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Theorem hypcgrlem2 25492
 Description: Lemma for hypcgr 25493, 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 480 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐺 ∈ TarskiG)
6 hypcgr.h . . . . 5 (𝜑𝐺DimTarskiG≥2)
76adantr 480 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐺DimTarskiG≥2)
8 hypcgr.a . . . . 5 (𝜑𝐴𝑃)
98adantr 480 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐴𝑃)
10 hypcgr.b . . . . 5 (𝜑𝐵𝑃)
1110adantr 480 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐵𝑃)
12 hypcgr.c . . . . 5 (𝜑𝐶𝑃)
1312adantr 480 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐶𝑃)
14 eqid 2610 . . . . 5 (LineG‘𝐺) = (LineG‘𝐺)
15 eqid 2610 . . . . 5 (pInvG‘𝐺) = (pInvG‘𝐺)
16 eqid 2610 . . . . 5 ((pInvG‘𝐺)‘𝐵) = ((pInvG‘𝐺)‘𝐵)
17 hypcgr.d . . . . . 6 (𝜑𝐷𝑃)
1817adantr 480 . . . . 5 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐷𝑃)
191, 2, 3, 14, 15, 5, 11, 16, 18mircl 25356 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (((pInvG‘𝐺)‘𝐵)‘𝐷) ∈ 𝑃)
20 hypcgr.e . . . . 5 (𝜑𝐸𝑃)
2120adantr 480 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐸𝑃)
22 hypcgr.1 . . . . 5 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
2322adantr 480 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
24 eqidd 2611 . . . . . 6 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (((pInvG‘𝐺)‘𝐵)‘𝐷) = (((pInvG‘𝐺)‘𝐵)‘𝐷))
25 hypcgrlem2.b . . . . . . . . 9 (𝜑𝐵 = 𝐸)
2625adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐵 = 𝐸)
271, 2, 3, 14, 15, 5, 11, 16, 21mirinv 25361 . . . . . . . 8 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐸) = 𝐸𝐵 = 𝐸))
2826, 27mpbird 246 . . . . . . 7 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (((pInvG‘𝐺)‘𝐵)‘𝐸) = 𝐸)
2928eqcomd 2616 . . . . . 6 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐸 = (((pInvG‘𝐺)‘𝐵)‘𝐸))
30 hypcgr.f . . . . . . . . . 10 (𝜑𝐹𝑃)
3130adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐹𝑃)
321, 2, 3, 5, 7, 13, 31midcom 25474 . . . . . . . 8 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐶(midG‘𝐺)𝐹) = (𝐹(midG‘𝐺)𝐶))
33 simpr 476 . . . . . . . 8 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐶(midG‘𝐺)𝐹) = 𝐵)
3432, 33eqtr3d 2646 . . . . . . 7 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐹(midG‘𝐺)𝐶) = 𝐵)
351, 2, 3, 5, 7, 31, 13, 15, 11ismidb 25470 . . . . . . 7 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐶 = (((pInvG‘𝐺)‘𝐵)‘𝐹) ↔ (𝐹(midG‘𝐺)𝐶) = 𝐵))
3634, 35mpbird 246 . . . . . 6 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐶 = (((pInvG‘𝐺)‘𝐵)‘𝐹))
3724, 29, 36s3eqd 13460 . . . . 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 480 . . . . . 6 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
401, 2, 3, 14, 15, 5, 18, 21, 31, 39, 16, 11mirrag 25396 . . . . 5 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝐷)(((pInvG‘𝐺)‘𝐵)‘𝐸)(((pInvG‘𝐺)‘𝐵)‘𝐹)”⟩ ∈ (∟G‘𝐺))
4137, 40eqeltrd 2688 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝐷)𝐸𝐶”⟩ ∈ (∟G‘𝐺))
42 hypcgr.3 . . . . . 6 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
4342adantr 480 . . . . 5 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 𝐵) = (𝐷 𝐸))
441, 2, 3, 14, 15, 5, 11, 16, 18, 21miriso 25365 . . . . 5 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) (((pInvG‘𝐺)‘𝐵)‘𝐸)) = (𝐷 𝐸))
4528oveq2d 6565 . . . . 5 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) (((pInvG‘𝐺)‘𝐵)‘𝐸)) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) 𝐸))
4643, 44, 453eqtr2d 2650 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 𝐵) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) 𝐸))
4726oveq1d 6564 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐵 𝐶) = (𝐸 𝐶))
48 eqid 2610 . . . 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 2611 . . . 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 25491 . . 3 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 𝐶) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) 𝐶))
5136oveq2d 6565 . . 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 25365 . . 3 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) (((pInvG‘𝐺)‘𝐵)‘𝐹)) = (𝐷 𝐹))
5350, 51, 523eqtrd 2648 . 2 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 𝐶) = (𝐷 𝐹))
544ad2antrr 758 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐺 ∈ TarskiG)
556ad2antrr 758 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐺DimTarskiG≥2)
568ad2antrr 758 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐴𝑃)
5710ad2antrr 758 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐵𝑃)
5812ad2antrr 758 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐶𝑃)
