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Theorem hypcgrlem2 25592
Description: Lemma for hypcgr 25593, 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 481 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐺 ∈ TarskiG)
6 hypcgr.h . . . . 5 (𝜑𝐺DimTarskiG≥2)
76adantr 481 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐺DimTarskiG≥2)
8 hypcgr.a . . . . 5 (𝜑𝐴𝑃)
98adantr 481 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐴𝑃)
10 hypcgr.b . . . . 5 (𝜑𝐵𝑃)
1110adantr 481 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐵𝑃)
12 hypcgr.c . . . . 5 (𝜑𝐶𝑃)
1312adantr 481 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐶𝑃)
14 eqid 2621 . . . . 5 (LineG‘𝐺) = (LineG‘𝐺)
15 eqid 2621 . . . . 5 (pInvG‘𝐺) = (pInvG‘𝐺)
16 eqid 2621 . . . . 5 ((pInvG‘𝐺)‘𝐵) = ((pInvG‘𝐺)‘𝐵)
17 hypcgr.d . . . . . 6 (𝜑𝐷𝑃)
1817adantr 481 . . . . 5 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐷𝑃)
191, 2, 3, 14, 15, 5, 11, 16, 18mircl 25456 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (((pInvG‘𝐺)‘𝐵)‘𝐷) ∈ 𝑃)
20 hypcgr.e . . . . 5 (𝜑𝐸𝑃)
2120adantr 481 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐸𝑃)
22 hypcgr.1 . . . . 5 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
2322adantr 481 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
24 eqidd 2622 . . . . . 6 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (((pInvG‘𝐺)‘𝐵)‘𝐷) = (((pInvG‘𝐺)‘𝐵)‘𝐷))
25 hypcgrlem2.b . . . . . . . . 9 (𝜑𝐵 = 𝐸)
2625adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐵 = 𝐸)
271, 2, 3, 14, 15, 5, 11, 16, 21mirinv 25461 . . . . . . . 8 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐸) = 𝐸𝐵 = 𝐸))
2826, 27mpbird 247 . . . . . . 7 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (((pInvG‘𝐺)‘𝐵)‘𝐸) = 𝐸)
2928eqcomd 2627 . . . . . 6 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐸 = (((pInvG‘𝐺)‘𝐵)‘𝐸))
30 hypcgr.f . . . . . . . . . 10 (𝜑𝐹𝑃)
3130adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐹𝑃)
321, 2, 3, 5, 7, 13, 31midcom 25574 . . . . . . . 8 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐶(midG‘𝐺)𝐹) = (𝐹(midG‘𝐺)𝐶))
33 simpr 477 . . . . . . . 8 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐶(midG‘𝐺)𝐹) = 𝐵)
3432, 33eqtr3d 2657 . . . . . . 7 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐹(midG‘𝐺)𝐶) = 𝐵)
351, 2, 3, 5, 7, 31, 13, 15, 11ismidb 25570 . . . . . . 7 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐶 = (((pInvG‘𝐺)‘𝐵)‘𝐹) ↔ (𝐹(midG‘𝐺)𝐶) = 𝐵))
3634, 35mpbird 247 . . . . . 6 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐶 = (((pInvG‘𝐺)‘𝐵)‘𝐹))
3724, 29, 36s3eqd 13546 . . . . 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 481 . . . . . 6 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
401, 2, 3, 14, 15, 5, 18, 21, 31, 39, 16, 11mirrag 25496 . . . . 5 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝐷)(((pInvG‘𝐺)‘𝐵)‘𝐸)(((pInvG‘𝐺)‘𝐵)‘𝐹)”⟩ ∈ (∟G‘𝐺))
4137, 40eqeltrd 2698 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝐷)𝐸𝐶”⟩ ∈ (∟G‘𝐺))
42 hypcgr.3 . . . . . 6 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
4342adantr 481 . . . . 5 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 𝐵) = (𝐷 𝐸))
441, 2, 3, 14, 15, 5, 11, 16, 18, 21miriso 25465 . . . . 5 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) (((pInvG‘𝐺)‘𝐵)‘𝐸)) = (𝐷 𝐸))
4528oveq2d 6620 . . . . 5 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) (((pInvG‘𝐺)‘𝐵)‘𝐸)) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) 𝐸))
4643, 44, 453eqtr2d 2661 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 𝐵) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) 𝐸))
4726oveq1d 6619 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐵 𝐶) = (𝐸 𝐶))
48 eqid 2621 . . . 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 2622 . . . 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 25591 . . 3 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 𝐶) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) 𝐶))
5136oveq2d 6620 . . 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 25465 . . 3 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) (((pInvG‘𝐺)‘𝐵)‘𝐹)) = (𝐷 𝐹))
5350, 51, 523eqtrd 2659 . 2 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 𝐶) = (𝐷 𝐹))
544ad2antrr 761 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐺 ∈ TarskiG)
556ad2antrr 761 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐺DimTarskiG≥2)
568ad2antrr 761 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐴𝑃)
