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Theorem cgratr 25936
Description: Angle congruence is transitive. Theorem 11.8 of [Schwabhauser] p. 97. (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)
cgracom.d (𝜑𝐷𝑃)
cgracom.e (𝜑𝐸𝑃)
cgracom.f (𝜑𝐹𝑃)
cgracom.1 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
cgratr.h (𝜑𝐻𝑃)
cgratr.i (𝜑𝑈𝑃)
cgratr.j (𝜑𝐽𝑃)
cgratr.1 (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩)
Assertion
Ref Expression
cgratr (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩)

Proof of Theorem cgratr
Dummy variables 𝑥 𝑦 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cgraid.p . . . . 5 𝑃 = (Base‘𝐺)
2 eqid 2771 . . . . 5 (dist‘𝐺) = (dist‘𝐺)
3 eqid 2771 . . . . 5 (cgrG‘𝐺) = (cgrG‘𝐺)
4 cgraid.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54ad3antrrr 709 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝐺 ∈ TarskiG)
6 cgraid.a . . . . . 6 (𝜑𝐴𝑃)
76ad3antrrr 709 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝐴𝑃)
8 cgraid.b . . . . . 6 (𝜑𝐵𝑃)
98ad3antrrr 709 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝐵𝑃)
10 cgraid.c . . . . . 6 (𝜑𝐶𝑃)
1110ad3antrrr 709 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝐶𝑃)
12 simpllr 760 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝑥𝑃)
13 cgratr.i . . . . . 6 (𝜑𝑈𝑃)
1413ad3antrrr 709 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝑈𝑃)
15 simplr 752 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝑦𝑃)
16 cgraid.i . . . . . 6 𝐼 = (Itv‘𝐺)
17 simprlr 765 . . . . . . 7 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴))
1817eqcomd 2777 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐵(dist‘𝐺)𝐴) = (𝑈(dist‘𝐺)𝑥))
191, 2, 16, 5, 9, 7, 14, 12, 18tgcgrcomlr 25596 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐴(dist‘𝐺)𝐵) = (𝑥(dist‘𝐺)𝑈))
20 simprrr 767 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶))
2120eqcomd 2777 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐵(dist‘𝐺)𝐶) = (𝑈(dist‘𝐺)𝑦))
225ad3antrrr 709 . . . . . . . 8 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → 𝐺 ∈ TarskiG)
237ad3antrrr 709 . . . . . . . 8 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → 𝐴𝑃)
249ad3antrrr 709 . . . . . . . 8 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → 𝐵𝑃)
2511ad3antrrr 709 . . . . . . . 8 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → 𝐶𝑃)
26 simpllr 760 . . . . . . . 8 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → 𝑢𝑃)
27 cgracom.e . . . . . . . . 9 (𝜑𝐸𝑃)
2827ad6antr 720 . . . . . . . 8 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → 𝐸𝑃)
29 simplr 752 . . . . . . . 8 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → 𝑣𝑃)
30 simpr1 1233 . . . . . . . 8 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩)
311, 2, 16, 3, 22, 23, 24, 25, 26, 28, 29, 30cgr3simp3 25638 . . . . . . 7 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → (𝐶(dist‘𝐺)𝐴) = (𝑣(dist‘𝐺)𝑢))
3212ad3antrrr 709 . . . . . . . 8 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → 𝑥𝑃)
3315ad3antrrr 709 . . . . . . . 8 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → 𝑦𝑃)
34 cgraid.k . . . . . . . . 9 𝐾 = (hlG‘𝐺)
35 cgracom.d . . . . . . . . . 10 (𝜑𝐷𝑃)
3635ad6antr 720 . . . . . . . . 9 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → 𝐷𝑃)
