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Theorem cgraswap 26778
Description: Swap rays in a congruence relation. Theorem 11.9 of [Schwabhauser] p. 96. (Contributed by Thierry Arnoux, 5-Mar-2020.)
Hypotheses
Ref Expression
cgraid.p 𝑃 = (Base‘𝐺)
cgraid.i 𝐼 = (Itv‘𝐺)
cgraid.g (𝜑𝐺 ∈ TarskiG)
cgraid.k 𝐾 = (hlG‘𝐺)
cgraid.a (𝜑𝐴𝑃)
cgraid.b (𝜑𝐵𝑃)
cgraid.c (𝜑𝐶𝑃)
cgraid.1 (𝜑𝐴𝐵)
cgraid.2 (𝜑𝐵𝐶)
Assertion
Ref Expression
cgraswap (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐶𝐵𝐴”⟩)

Proof of Theorem cgraswap
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cgraid.p . . . . 5 𝑃 = (Base‘𝐺)
2 eqid 2739 . . . . 5 (dist‘𝐺) = (dist‘𝐺)
3 eqid 2739 . . . . 5 (cgrG‘𝐺) = (cgrG‘𝐺)
4 cgraid.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54ad3antrrr 730 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝐺 ∈ TarskiG)
6 cgraid.a . . . . . 6 (𝜑𝐴𝑃)
76ad3antrrr 730 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝐴𝑃)
8 cgraid.b . . . . . 6 (𝜑𝐵𝑃)
98ad3antrrr 730 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝐵𝑃)
10 cgraid.c . . . . . 6 (𝜑𝐶𝑃)
1110ad3antrrr 730 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝐶𝑃)
12 simpllr 776 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝑥𝑃)
13 simplr 769 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝑦𝑃)
14 cgraid.i . . . . . . 7 𝐼 = (Itv‘𝐺)
15 simprlr 780 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴))
161, 2, 14, 5, 9, 12, 9, 7, 15tgcgrcomlr 26438 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑥(dist‘𝐺)𝐵) = (𝐴(dist‘𝐺)𝐵))
1716eqcomd 2745 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐴(dist‘𝐺)𝐵) = (𝑥(dist‘𝐺)𝐵))
18 simprrr 782 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶))
1918eqcomd 2745 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐵(dist‘𝐺)𝐶) = (𝐵(dist‘𝐺)𝑦))
20 eqid 2739 . . . . . . . 8 (LineG‘𝐺) = (LineG‘𝐺)
21 cgraid.k . . . . . . . . . . 11 𝐾 = (hlG‘𝐺)
22 simprll 779 . . . . . . . . . . 11 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝑥(𝐾𝐵)𝐶)
231, 14, 21, 12, 11, 9, 5, 20, 22hlln 26565 . . . . . . . . . 10 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝑥 ∈ (𝐶(LineG‘𝐺)𝐵))
2423orcd 872 . . . . . . . . 9 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑥 ∈ (𝐶(LineG‘𝐺)𝐵) ∨ 𝐶 = 𝐵))
251, 20, 14, 5, 11, 9, 12, 24colrot1 26517 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐶 ∈ (𝐵(LineG‘𝐺)𝑥) ∨ 𝐵 = 𝑥))
26 eqid 2739 . . . . . . . . . 10 (≤G‘𝐺) = (≤G‘𝐺)
271, 14, 21, 12, 11, 9, 5ishlg 26560 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑥(𝐾𝐵)𝐶 ↔ (𝑥𝐵𝐶𝐵 ∧ (𝑥 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝑥)))))
2822, 27mpbid 235 . . . . . . . . . . . 12 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑥𝐵𝐶𝐵 ∧ (𝑥 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝑥))))
2928simp3d 1145 . . . . . . . . . . 11 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑥 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝑥)))
3029orcomd 870 . . . . . . . . . 10 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐶 ∈ (𝐵𝐼𝑥) ∨ 𝑥 ∈ (𝐵𝐼𝐶)))
31 simprrl 781 . . . . . . . . . . . 12 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝑦(𝐾𝐵)𝐴)
321, 14, 21, 13, 7, 9, 5ishlg 26560 . . . . . . . . . . . 12 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑦(𝐾𝐵)𝐴 ↔ (𝑦𝐵𝐴𝐵 ∧ (𝑦 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝑦)))))
3331, 32mpbid 235 . . . . . . . . . . 11 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑦𝐵𝐴𝐵 ∧ (𝑦 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝑦))))
3433simp3d 1145 . . . . . . . . . 10 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑦 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝑦)))
351, 2, 14, 26, 5, 9, 11, 12, 9, 9, 13, 7, 30, 34, 19, 15tgcgrsub2 26553 . . . . . . . . 9 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐶(dist‘𝐺)𝑥) = (𝑦(dist‘𝐺)𝐴))
