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Theorem cgraswap 27181
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 2738 . . . . 5 (dist‘𝐺) = (dist‘𝐺)
3 eqid 2738 . . . . 5 (cgrG‘𝐺) = (cgrG‘𝐺)
4 cgraid.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54ad3antrrr 727 . . . . 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 727 . . . . 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 727 . . . . 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 727 . . . . 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 773 . . . . 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 766 . . . . 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 777 . . . . . . 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 26841 . . . . . 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 2744 . . . . 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 779 . . . . . 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 2744 . . . . 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 2738 . . . . . . . 8 (LineG‘𝐺) = (LineG‘𝐺)
21 cgraid.k . . . . . . . . . . 11 𝐾 = (hlG‘𝐺)
22 simprll 776 . . . . . . . . . . 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 26968 . . . . . . . . . 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 870 . . . . . . . . 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 26920 . . . . . . . 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 2738 . . . . . . . . . 10 (≤G‘𝐺) = (≤G‘𝐺)
271, 14, 21, 12, 11, 9, 5ishlg 26963 . . . . . . . . . . . . 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 231 . . . . . . . . . . . 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 1143 . . . . . . . . . . 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 868 . . . . . . . . . 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 778 . . . . . . . . . . . 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 26963 . . . . . . . . . . . 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 231 . . . . . . . . . . 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 1143 . . . . . . . . . 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 26956 . . . . . . . . 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 26877 . . . . . . . 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 26823 . . . . . . . 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 727 . . . . . . . 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 26929 . . . . . . 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 26841 . . . . . 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 2744 . . . . 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 26877 . . . 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 1127 . . 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 2999 . . . . 5 (𝜑𝐶𝐵)
46 cgraid.1 . . . . . 6 (𝜑𝐴𝐵)
4746necomd 2999 . . . . 5 (𝜑𝐵𝐴)
481, 14, 21, 8, 8, 6, 4, 10, 2, 45, 47hlcgrex 26977 . . . 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 26977 . . . 4 (𝜑 → ∃𝑦𝑃 (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))
50 reeanv 3294 . . . 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 583 . . 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 3207 . 2 (𝜑 → ∃𝑥𝑃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐵𝑦”⟩ ∧ 𝑥(𝐾𝐵)𝐶𝑦(𝐾𝐵)𝐴))
531, 14, 21, 4, 6, 8, 10, 10, 8, 6iscgra 27170 . 2 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐶𝐵𝐴”⟩ ↔ ∃𝑥𝑃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐵𝑦”⟩ ∧ 𝑥(𝐾𝐵)𝐶𝑦(𝐾𝐵)𝐴)))
5452, 53mpbird 256 1 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐶𝐵𝐴”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 844  w3a 1086   = wceq 1539  wcel 2106  wne 2943  wrex 3065   class class class wbr 5074  cfv 6433  (class class class)co 7275  ⟨“cs3 14555  Basecbs 16912  distcds 16971  TarskiGcstrkg 26788  Itvcitv 26794  LineGclng 26795  cgrGccgrg 26871  ≤Gcleg 26943  hlGchlg 26961  cgrAccgra 27168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-hash 14045  df-word 14218  df-concat 14274  df-s1 14301  df-s2 14561  df-s3 14562  df-trkgc 26809  df-trkgb 26810  df-trkgcb 26811  df-trkg 26814  df-cgrg 26872  df-leg 26944  df-hlg 26962  df-cgra 27169
This theorem is referenced by:  cgraswaplr  27186  oacgr  27193  tgasa1  27219
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