37 cgracom.f . . . . . . . . . 10 (𝜑𝐹𝑃)
3837ad6antr 720 . . . . . . . . 9 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → 𝐹𝑃)
3914ad3antrrr 709 . . . . . . . . 9 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → 𝑈𝑃)
40 cgratr.j . . . . . . . . . . 11 (𝜑𝐽𝑃)
4140ad6antr 720 . . . . . . . . . 10 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → 𝐽𝑃)
42 cgratr.h . . . . . . . . . . . 12 (𝜑𝐻𝑃)
4342ad6antr 720 . . . . . . . . . . 11 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → 𝐻𝑃)
44 cgratr.1 . . . . . . . . . . . 12 (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩)
4544ad6antr 720 . . . . . . . . . . 11 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩)
46 simprll 764 . . . . . . . . . . . 12 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝑥(𝐾𝑈)𝐻)
4746ad3antrrr 709 . . . . . . . . . . 11 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → 𝑥(𝐾𝑈)𝐻)
481, 16, 34, 22, 36, 28, 38, 43, 39, 41, 45, 32, 47cgrahl1 25929 . . . . . . . . . 10 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑥𝑈𝐽”⟩)
49 simprrl 766 . . . . . . . . . . 11 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → 𝑦(𝐾𝑈)𝐽)
5049ad3antrrr 709 . . . . . . . . . 10 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → 𝑦(𝐾𝑈)𝐽)
511, 16, 34, 22, 36, 28, 38, 32, 39, 41, 48, 33, 50cgrahl2 25930 . . . . . . . . 9 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑥𝑈𝑦”⟩)
52 simpr2 1235 . . . . . . . . 9 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → 𝑢(𝐾𝐸)𝐷)
53 simpr3 1237 . . . . . . . . 9 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → 𝑣(𝐾𝐸)𝐹)
541, 2, 16, 3, 22, 23, 24, 25, 26, 28, 29, 30cgr3simp1 25636 . . . . . . . . . . . 12 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → (𝐴(dist‘𝐺)𝐵) = (𝑢(dist‘𝐺)𝐸))
5554eqcomd 2777 . . . . . . . . . . 11 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → (𝑢(dist‘𝐺)𝐸) = (𝐴(dist‘𝐺)𝐵))
561, 2, 16, 22, 26, 28, 23, 24, 55tgcgrcomlr 25596 . . . . . . . . . 10 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → (𝐸(dist‘𝐺)𝑢) = (𝐵(dist‘𝐺)𝐴))
5718ad3antrrr 709 . . . . . . . . . 10 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → (𝐵(dist‘𝐺)𝐴) = (𝑈(dist‘𝐺)𝑥))
5856, 57eqtrd 2805 . . . . . . . . 9 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → (𝐸(dist‘𝐺)𝑢) = (𝑈(dist‘𝐺)𝑥))
591, 2, 16, 3, 22, 23, 24, 25, 26, 28, 29, 30cgr3simp2 25637 . . . . . . . . . . 11 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → (𝐵(dist‘𝐺)𝐶) = (𝐸(dist‘𝐺)𝑣))
6059eqcomd 2777 . . . . . . . . . 10 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → (𝐸(dist‘𝐺)𝑣) = (𝐵(dist‘𝐺)𝐶))
6121ad3antrrr 709 . . . . . . . . . 10 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → (𝐵(dist‘𝐺)𝐶) = (𝑈(dist‘𝐺)𝑦))
6260, 61eqtrd 2805 . . . . . . . . 9 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → (𝐸(dist‘𝐺)𝑣) = (𝑈(dist‘𝐺)𝑦))
631, 16, 34, 22, 36, 28, 38, 32, 39, 33, 51, 26, 2, 29, 52, 53, 58, 62cgracgr 25931 . . . . . . . 8 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → (𝑢(dist‘𝐺)𝑣) = (𝑥(dist‘𝐺)𝑦))
641, 2, 16, 22, 26, 29, 32, 33, 63tgcgrcomlr 25596 . . . . . . 7 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → (𝑣(dist‘𝐺)𝑢) = (𝑦(dist‘𝐺)𝑥))
6531, 64eqtrd 2805 . . . . . 6 (((((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) ∧ 𝑢𝑃) ∧ 𝑣𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)) → (𝐶(dist‘𝐺)𝐴) = (𝑦(dist‘𝐺)𝑥))