5917ad2antrr 758 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐷𝑃)
6020ad2antrr 758 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐸𝑃)
6130ad2antrr 758 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐹𝑃)
6222ad2antrr 758 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
6338ad2antrr 758 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
6442ad2antrr 758 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → (𝐴 𝐵) = (𝐷 𝐸))
65 hypcgr.4 . . . . 5 (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))
6665ad2antrr 758 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → (𝐵 𝐶) = (𝐸 𝐹))
6725ad2antrr 758 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐵 = 𝐸)
68 eqid 2610 . . . 4 ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵)) = ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵))
69 simpr 476 . . . 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 25491 . . 3 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → (𝐴 𝐶) = (𝐷 𝐹))
714ad2antrr 758 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐺 ∈ TarskiG)
726ad2antrr 758 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐺DimTarskiG≥2)
738ad2antrr 758 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐴𝑃)
7410ad2antrr 758 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐵𝑃)
7512ad2antrr 758 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐶𝑃)
76 hypcgrlem2.s . . . . . 6 𝑆 = ((lInvG‘𝐺)‘((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
7730ad2antrr 758 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐹𝑃)
781, 2, 3, 71, 72, 75, 77midcl 25469 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ 𝑃)
79 simplr 788 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ≠ 𝐵)
801, 3, 14, 71, 78, 74, 79tgelrnln 25325 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵) ∈ ran (LineG‘𝐺))
8117ad2antrr 758 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐷𝑃)
821, 2, 3, 71, 72, 76, 14, 80, 81lmicl 25478 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝑆𝐷) ∈ 𝑃)
8320ad2antrr 758 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐸𝑃)
841, 2, 3, 71, 72, 76, 14, 80, 83lmicl 25478 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝑆𝐸) ∈ 𝑃)
851, 2, 3, 71, 72, 76, 14, 80, 77lmicl 25478 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝑆𝐹) ∈ 𝑃)
8622ad2antrr 758 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
871, 2, 3, 71, 72, 76, 14, 80lmimot 25490 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝑆 ∈ (𝐺Ismt𝐺))
8838ad2antrr 758 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
891, 2, 3, 14, 15, 71, 81, 83, 77, 87, 88motrag 25403 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ⟨“(𝑆𝐷)(𝑆𝐸)(𝑆𝐹)”⟩ ∈ (∟G‘𝐺))
9042ad2antrr 758 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐴 𝐵) = (𝐷 𝐸))
911, 2, 3, 71, 72, 76, 14, 80, 81, 83lmiiso 25489 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝑆𝐷) (𝑆𝐸)) = (𝐷 𝐸))
9290, 91eqtr4d 2647 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐴 𝐵) = ((𝑆𝐷) (𝑆𝐸)))
9365ad2antrr 758 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐵 𝐶) = (𝐸 𝐹))
941, 2, 3, 71, 72, 76, 14, 80, 83, 77lmiiso 25489 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝑆𝐸) (𝑆𝐹)) = (𝐸 𝐹))
9593, 94eqtr4d 2647 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐵 𝐶) = ((𝑆𝐸) (𝑆𝐹)))
961, 3, 14, 71, 78, 74, 79tglinerflx2 25329 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐵 ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
971, 2, 3, 71, 72, 76, 14, 80, 74lmiinv 25484 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝑆𝐵) = 𝐵𝐵 ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵)))
9896, 97mpbird 246 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝑆𝐵) = 𝐵)
9925ad2antrr 758 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐵 = 𝐸)
10099fveq2d 6107 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝑆𝐵) = (𝑆𝐸))
10198, 100eqtr3d 2646 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐵 = (𝑆𝐸))
102 eqid 2610 . . . . 5 ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)(𝑆𝐷))(LineG‘𝐺)𝐵)) = ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)(𝑆𝐷))(LineG‘𝐺)𝐵))
1031, 2, 3, 71, 72, 75, 77midcom 25474 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) = (𝐹(midG‘𝐺)𝐶))
1041, 3, 14, 71, 78, 74, 79tglinerflx1 25328 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
105103, 104eqeltrrd 2689 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐹(midG‘𝐺)𝐶) ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
106 simpr 476 . . . . . . . . . 10 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐶𝐹)