361, 2, 3, 5, 9, 11, 12, 9, 13, 7, 19, 35, 16trgcgr 26474 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → ⟨“𝐵𝐶𝑥”⟩(cgrG‘𝐺)⟨“𝐵𝑦𝐴”⟩)
371, 2, 14, 5, 11, 13axtgcgrrflx 26420 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐶(dist‘𝐺)𝑦) = (𝑦(dist‘𝐺)𝐶))
38 cgraid.2 . . . . . . . . 9 (𝜑𝐵𝐶)
3938ad3antrrr 730 . . . . . . . 8 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝐵𝐶)
401, 20, 14, 5, 9, 11, 12, 3, 9, 13, 2, 13, 7, 11, 25, 36, 18, 37, 39tgfscgr 26526 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑥(dist‘𝐺)𝑦) = (𝐴(dist‘𝐺)𝐶))
411, 2, 14, 5, 12, 13, 7, 11, 40tgcgrcomlr 26438 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑦(dist‘𝐺)𝑥) = (𝐶(dist‘𝐺)𝐴))
4241eqcomd 2745 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐶(dist‘𝐺)𝐴) = (𝑦(dist‘𝐺)𝑥))
431, 2, 3, 5, 7, 9, 11, 12, 9, 13, 17, 19, 42trgcgr 26474 . . . 4 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐵𝑦”⟩)
4443, 22, 313jca 1129 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐵𝑦”⟩ ∧ 𝑥(𝐾𝐵)𝐶𝑦(𝐾𝐵)𝐴))
4538necomd 2990 . . . . 5 (𝜑𝐶𝐵)
46 cgraid.1 . . . . . 6 (𝜑𝐴𝐵)
4746necomd 2990 . . . . 5 (𝜑𝐵𝐴)
481, 14, 21, 8, 8, 6, 4, 10, 2, 45, 47hlcgrex 26574 . . . 4 (𝜑 → ∃𝑥𝑃 (𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)))
491, 14, 21, 8, 8, 10, 4, 6, 2, 46, 38hlcgrex 26574 . . . 4 (𝜑 → ∃𝑦𝑃 (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))
50 reeanv 3271 . . . 4 (∃𝑥𝑃𝑦𝑃 ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶))) ↔ (∃𝑥𝑃 (𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ ∃𝑦𝑃 (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶))))
5148, 49, 50sylanbrc 586 . . 3 (𝜑 → ∃𝑥𝑃𝑦𝑃 ((𝑥(𝐾𝐵)𝐶 ∧ (𝐵(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶))))
5244, 51reximddv2 3189 . 2 (𝜑 → ∃𝑥𝑃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐵𝑦”⟩ ∧ 𝑥(𝐾𝐵)𝐶𝑦(𝐾𝐵)𝐴))
531, 14, 21, 4, 6, 8, 10, 10, 8, 6iscgra 26767 . 2 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐶𝐵𝐴”⟩ ↔ ∃𝑥𝑃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐵𝑦”⟩ ∧ 𝑥(𝐾𝐵)𝐶𝑦(𝐾𝐵)𝐴)))
5452, 53mpbird 260 1 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐶𝐵𝐴”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 846  w3a 1088   = wceq 1542  wcel 2114  wne 2935  wrex 3055   class class class wbr 5040  cfv 6349  (class class class)co 7182  ⟨“cs3 14305  Basecbs 16598  distcds 16689  TarskiGcstrkg 26388  Itvcitv 26394  LineGclng 26395  cgrGccgrg 26468  ≤Gcleg 26540  hlGchlg 26558  cgrAccgra 26765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491  ax-cnex 10683  ax-resscn 10684  ax-1cn 10685  ax-icn 10686  ax-addcl 10687  ax-addrcl 10688  ax-mulcl 10689  ax-mulrcl 10690  ax-mulcom 10691  ax-addass 10692  ax-mulass 10693  ax-distr 10694  ax-i2m1 10695  ax-1ne0 10696  ax-1rid 10697  ax-rnegex 10698  ax-rrecex 10699  ax-cnre 10700  ax-pre-lttri 10701  ax-pre-lttrn 10702  ax-pre-ltadd 10703  ax-pre-mulgt0 10704
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-int 4847  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-pred 6139  df-ord 6185  df-on 6186  df-lim 6187  df-suc 6188  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7139  df-ov 7185  df-oprab 7186  df-mpo 7187  df-om 7612  df-1st 7726  df-2nd 7727  df-wrecs 7988  df-recs 8049  df-rdg 8087  df-1o 8143  df-oadd 8147  df-er 8332  df-map 8451  df-pm 8452  df-en 8568  df-dom 8569  df-sdom 8570  df-fin 8571  df-dju 9415  df-card 9453  df-pnf 10767  df-mnf 10768  df-xr 10769  df-ltxr 10770  df-le 10771  df-sub 10962  df-neg 10963  df-nn 11729  df-2 11791  df-3 11792  df-n0 11989  df-xnn0 12061  df-z 12075  df-uz 12337  df-fz 12994  df-fzo 13137  df-hash 13795  df-word 13968  df-concat 14024  df-s1 14051  df-s2 14311  df-s3 14312  df-trkgc 26406  df-trkgb 26407  df-trkgcb 26408  df-trkg 26411  df-cgrg 26469  df-leg 26541  df-hlg 26559  df-cgra 26766
This theorem is referenced by:  cgraswaplr  26783  oacgr  26790  tgasa1  26816
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