5710ad2antrr 761 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐵𝑃)
5812ad2antrr 761 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐶𝑃)
5917ad2antrr 761 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐷𝑃)
6020ad2antrr 761 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐸𝑃)
6130ad2antrr 761 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐹𝑃)
6222ad2antrr 761 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
6338ad2antrr 761 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
6442ad2antrr 761 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → (𝐴 𝐵) = (𝐷 𝐸))
65 hypcgr.4 . . . . 5 (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))
6665ad2antrr 761 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → (𝐵 𝐶) = (𝐸 𝐹))
6725ad2antrr 761 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐵 = 𝐸)
68 eqid 2621 . . . 4 ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵)) = ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵))
69 simpr 477 . . . 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 25591 . . 3 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → (𝐴 𝐶) = (𝐷 𝐹))
714ad2antrr 761 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐺 ∈ TarskiG)
726ad2antrr 761 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐺DimTarskiG≥2)
738ad2antrr 761 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐴𝑃)
7410ad2antrr 761 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐵𝑃)
7512ad2antrr 761 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐶𝑃)
76 hypcgrlem2.s . . . . . 6 𝑆 = ((lInvG‘𝐺)‘((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
7730ad2antrr 761 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐹𝑃)
781, 2, 3, 71, 72, 75, 77midcl 25569 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ 𝑃)
79 simplr 791 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ≠ 𝐵)
801, 3, 14, 71, 78, 74, 79tgelrnln 25425 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵) ∈ ran (LineG‘𝐺))
8117ad2antrr 761 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐷𝑃)
821, 2, 3, 71, 72, 76, 14, 80, 81lmicl 25578 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝑆𝐷) ∈ 𝑃)
8320ad2antrr 761 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐸𝑃)
841, 2, 3, 71, 72, 76, 14, 80, 83lmicl 25578 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝑆𝐸) ∈ 𝑃)
851, 2, 3, 71, 72, 76, 14, 80, 77lmicl 25578 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝑆𝐹) ∈ 𝑃)
8622ad2antrr 761 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
871, 2, 3, 71, 72, 76, 14, 80lmimot 25590 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝑆 ∈ (𝐺Ismt𝐺))
8838ad2antrr 761 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
891, 2, 3, 14, 15, 71, 81, 83, 77, 87, 88motrag 25503 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ⟨“(𝑆𝐷)(𝑆𝐸)(𝑆𝐹)”⟩ ∈ (∟G‘𝐺))
9042ad2antrr 761 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐴 𝐵) = (𝐷 𝐸))
911, 2, 3, 71, 72, 76, 14, 80, 81, 83lmiiso 25589 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝑆𝐷) (𝑆𝐸)) = (𝐷 𝐸))
9290, 91eqtr4d 2658 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐴 𝐵) = ((𝑆𝐷) (𝑆𝐸)))
9365ad2antrr 761 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐵 𝐶) = (𝐸 𝐹))
941, 2, 3, 71, 72, 76, 14, 80, 83, 77lmiiso 25589 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝑆𝐸) (𝑆𝐹)) = (𝐸 𝐹))
9593, 94eqtr4d 2658 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐵 𝐶) = ((𝑆𝐸) (𝑆𝐹)))
961, 3, 14, 71, 78, 74, 79tglinerflx2 25429 . . . . . . 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 25584 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝑆𝐵) = 𝐵𝐵 ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵)))
9896, 97mpbird 247 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝑆𝐵) = 𝐵)
9925ad2antrr 761 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐵 = 𝐸)
10099fveq2d 6152 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝑆𝐵) = (𝑆𝐸))
10198, 100eqtr3d 2657 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐵 = (𝑆𝐸))
102 eqid 2621 . . . . 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 25574 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) = (𝐹(midG‘𝐺)𝐶))
1041, 3, 14, 71, 78, 74, 79tglinerflx1 25428 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
105103, 104eqeltrrd 2699 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐹(midG‘𝐺)𝐶) ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
