(midG‘𝐺)𝐹)))
129127, 128mpbird 260 . . . . . . . . . . 11 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐹 = (((pInvG‘𝐺)‘(𝐶(midG‘𝐺)𝐹))‘𝐶))
130126, 129oveq12d 7169 . . . . . . . . . 10 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐸 𝐹) = (𝐵 (((pInvG‘𝐺)‘(𝐶(midG‘𝐺)𝐹))‘𝐶)))
13193, 130eqtrd 2794 . . . . . . . . 9 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐵 𝐶) = (𝐵 (((pInvG‘𝐺)‘(𝐶(midG‘𝐺)𝐹))‘𝐶)))
1321, 2, 3, 14, 15, 71, 74, 78, 75israg 26583 . . . . . . . . 9 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (⟨“𝐵(𝐶(midG‘𝐺)𝐹)𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 𝐶) = (𝐵 (((pInvG‘𝐺)‘(𝐶(midG‘𝐺)𝐹))‘𝐶))))
133131, 132mpbird 260 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ⟨“𝐵(𝐶(midG‘𝐺)𝐹)𝐶”⟩ ∈ (∟G‘𝐺))
1341, 2, 3, 14, 71, 80, 107, 111, 96, 112, 113, 125, 133ragperp 26603 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵)(⟂G‘𝐺)(𝐹(LineG‘𝐺)𝐶))
135134orcd 871 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵)(⟂G‘𝐺)(𝐹(LineG‘𝐺)𝐶) ∨ 𝐹 = 𝐶))
1361, 2, 3, 71, 72, 76, 14, 80, 77, 75islmib 26673 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶 = (𝑆𝐹) ↔ ((𝐹(midG‘𝐺)𝐶) ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵) ∧ (((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵)(⟂G‘𝐺)(𝐹(LineG‘𝐺)𝐶) ∨ 𝐹 = 𝐶))))
137104, 135, 136mpbir2and 713 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐶 = (𝑆𝐹))
1381, 2, 3, 71, 72, 73, 74, 75, 82, 84, 85, 86, 89, 92, 95, 100, 101, 137hypcgrlem1 26685 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐴 𝐶) = ((𝑆𝐷) (𝑆𝐹)))
1391, 2, 3, 71, 72, 76, 14, 80, 81, 77lmiiso 26683 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝑆𝐷) (𝑆𝐹)) = (𝐷 𝐹))
140138, 139eqtrd 2794 . . 3 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐴 𝐶) = (𝐷 𝐹))
14170, 140pm2.61dane 3039 . 2 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐴 𝐶) = (𝐷 𝐹))
14253, 141pm2.61dane 3039 1 (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 845   = wceq 1539   ∈ wcel 2112   ≠ wne 2952   class class class wbr 5033  ‘cfv 6336  (class class class)co 7151  2c2 11722  ⟨“cs3 14244  Basecbs 16534  distcds 16625  TarskiGcstrkg 26316  DimTarskiG≥cstrkgld 26320  Itvcitv 26322  LineGclng 26323  pInvGcmir 26538  ∟Gcrag 26579  ⟂Gcperpg 26581  midGcmid 26658  lInvGclmi 26659 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-cnex 10624  ax-resscn 10625  ax-1cn 10626  ax-icn 10627  ax-addcl 10628  ax-addrcl 10629  ax-mulcl 10630  ax-mulrcl 10631  ax-mulcom 10632  ax-addass 10633  ax-mulass 10634  ax-distr 10635  ax-i2m1 10636  ax-1ne0 10637  ax-1rid 10638  ax-rnegex 10639  ax-rrecex 10640  ax-cnre 10641  ax-pre-lttri 10642  ax-pre-lttrn 10643  ax-pre-ltadd 10644  ax-pre-mulgt0 10645 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7581  df-1st 7694  df-2nd 7695  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-1o 8113  df-oadd 8117  df-er 8300  df-map 8419  df-pm 8420  df-en 8529  df-dom 8530  df-sdom 8531  df-fin 8532  df-dju 9356  df-card 9394  df-pnf 10708  df-mnf 10709  df-xr 10710  df-ltxr 10711  df-le 10712  df-sub 10903  df-neg 10904  df-nn 11668  df-2 11730  df-3 11731  df-n0 11928  df-xnn0 12000  df-z 12014  df-uz 12276  df-fz 12933  df-fzo 13076  df-hash 13734  df-word 13907  df-concat 13963  df-s1 13990  df-s2 14250  df-s3 14251  df-trkgc 26334  df-trkgb 26335  df-trkgcb 26336  df-trkgld 26338  df-trkg 26339  df-cgrg 26397  df-ismt 26419  df-leg 26469  df-mir 26539  df-rag 26580  df-perpg 26582  df-mid 26660  df-lmi 26661 This theorem is referenced by:  hypcgr  26687
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