MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hypcgrlem2 Structured version   Visualization version   GIF version

Theorem hypcgrlem2 26500
Description: Lemma for hypcgr 26501, 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 2826 . . . . 5 (LineG‘𝐺) = (LineG‘𝐺)
15 eqid 2826 . . . . 5 (pInvG‘𝐺) = (pInvG‘𝐺)
16 eqid 2826 . . . . 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 26361 . . . 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 2827 . . . . . 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 26366 . . . . . . . 8 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐸) = 𝐸𝐵 = 𝐸))
2826, 27mpbird 258 . . . . . . 7 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (((pInvG‘𝐺)‘𝐵)‘𝐸) = 𝐸)
2928eqcomd 2832 . . . . . 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 26482 . . . . . . . 8 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐶(midG‘𝐺)𝐹) = (𝐹(midG‘𝐺)𝐶))
33 simpr 485 . . . . . . . 8 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐶(midG‘𝐺)𝐹) = 𝐵)
3432, 33eqtr3d 2863 . . . . . . 7 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐹(midG‘𝐺)𝐶) = 𝐵)
351, 2, 3, 5, 7, 31, 13, 15, 11ismidb 26478 . . . . . . 7 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐶 = (((pInvG‘𝐺)‘𝐵)‘𝐹) ↔ (𝐹(midG‘𝐺)𝐶) = 𝐵))
3634, 35mpbird 258 . . . . . 6 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → 𝐶 = (((pInvG‘𝐺)‘𝐵)‘𝐹))
3724, 29, 36s3eqd 14216 . . . . 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 26401 . . . . 5 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ⟨“(((pInvG‘𝐺)‘𝐵)‘𝐷)(((pInvG‘𝐺)‘𝐵)‘𝐸)(((pInvG‘𝐺)‘𝐵)‘𝐹)”⟩ ∈ (∟G‘𝐺))
4137, 40eqeltrd 2918 . . . 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 26370 . . . . 5 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) (((pInvG‘𝐺)‘𝐵)‘𝐸)) = (𝐷 𝐸))
4528oveq2d 7164 . . . . 5 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) (((pInvG‘𝐺)‘𝐵)‘𝐸)) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) 𝐸))
4643, 44, 453eqtr2d 2867 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 𝐵) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) 𝐸))
4726oveq1d 7163 . . . 4 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐵 𝐶) = (𝐸 𝐶))
48 eqid 2826 . . . 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 2827 . . . 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 26499 . . 3 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 𝐶) = ((((pInvG‘𝐺)‘𝐵)‘𝐷) 𝐶))
5136oveq2d 7164 . . 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 26370 . . 3 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → ((((pInvG‘𝐺)‘𝐵)‘𝐷) (((pInvG‘𝐺)‘𝐵)‘𝐹)) = (𝐷 𝐹))
5350, 51, 523eqtrd 2865 . 2 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) = 𝐵) → (𝐴 𝐶) = (𝐷 𝐹))
544ad2antrr 722 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐺 ∈ TarskiG)
556ad2antrr 722 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐺DimTarskiG≥2)
568ad2antrr 722 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐴𝑃)
5710ad2antrr 722 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐵𝑃)
5812ad2antrr 722 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐶𝑃)
5917ad2antrr 722 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐷𝑃)
6020ad2antrr 722 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐸𝑃)
6130ad2antrr 722 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐹𝑃)
6222ad2antrr 722 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
6338ad2antrr 722 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
6442ad2antrr 722 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → (𝐴 𝐵) = (𝐷 𝐸))
65 hypcgr.4 . . . . 5 (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))
6665ad2antrr 722 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → (𝐵 𝐶) = (𝐸 𝐹))
6725ad2antrr 722 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → 𝐵 = 𝐸)
68 eqid 2826 . . . 4 ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵)) = ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵))