106 simpr 477 . . . . . . . . . 10 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐶𝐹)
107106necomd 2845 . . . . . . . . 9 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐹𝐶)
1081, 3, 14, 71, 77, 75, 107tgelrnln 25425 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐹(LineG‘𝐺)𝐶) ∈ ran (LineG‘𝐺))
1091, 2, 3, 71, 72, 75, 77midbtwn 25571 . . . . . . . . . . 11 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (𝐶𝐼𝐹))
1101, 2, 3, 71, 75, 78, 77, 109tgbtwncom 25283 . . . . . . . . . 10 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (𝐹𝐼𝐶))
1111, 3, 14, 71, 77, 75, 78, 107, 110btwnlng1 25414 . . . . . . . . 9 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (𝐹(LineG‘𝐺)𝐶))
112104, 111elind 3776 . . . . . . . 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 25429 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐶 ∈ (𝐹(LineG‘𝐺)𝐶))
11479necomd 2845 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐵 ≠ (𝐶(midG‘𝐺)𝐹))
1154ad2antrr 761 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐺 ∈ TarskiG)
11612ad2antrr 761 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐶𝑃)
11730ad2antrr 761 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐹𝑃)
1186ad2antrr 761 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐺DimTarskiG≥2)
119 simpr 477 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐶 = (𝐶(midG‘𝐺)𝐹))
120119eqcomd 2627 . . . . . . . . . . . . . 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 25572 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → (𝐶 𝐶) = (𝐶 𝐹))
122121eqcomd 2627 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → (𝐶 𝐹) = (𝐶 𝐶))
1231, 2, 3, 115, 116, 117, 116, 122axtgcgrid 25262 . . . . . . . . . . 11 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐶 = 𝐹)
124123ex 450 . . . . . . . . . 10 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐶 = (𝐶(midG‘𝐺)𝐹) → 𝐶 = 𝐹))
125124necon3d 2811 . . . . . . . . 9 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐶𝐹𝐶 ≠ (𝐶(midG‘𝐺)𝐹)))
126125imp 445 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐶 ≠ (𝐶(midG‘𝐺)𝐹))
12799eqcomd 2627 . . . . . . . . . . 11 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐸 = 𝐵)
128 eqidd 2622 . . . . . . . . . . . 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 25570 . . . . . . . . . . . 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 247 . . . . . . . . . . 11 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐹 = (((pInvG‘𝐺)‘(𝐶(midG‘𝐺)𝐹))‘𝐶))
131127, 130oveq12d 6622 . . . . . . . . . 10 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐸 𝐹) = (𝐵 (((pInvG‘𝐺)‘(𝐶(midG‘𝐺)𝐹))‘𝐶)))
13293, 131eqtrd 2655 . . . . . . . . 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 25492 . . . . . . . . 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 247 . . . . . . . 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 25512 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵)(⟂G‘𝐺)(𝐹(LineG‘𝐺)𝐶))
136135orcd 407 . . . . . 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 25579 . . . . . 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 956 . . . . 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 25591 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐴 𝐶) = ((𝑆𝐷) (𝑆𝐹)))
1401, 2, 3, 71, 72, 76, 14, 80, 81, 77lmiiso 25589 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝑆𝐷) (𝑆𝐹)) = (𝐷 𝐹))
141139, 140eqtrd 2655 . . 3 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐴 𝐶) = (𝐷 𝐹))
14270, 141pm2.61dane 2877 . 2 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐴 𝐶) = (𝐷 𝐹))
14353, 142pm2.61dane 2877 1 (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384   = wceq 1480  wcel 1987  wne 2790   class class class wbr 4613  cfv 5847  (class class class)co 6604  2c2 11014  ⟨“cs3 13524  Basecbs 15781  distcds 15871  TarskiGcstrkg 25229  DimTarskiGcstrkgld 25233  Itvcitv 25235  LineGclng 25236  pInvGcmir 25447  ∟Gcrag 25488  ⟂Gcperpg 25490  midGcmid 25564  lInvGclmi 25565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-xnn0 11308  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-concat 13240  df-s1 13241  df-s2 13530  df-s3 13531  df-trkgc 25247  df-trkgb 25248  df-trkgcb 25249  df-trkgld 25251  df-trkg 25252  df-cgrg 25306  df-ismt 25328  df-leg 25378  df-mir 25448  df-rag 25489  df-perpg 25491  df-mid 25566  df-lmi 25567
This theorem is referenced by:  hypcgr  25593
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