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Theorem cgracgr 25510
Description: First direction of proposition 11.4 of [Schwabhauser] p. 95. Again, this is "half" of the proposition, i.e. only two additional points are used, while Schwabhauser has four. (Contributed by Thierry Arnoux, 31-Jul-2020.)
Hypotheses
Ref Expression
iscgra.p 𝑃 = (Base‘𝐺)
iscgra.i 𝐼 = (Itv‘𝐺)
iscgra.k 𝐾 = (hlG‘𝐺)
iscgra.g (𝜑𝐺 ∈ TarskiG)
iscgra.a (𝜑𝐴𝑃)
iscgra.b (𝜑𝐵𝑃)
iscgra.c (𝜑𝐶𝑃)
iscgra.d (𝜑𝐷𝑃)
iscgra.e (𝜑𝐸𝑃)
iscgra.f (𝜑𝐹𝑃)
cgrahl1.2 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
cgrahl1.x (𝜑𝑋𝑃)
cgracgr.m = (dist‘𝐺)
cgracgr.y (𝜑𝑌𝑃)
cgracgr.1 (𝜑𝑋(𝐾𝐵)𝐴)
cgracgr.2 (𝜑𝑌(𝐾𝐵)𝐶)
cgracgr.3 (𝜑 → (𝐵 𝑋) = (𝐸 𝐷))
cgracgr.4 (𝜑 → (𝐵 𝑌) = (𝐸 𝐹))
Assertion
Ref Expression
cgracgr (𝜑 → (𝑋 𝑌) = (𝐷 𝐹))

Proof of Theorem cgracgr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscgra.p . . 3 𝑃 = (Base‘𝐺)
2 eqid 2610 . . 3 (LineG‘𝐺) = (LineG‘𝐺)
3 iscgra.i . . 3 𝐼 = (Itv‘𝐺)
4 iscgra.g . . . 4 (𝜑𝐺 ∈ TarskiG)
54ad3antrrr 762 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → 𝐺 ∈ TarskiG)
6 iscgra.a . . . 4 (𝜑𝐴𝑃)
76ad3antrrr 762 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → 𝐴𝑃)
8 iscgra.b . . . 4 (𝜑𝐵𝑃)
98ad3antrrr 762 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → 𝐵𝑃)
10 cgrahl1.x . . . 4 (𝜑𝑋𝑃)
1110ad3antrrr 762 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → 𝑋𝑃)
12 eqid 2610 . . 3 (cgrG‘𝐺) = (cgrG‘𝐺)
13 simpllr 795 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → 𝑥𝑃)
14 iscgra.e . . . 4 (𝜑𝐸𝑃)
1514ad3antrrr 762 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → 𝐸𝑃)
16 cgracgr.m . . 3 = (dist‘𝐺)
17 cgracgr.y . . . 4 (𝜑𝑌𝑃)
1817ad3antrrr 762 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → 𝑌𝑃)
19 iscgra.d . . . 4 (𝜑𝐷𝑃)
2019ad3antrrr 762 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → 𝐷𝑃)
21 iscgra.f . . . 4 (𝜑𝐹𝑃)
2221ad3antrrr 762 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → 𝐹𝑃)
23 iscgra.k . . . . . . . . 9 𝐾 = (hlG‘𝐺)
24 iscgra.c . . . . . . . . 9 (𝜑𝐶𝑃)
25 cgrahl1.2 . . . . . . . . 9 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
261, 3, 23, 4, 6, 8, 24, 19, 14, 21, 25cgrane1 25504 . . . . . . . 8 (𝜑𝐴𝐵)
2726necomd 2837 . . . . . . 7 (𝜑𝐵𝐴)
28 cgracgr.1 . . . . . . . 8 (𝜑𝑋(𝐾𝐵)𝐴)
291, 3, 23, 10, 6, 8, 4, 2, 28hlln 25302 . . . . . . 7 (𝜑𝑋 ∈ (𝐴(LineG‘𝐺)𝐵))
301, 3, 2, 4, 8, 6, 10, 27, 29lncom 25317 . . . . . 6 (𝜑𝑋 ∈ (𝐵(LineG‘𝐺)𝐴))
3130orcd 406 . . . . 5 (𝜑 → (𝑋 ∈ (𝐵(LineG‘𝐺)𝐴) ∨ 𝐵 = 𝐴))
321, 2, 3, 4, 8, 6, 10, 31colrot1 25254 . . . 4 (𝜑 → (𝐵 ∈ (𝐴(LineG‘𝐺)𝑋) ∨ 𝐴 = 𝑋))
3332ad3antrrr 762 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝐵 ∈ (𝐴(LineG‘𝐺)𝑋) ∨ 𝐴 = 𝑋))
3424ad3antrrr 762 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → 𝐶𝑃)
35 simplr 788 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → 𝑦𝑃)
36 simpr1 1060 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩)
371, 16, 3, 12, 5, 7, 9, 34, 13, 15, 35, 36cgr3simp1 25215 . . . 4 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝐴 𝐵) = (𝑥 𝐸))