107106necomd 2837 . . . . . . . . 9 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐹𝐶)
1081, 3, 14, 71, 77, 75, 107tgelrnln 25325 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐹(LineG‘𝐺)𝐶) ∈ ran (LineG‘𝐺))
1091, 2, 3, 71, 72, 75, 77midbtwn 25471 . . . . . . . . . . 11 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (𝐶𝐼𝐹))
1101, 2, 3, 71, 75, 78, 77, 109tgbtwncom 25183 . . . . . . . . . 10 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (𝐹𝐼𝐶))
1111, 3, 14, 71, 77, 75, 78, 107, 110btwnlng1 25314 . . . . . . . . 9 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (𝐹(LineG‘𝐺)𝐶))
112104, 111elind 3760 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵) ∩ (𝐹(LineG‘𝐺)𝐶)))
1131, 3, 14, 71, 77, 75, 107tglinerflx2 25329 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐶 ∈ (𝐹(LineG‘𝐺)𝐶))
11479necomd 2837 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐵 ≠ (𝐶(midG‘𝐺)𝐹))
1154ad2antrr 758 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐺 ∈ TarskiG)
11612ad2antrr 758 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐶𝑃)
11730ad2antrr 758 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐹𝑃)
1186ad2antrr 758 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐺DimTarskiG≥2)
119 simpr 476 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐶 = (𝐶(midG‘𝐺)𝐹))
120119eqcomd 2616 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → (𝐶(midG‘𝐺)𝐹) = 𝐶)
1211, 2, 3, 115, 118, 116, 117, 120midcgr 25472 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → (𝐶 𝐶) = (𝐶 𝐹))
122121eqcomd 2616 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → (𝐶 𝐹) = (𝐶 𝐶))
1231, 2, 3, 115, 116, 117, 116, 122axtgcgrid 25162 . . . . . . . . . . 11 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐶 = 𝐹)
124123ex 449 . . . . . . . . . 10 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐶 = (𝐶(midG‘𝐺)𝐹) → 𝐶 = 𝐹))
125124necon3d 2803 . . . . . . . . 9 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐶𝐹𝐶 ≠ (𝐶(midG‘𝐺)𝐹)))
126125imp 444 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐶 ≠ (𝐶(midG‘𝐺)𝐹))
12799eqcomd 2616 . . . . . . . . . . 11 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐸 = 𝐵)
128 eqidd 2611 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) = (𝐶(midG‘𝐺)𝐹))
1291, 2, 3, 71, 72, 75, 77, 15, 78ismidb 25470 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐹 = (((pInvG‘𝐺)‘(𝐶(midG‘𝐺)𝐹))‘𝐶) ↔ (𝐶(midG‘𝐺)𝐹) = (𝐶(midG‘𝐺)𝐹)))
130128, 129mpbird 246 . . . . . . . . . . 11 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐹 = (((pInvG‘𝐺)‘(𝐶(midG‘𝐺)𝐹))‘𝐶))
131127, 130oveq12d 6567 . . . . . . . . . 10 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐸 𝐹) = (𝐵 (((pInvG‘𝐺)‘(𝐶(midG‘𝐺)𝐹))‘𝐶)))
13293, 131eqtrd 2644 . . . . . . . . 9 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐵 𝐶) = (𝐵 (((pInvG‘𝐺)‘(𝐶(midG‘𝐺)𝐹))‘𝐶)))
1331, 2, 3, 14, 15, 71, 74, 78, 75israg 25392 . . . . . . . . 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))))
134132, 133mpbird 246 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ⟨“𝐵(𝐶(midG‘𝐺)𝐹)𝐶”⟩ ∈ (∟G‘𝐺))
1351, 2, 3, 14, 71, 80, 108, 112, 96, 113, 114, 126, 134ragperp 25412 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵)(⟂G‘𝐺)(𝐹(LineG‘𝐺)𝐶))
136135orcd 406 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵)(⟂G‘𝐺)(𝐹(LineG‘𝐺)𝐶) ∨ 𝐹 = 𝐶))
1371, 2, 3, 71, 72, 76, 14, 80, 77, 75islmib 25479 . . . . . 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))))
138105, 136, 137mpbir2and 959 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐶 = (𝑆𝐹))
1391, 2, 3, 71, 72, 73, 74, 75, 82, 84, 85, 86, 89, 92, 95, 101, 102, 138hypcgrlem1 25491 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐴 𝐶) = ((𝑆𝐷) (𝑆𝐹)))
1401, 2, 3, 71, 72, 76, 14, 80, 81, 77lmiiso 25489 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝑆𝐷) (𝑆𝐹)) = (𝐷 𝐹))
141139, 140eqtrd 2644 . . 3 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐴 𝐶) = (𝐷 𝐹))
14270, 141pm2.61dane 2869 . 2 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐴 𝐶) = (𝐷 𝐹))
14353, 142pm2.61dane 2869 1 (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  2c2 10947  ⟨“cs3 13438  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  DimTarskiG≥cstrkgld 25133  Itvcitv 25135  LineGclng 25136  pInvGcmir 25347  ∟Gcrag 25388  ⟂Gcperpg 25390  midGcmid 25464  lInvGclmi 25465 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445  df-trkgc 25147  df-trkgb 25148  df-trkgcb 25149  df-trkgld 25151  df-trkg 25152  df-cgrg 25206  df-ismt 25228  df-leg 25278  df-mir 25348  df-rag 25389  df-perpg 25391  df-mid 25466  df-lmi 25467 This theorem is referenced by:  hypcgr  25493
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