69 simpr 485 . . . 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 26499 . . 3 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = 𝐹) → (𝐴 𝐶) = (𝐷 𝐹))
714ad2antrr 722 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐺 ∈ TarskiG)
726ad2antrr 722 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐺DimTarskiG≥2)
738ad2antrr 722 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐴𝑃)
7410ad2antrr 722 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐵𝑃)
7512ad2antrr 722 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐶𝑃)
76 hypcgrlem2.s . . . . . 6 𝑆 = ((lInvG‘𝐺)‘((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
7730ad2antrr 722 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐹𝑃)
781, 2, 3, 71, 72, 75, 77midcl 26477 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ 𝑃)
79 simplr 765 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ≠ 𝐵)
801, 3, 14, 71, 78, 74, 79tgelrnln 26330 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵) ∈ ran (LineG‘𝐺))
8117ad2antrr 722 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐷𝑃)
821, 2, 3, 71, 72, 76, 14, 80, 81lmicl 26486 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝑆𝐷) ∈ 𝑃)
8320ad2antrr 722 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐸𝑃)
841, 2, 3, 71, 72, 76, 14, 80, 83lmicl 26486 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝑆𝐸) ∈ 𝑃)
851, 2, 3, 71, 72, 76, 14, 80, 77lmicl 26486 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝑆𝐹) ∈ 𝑃)
8622ad2antrr 722 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
871, 2, 3, 71, 72, 76, 14, 80lmimot 26498 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝑆 ∈ (𝐺Ismt𝐺))
8838ad2antrr 722 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
891, 2, 3, 14, 15, 71, 81, 83, 77, 87, 88motrag 26408 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ⟨“(𝑆𝐷)(𝑆𝐸)(𝑆𝐹)”⟩ ∈ (∟G‘𝐺))
9042ad2antrr 722 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐴 𝐵) = (𝐷 𝐸))
911, 2, 3, 71, 72, 76, 14, 80, 81, 83lmiiso 26497 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝑆𝐷) (𝑆𝐸)) = (𝐷 𝐸))
9290, 91eqtr4d 2864 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐴 𝐵) = ((𝑆𝐷) (𝑆𝐸)))
9365ad2antrr 722 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐵 𝐶) = (𝐸 𝐹))
941, 2, 3, 71, 72, 76, 14, 80, 83, 77lmiiso 26497 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝑆𝐸) (𝑆𝐹)) = (𝐸 𝐹))
9593, 94eqtr4d 2864 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐵 𝐶) = ((𝑆𝐸) (𝑆𝐹)))
961, 3, 14, 71, 78, 74, 79tglinerflx2 26334 . . . . . . 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 26493 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝑆𝐵) = 𝐵)
9825ad2antrr 722 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐵 = 𝐸)
9998fveq2d 6671 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝑆𝐵) = (𝑆𝐸))
10097, 99eqtr3d 2863 . . . . 5 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐵 = (𝑆𝐸))
101 eqid 2826 . . . . 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 26482 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) = (𝐹(midG‘𝐺)𝐶))
1031, 3, 14, 71, 78, 74, 79tglinerflx1 26333 . . . . . . 7 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
104102, 103eqeltrrd 2919 . . . . . 6 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐹(midG‘𝐺)𝐶) ∈ ((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))
105 simpr 485 . . . . . . . . . 10 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐶𝐹)
106105necomd 3076 . . . . . . . . 9 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐹𝐶)
1071, 3, 14, 71, 77, 75, 106tgelrnln 26330 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐹(LineG‘𝐺)𝐶) ∈ ran (LineG‘𝐺))
1081, 2, 3, 71, 72, 75, 77midbtwn 26479 . . . . . . . . . . 11 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (𝐶𝐼𝐹))
1091, 2, 3, 71, 75, 78, 77, 108tgbtwncom 26188 . . . . . . . . . 10 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (𝐹𝐼𝐶))