38 cgracgr.3 . . . . 5 (𝜑 → (𝐵 𝑋) = (𝐸 𝐷))
3938ad3antrrr 762 . . . 4 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝐵 𝑋) = (𝐸 𝐷))
40 eqid 2610 . . . . . . 7 (≤G‘𝐺) = (≤G‘𝐺)
41 simpr2 1061 . . . . . . . . 9 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → 𝑥(𝐾𝐸)𝐷)
421, 3, 23, 13, 20, 15, 5ishlg 25297 . . . . . . . . 9 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝑥(𝐾𝐸)𝐷 ↔ (𝑥𝐸𝐷𝐸 ∧ (𝑥 ∈ (𝐸𝐼𝐷) ∨ 𝐷 ∈ (𝐸𝐼𝑥)))))
4341, 42mpbid 221 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝑥𝐸𝐷𝐸 ∧ (𝑥 ∈ (𝐸𝐼𝐷) ∨ 𝐷 ∈ (𝐸𝐼𝑥))))
4443simp3d 1068 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝑥 ∈ (𝐸𝐼𝐷) ∨ 𝐷 ∈ (𝐸𝐼𝑥)))
451, 3, 23, 10, 6, 8, 4ishlg 25297 . . . . . . . . . . 11 (𝜑 → (𝑋(𝐾𝐵)𝐴 ↔ (𝑋𝐵𝐴𝐵 ∧ (𝑋 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝑋)))))
4628, 45mpbid 221 . . . . . . . . . 10 (𝜑 → (𝑋𝐵𝐴𝐵 ∧ (𝑋 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝑋))))
4746simp3d 1068 . . . . . . . . 9 (𝜑 → (𝑋 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝑋)))
4847orcomd 402 . . . . . . . 8 (𝜑 → (𝐴 ∈ (𝐵𝐼𝑋) ∨ 𝑋 ∈ (𝐵𝐼𝐴)))
4948ad3antrrr 762 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝐴 ∈ (𝐵𝐼𝑋) ∨ 𝑋 ∈ (𝐵𝐼𝐴)))
5037eqcomd 2616 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝑥 𝐸) = (𝐴 𝐵))
511, 16, 3, 5, 13, 15, 7, 9, 50tgcgrcomlr 25175 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝐸 𝑥) = (𝐵 𝐴))
5239eqcomd 2616 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝐸 𝐷) = (𝐵 𝑋))
531, 16, 3, 40, 5, 15, 13, 20, 9, 9, 7, 11, 44, 49, 51, 52tgcgrsub2 25290 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝑥 𝐷) = (𝐴 𝑋))
5453eqcomd 2616 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝐴 𝑋) = (𝑥 𝐷))
551, 16, 3, 5, 7, 11, 13, 20, 54tgcgrcomlr 25175 . . . 4 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝑋 𝐴) = (𝐷 𝑥))
561, 16, 12, 5, 7, 9, 11, 13, 15, 20, 37, 39, 55trgcgr 25211 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → ⟨“𝐴𝐵𝑋”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝐷”⟩)
57 cgracgr.2 . . . . . . . . 9 (𝜑𝑌(𝐾𝐵)𝐶)
581, 3, 23, 17, 24, 8, 4, 2, 57hlln 25302 . . . . . . . 8 (𝜑𝑌 ∈ (𝐶(LineG‘𝐺)𝐵))
5958orcd 406 . . . . . . 7 (𝜑 → (𝑌 ∈ (𝐶(LineG‘𝐺)𝐵) ∨ 𝐶 = 𝐵))
601, 2, 3, 4, 24, 8, 17, 59colrot1 25254 . . . . . 6 (𝜑 → (𝐶 ∈ (𝐵(LineG‘𝐺)𝑌) ∨ 𝐵 = 𝑌))
6160ad3antrrr 762 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝐶 ∈ (𝐵(LineG‘𝐺)𝑌) ∨ 𝐵 = 𝑌))
621, 16, 3, 12, 5, 7, 9, 34, 13, 15, 35, 36cgr3simp2 25216 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝐵 𝐶) = (𝐸 𝑦))
631, 3, 23, 17, 24, 8, 4ishlg 25297 . . . . . . . . . . 11 (𝜑 → (𝑌(𝐾𝐵)𝐶 ↔ (𝑌𝐵𝐶𝐵 ∧ (𝑌 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝑌)))))
6457, 63mpbid 221 . . . . . . . . . 10 (𝜑 → (𝑌𝐵𝐶𝐵 ∧ (𝑌 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝑌))))
6564simp3d 1068 . . . . . . . . 9 (𝜑 → (𝑌 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝑌)))
6665orcomd 402 . . . . . . . 8 (𝜑 → (𝐶 ∈ (𝐵𝐼𝑌) ∨ 𝑌 ∈ (𝐵𝐼𝐶)))
