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Theorem cgratr 26139
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 2825 . . . . 5 (dist‘𝐺) = (dist‘𝐺)
3 eqid 2825 . . . . 5 (cgrG‘𝐺) = (cgrG‘𝐺)
4 cgraid.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54ad3antrrr 721 . . . . 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 721 . . . . 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 721 . . . . 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 721 . . . . 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 793 . . . . 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 721 . . . . 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 785 . . . . 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 798 . . . . . . 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 2831 . . . . . 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 25799 . . . . 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 800 . . . . . 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 2831 . . . . 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 721 . . . . . . . 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 721 . . . . . . . 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 721 . . . . . . . 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 721 . . . . . . . 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 793 . . . . . . . 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 732 . . . . . . . 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 785 . . . . . . . 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 1252 . . . . . . . 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 25841 . . . . . . 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 721 . . . . . . . 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 721 . . . . . . . 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 732 . . . . . . . . 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 732 . . . . . . . . 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 721 . . . . . . . . 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 732 . . . . . . . . . 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 732 . . . . . . . . . . 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 732 . . . . . . . . . . 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 797 . . . . . . . . . . . 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 721 . . . . . . . . . . 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 26132 . . . . . . . . . 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 799 . . . . . . . . . . 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 721 . . . . . . . . . 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 26133 . . . . . . . . 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 1254 . . . . . . . . 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 1256 . . . . . . . . 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 25839 . . . . . . . . . . . 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 2831 . . . . . . . . . . 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 25799 . . . . . . . . . 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 721 . . . . . . . . . 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 2861 . . . . . . . . 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 25840 . . . . . . . . . . 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 2831 . . . . . . . . . 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 721 . . . . . . . . . 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 2861 . . . . . . . . 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 26134 . . . . . . . 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 25799 . . . . . . 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 2861 . . . . . 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 26125 . . . . . . . 8 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑢𝑃𝑣𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)))
6866, 67mpbid 224 . . . . . . 7 (𝜑 → ∃𝑢𝑃𝑣𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹))
6968ad3antrrr 721 . . . . . 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 3291 . . . . 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 25835 . . . 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 1162 . . 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 26130 . . . . . 6 (𝜑𝑈𝐻)
7473necomd 3054 . . . . 5 (𝜑𝐻𝑈)
751, 16, 34, 4, 6, 8, 10, 35, 27, 37, 66cgrane1 26128 . . . . . 6 (𝜑𝐴𝐵)
7675necomd 3054 . . . . 5 (𝜑𝐵𝐴)
771, 16, 34, 13, 8, 6, 4, 42, 2, 74, 76hlcgrex 25935 . . . 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 26131 . . . . . 6 (𝜑𝑈𝐽)
7978necomd 3054 . . . . 5 (𝜑𝐽𝑈)
801, 16, 34, 4, 6, 8, 10, 35, 27, 37, 66cgrane2 26129 . . . . 5 (𝜑𝐵𝐶)
811, 16, 34, 13, 8, 10, 4, 40, 2, 79, 80hlcgrex 25935 . . . 4 (𝜑 → ∃𝑦𝑃 (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))
82 reeanv 3317 . . . 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 578 . . 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 3229 . 2 (𝜑 → ∃𝑥𝑃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝑈𝑦”⟩ ∧ 𝑥(𝐾𝑈)𝐻𝑦(𝐾𝑈)𝐽))
851, 16, 34, 4, 6, 8, 10, 42, 13, 40iscgra 26125 . 2 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩ ↔ ∃𝑥𝑃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝑈𝑦”⟩ ∧ 𝑥(𝐾𝑈)𝐻𝑦(𝐾𝑈)𝐽)))
8684, 85mpbird 249 1 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1111   = wceq 1656  wcel 2164  wrex 3118   class class class wbr 4875  cfv 6127  (class class class)co 6910  ⟨“cs3 13970  Basecbs 16229  distcds 16321  TarskiGcstrkg 25749  Itvcitv 25755  cgrGccgrg 25829  hlGchlg 25919  cgrAccgra 26123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-oadd 7835  df-er 8014  df-map 8129  df-pm 8130  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-card 9085  df-cda 9312  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-2 11421  df-3 11422  df-n0 11626  df-xnn0 11698  df-z 11712  df-uz 11976  df-fz 12627  df-fzo 12768  df-hash 13418  df-word 13582  df-concat 13638  df-s1 13663  df-s2 13976  df-s3 13977  df-trkgc 25767  df-trkgb 25768  df-trkgcb 25769  df-trkg 25772  df-cgrg 25830  df-leg 25902  df-hlg 25920  df-cgra 26124
This theorem is referenced by:  cgraswaplr  26140  sacgr  26146  sacgrOLD  26147  oacgr  26148  tgasa1  26164  isoas  26170
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