1101, 3, 14, 71, 77, 75, 78, 106, 109btwnlng1 26319 . . . . . . . . 9 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐶(midG‘𝐺)𝐹) ∈ (𝐹(LineG‘𝐺)𝐶))
111103, 110elind 4175 . . . . . . . 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 26334 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐶 ∈ (𝐹(LineG‘𝐺)𝐶))
11379necomd 3076 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐵 ≠ (𝐶(midG‘𝐺)𝐹))
1144ad2antrr 722 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐺 ∈ TarskiG)
11512ad2antrr 722 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐶𝑃)
11630ad2antrr 722 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐹𝑃)
1176ad2antrr 722 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐺DimTarskiG≥2)
118 simpr 485 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐶 = (𝐶(midG‘𝐺)𝐹))
119118eqcomd 2832 . . . . . . . . . . . . . 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 26480 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → (𝐶 𝐶) = (𝐶 𝐹))
121120eqcomd 2832 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → (𝐶 𝐹) = (𝐶 𝐶))
1221, 2, 3, 114, 115, 116, 115, 121axtgcgrid 26163 . . . . . . . . . . 11 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶 = (𝐶(midG‘𝐺)𝐹)) → 𝐶 = 𝐹)
123122ex 413 . . . . . . . . . 10 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐶 = (𝐶(midG‘𝐺)𝐹) → 𝐶 = 𝐹))
124123necon3d 3042 . . . . . . . . 9 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐶𝐹𝐶 ≠ (𝐶(midG‘𝐺)𝐹)))
125124imp 407 . . . . . . . 8 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐶 ≠ (𝐶(midG‘𝐺)𝐹))
12698eqcomd 2832 . . . . . . . . . . 11 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐸 = 𝐵)
127 eqidd 2827 . . . . . . . . . . . 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 26478 . . . . . . . . . . . 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 258 . . . . . . . . . . 11 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → 𝐹 = (((pInvG‘𝐺)‘(𝐶(midG‘𝐺)𝐹))‘𝐶))
130126, 129oveq12d 7166 . . . . . . . . . 10 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐸 𝐹) = (𝐵 (((pInvG‘𝐺)‘(𝐶(midG‘𝐺)𝐹))‘𝐶)))
13193, 130eqtrd 2861 . . . . . . . . 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 26397 . . . . . . . . 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 258 . . . . . . . 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 26417 . . . . . . 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 26487 . . . . . 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 709 . . . . 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 26499 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐴 𝐶) = ((𝑆𝐷) (𝑆𝐹)))
1391, 2, 3, 71, 72, 76, 14, 80, 81, 77lmiiso 26497 . . . 4 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → ((𝑆𝐷) (𝑆𝐹)) = (𝐷 𝐹))
140138, 139eqtrd 2861 . . 3 (((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) ∧ 𝐶𝐹) → (𝐴 𝐶) = (𝐷 𝐹))
14170, 140pm2.61dane 3109 . 2 ((𝜑 ∧ (𝐶(midG‘𝐺)𝐹) ≠ 𝐵) → (𝐴 𝐶) = (𝐷 𝐹))
14253, 141pm2.61dane 3109 1 (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 843   = wceq 1530  wcel 2107  wne 3021   class class class wbr 5063  cfv 6352  (class class class)co 7148  2c2 11681  ⟨“cs3 14194  Basecbs 16473  distcds 16564  TarskiGcstrkg 26130  DimTarskiGcstrkgld 26134  Itvcitv 26136  LineGclng 26137  pInvGcmir 26352  ∟Gcrag 26393  ⟂Gcperpg 26395  midGcmid 26472  lInvGclmi 26473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-pm 8399  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-dju 9319  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-3 11690  df-n0 11887  df-xnn0 11957  df-z 11971  df-uz 12233  df-fz 12883  df-fzo 13024  df-hash 13681  df-word 13852  df-concat 13913  df-s1 13940  df-s2 14200  df-s3 14201  df-trkgc 26148  df-trkgb 26149  df-trkgcb 26150  df-trkgld 26152  df-trkg 26153  df-cgrg 26211  df-ismt 26233  df-leg 26283  df-mir 26353  df-rag 26394  df-perpg 26396  df-mid 26474  df-lmi 26475
This theorem is referenced by:  hypcgr  26501
  Copyright terms: Public domain W3C validator