6766ad3antrrr 762 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝐶 ∈ (𝐵𝐼𝑌) ∨ 𝑌 ∈ (𝐵𝐼𝐶)))
68 simpr3 1062 . . . . . . . . 9 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → 𝑦(𝐾𝐸)𝐹)
691, 3, 23, 35, 22, 15, 5ishlg 25297 . . . . . . . . 9 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝑦(𝐾𝐸)𝐹 ↔ (𝑦𝐸𝐹𝐸 ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝑦)))))
7068, 69mpbid 221 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝑦𝐸𝐹𝐸 ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝑦))))
7170simp3d 1068 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝑦 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝑦)))
72 cgracgr.4 . . . . . . . 8 (𝜑 → (𝐵 𝑌) = (𝐸 𝐹))
7372ad3antrrr 762 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝐵 𝑌) = (𝐸 𝐹))
741, 16, 3, 40, 5, 9, 34, 18, 15, 15, 35, 22, 67, 71, 62, 73tgcgrsub2 25290 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝐶 𝑌) = (𝑦 𝐹))
751, 16, 3, 5, 9, 18, 15, 22, 73tgcgrcomlr 25175 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝑌 𝐵) = (𝐹 𝐸))
761, 16, 12, 5, 9, 34, 18, 15, 35, 22, 62, 74, 75trgcgr 25211 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → ⟨“𝐵𝐶𝑌”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝐹”⟩)
7751eqcomd 2616 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝐵 𝐴) = (𝐸 𝑥))
781, 16, 3, 12, 5, 7, 9, 34, 13, 15, 35, 36cgr3simp3 25217 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝐶 𝐴) = (𝑦 𝑥))
791, 3, 23, 4, 6, 8, 24, 19, 14, 21, 25cgrane2 25505 . . . . . 6 (𝜑𝐵𝐶)
8079ad3antrrr 762 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → 𝐵𝐶)
811, 2, 3, 5, 9, 34, 18, 12, 15, 35, 16, 7, 22, 13, 61, 76, 77, 78, 80tgfscgr 25263 . . . 4 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝑌 𝐴) = (𝐹 𝑥))
821, 16, 3, 5, 18, 7, 22, 13, 81tgcgrcomlr 25175 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝐴 𝑌) = (𝑥 𝐹))
8326ad3antrrr 762 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → 𝐴𝐵)
841, 2, 3, 5, 7, 9, 11, 12, 13, 15, 16, 18, 20, 22, 33, 56, 82, 73, 83tgfscgr 25263 . 2 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)) → (𝑋 𝑌) = (𝐷 𝐹))
851, 3, 23, 4, 6, 8, 24, 19, 14, 21iscgra 25501 . . 3 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥𝑃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)))
8625, 85mpbid 221 . 2 (𝜑 → ∃𝑥𝑃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹))
8784, 86r19.29vva 3062 1 (𝜑 → (𝑋 𝑌) = (𝐷 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wrex 2897   class class class wbr 4583  cfv 5804  (class class class)co 6549  ⟨“cs3 13438  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135  LineGclng 25136  cgrGccgrg 25205  ≤Gcleg 25277  hlGchlg 25295  cgrAccgra 25499
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-trkg 25152  df-cgrg 25206  df-leg 25278  df-hlg 25296  df-cgra 25500
This theorem is referenced by:  cgracom  25514  cgratr  25515  dfcgra2  25521  tgsas1  25535  tgasa1  25539
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