66 cgracom.1 . . . . . . . 8 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
671, 16, 34, 4, 6, 8, 10, 35, 27, 37iscgra 25922 . . . . . . . 8 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑢𝑃𝑣𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)))
6866, 67mpbid 222 . . . . . . 7 (𝜑 → ∃𝑢𝑃𝑣𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹))
6968ad3antrrr 709 . . . . . 6 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → ∃𝑢𝑃𝑣𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹))
7065, 69r19.29vva 3229 . . . . 5 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (𝐶(dist‘𝐺)𝐴) = (𝑦(dist‘𝐺)𝑥))
711, 2, 3, 5, 7, 9, 11, 12, 14, 15, 19, 21, 70trgcgr 25632 . . . 4 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝑈𝑦”⟩)
7271, 46, 493jca 1122 . . 3 ((((𝜑𝑥𝑃) ∧ 𝑦𝑃) ∧ ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))) → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝑈𝑦”⟩ ∧ 𝑥(𝐾𝑈)𝐻𝑦(𝐾𝑈)𝐽))
731, 16, 34, 4, 35, 27, 37, 42, 13, 40, 44cgrane3 25927 . . . . . 6 (𝜑𝑈𝐻)
7473necomd 2998 . . . . 5 (𝜑𝐻𝑈)
751, 16, 34, 4, 6, 8, 10, 35, 27, 37, 66cgrane1 25925 . . . . . 6 (𝜑𝐴𝐵)
7675necomd 2998 . . . . 5 (𝜑𝐵𝐴)
771, 16, 34, 13, 8, 6, 4, 42, 2, 74, 76hlcgrex 25732 . . . 4 (𝜑 → ∃𝑥𝑃 (𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)))
781, 16, 34, 4, 35, 27, 37, 42, 13, 40, 44cgrane4 25928 . . . . . 6 (𝜑𝑈𝐽)
7978necomd 2998 . . . . 5 (𝜑𝐽𝑈)
801, 16, 34, 4, 6, 8, 10, 35, 27, 37, 66cgrane2 25926 . . . . 5 (𝜑𝐵𝐶)
811, 16, 34, 13, 8, 10, 4, 40, 2, 79, 80hlcgrex 25732 . . . 4 (𝜑 → ∃𝑦𝑃 (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))
82 reeanv 3255 . . . 4 (∃𝑥𝑃𝑦𝑃 ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶))) ↔ (∃𝑥𝑃 (𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ ∃𝑦𝑃 (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶))))
8377, 81, 82sylanbrc 572 . . 3 (𝜑 → ∃𝑥𝑃𝑦𝑃 ((𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)) ∧ (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶))))
8472, 83reximddv2 3168 . 2 (𝜑 → ∃𝑥𝑃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝑈𝑦”⟩ ∧ 𝑥(𝐾𝑈)𝐻𝑦(𝐾𝑈)𝐽))
851, 16, 34, 4, 6, 8, 10, 42, 13, 40iscgra 25922 . 2 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩ ↔ ∃𝑥𝑃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝑈𝑦”⟩ ∧ 𝑥(𝐾𝑈)𝐻𝑦(𝐾𝑈)𝐽)))
8684, 85mpbird 247 1 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wrex 3062   class class class wbr 4787  cfv 6030  (class class class)co 6796  ⟨“cs3 13796  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556  cgrGccgrg 25626  hlGchlg 25716  cgrAccgra 25920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7900  df-map 8015  df-pm 8016  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-card 8969  df-cda 9196  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-2 11285  df-3 11286  df-n0 11500  df-xnn0 11571  df-z 11585  df-uz 11894  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-concat 13497  df-s1 13498  df-s2 13802  df-s3 13803  df-trkgc 25568  df-trkgb 25569  df-trkgcb 25570  df-trkg 25573  df-cgrg 25627  df-leg 25699  df-hlg 25717  df-cgra 25921
This theorem is referenced by:  cgraswaplr  25937  sacgr  25943  oacgr  25944  tgasa1  25960  isoas  25965
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