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Theorem cgraswap 28060
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 2732 . . . . 5 (dist‘𝐺) = (dist‘𝐺)
3 eqid 2732 . . . . 5 (cgrG‘𝐺) = (cgrG‘𝐺)
4 cgraid.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54ad3antrrr 728 . . . . 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 728 . . . . 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 728 . . . . 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 728 . . . . 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 774 . . . . 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 767 . . . . 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 778 . . . . . . 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 27720 . . . . . 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 2738 . . . . 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 780 . . . . . 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 2738 . . . . 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 2732 . . . . . . . 8 (LineG‘𝐺) = (LineG‘𝐺)
21 cgraid.k . . . . . . . . . . 11 𝐾 = (hlG‘𝐺)
22 simprll 777 . . . . . . . . . . 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 27847 . . . . . . . . . 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 871 . . . . . . . . 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 27799 . . . . . . . 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 2732 . . . . . . . . . 10 (≤G‘𝐺) = (≤G‘𝐺)
271, 14, 21, 12, 11, 9, 5ishlg 27842 . . . . . . . . . . . . 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 1144 . . . . . . . . . . 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 869 . . . . . . . . . 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 779 . . . . . . . . . . . 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 27842 . . . . . . . . . . . 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 1144 . . . . . . . . . 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 27835 . . . . . . . . 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 27756 . . . . . . . 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 27702 . . . . . . . 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 728 . . . . . . . 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 27808 . . . . . . 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 27720 . . . . . 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 2738 . . . . 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 27756 . . . 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 1128 . . 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 2996 . . . . 5 (𝜑𝐶𝐵)
46 cgraid.1 . . . . . 6 (𝜑𝐴𝐵)
4746necomd 2996 . . . . 5 (𝜑𝐵𝐴)
481, 14, 21, 8, 8, 6, 4, 10, 2, 45, 47hlcgrex 27856 . . . 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 27856 . . . 4 (𝜑 → ∃𝑦𝑃 (𝑦(𝐾𝐵)𝐴 ∧ (𝐵(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))
50 reeanv 3226 . . . 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 3212 . 2 (𝜑 → ∃𝑥𝑃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐵𝑦”⟩ ∧ 𝑥(𝐾𝐵)𝐶𝑦(𝐾𝐵)𝐴))
531, 14, 21, 4, 6, 8, 10, 10, 8, 6iscgra 28049 . 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 845  w3a 1087   = wceq 1541  wcel 2106  wne 2940  wrex 3070   class class class wbr 5147  cfv 6540  (class class class)co 7405  ⟨“cs3 14789  Basecbs 17140  distcds 17202  TarskiGcstrkg 27667  Itvcitv 27673  LineGclng 27674  cgrGccgrg 27750  ≤Gcleg 27822  hlGchlg 27840  cgrAccgra 28047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-xnn0 12541  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-hash 14287  df-word 14461  df-concat 14517  df-s1 14542  df-s2 14795  df-s3 14796  df-trkgc 27688  df-trkgb 27689  df-trkgcb 27690  df-trkg 27693  df-cgrg 27751  df-leg 27823  df-hlg 27841  df-cgra 28048
This theorem is referenced by:  cgraswaplr  28065  oacgr  28072  tgasa1  28098
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