MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cgratr Structured version   Visualization version   GIF version

Theorem cgratr 26595
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 2821 . . . . 5 (dist‘𝐺) = (dist‘𝐺)
3 eqid 2821 . . . . 5 (cgrG‘𝐺) = (cgrG‘𝐺)
4 cgraid.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54ad3antrrr 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 775 . . . . 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 729 . . . . 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 768 . . . . 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 779 . . . . . . 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 2827 . . . . . 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 26252 . . . . 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 781 . . . . . 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 2827 . . . . 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 729 . . . . . . . 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 729 . . . . . . . 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 729 . . . . . . . 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 729 . . . . . . . 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 775 . . . . . . . 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 735 . . . . . . . 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 768 . . . . . . . 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 1191 . . . . . . . 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 26294 . . . . . . 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 729 . . . . . . . 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 729 . . . . . . . 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 735 . . . . . . . . 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 735 . . . . . . . . 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 729 . . . . . . . . 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 735 . . . . . . . . . 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 735 . . . . . . . . . . 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 735 . . . . . . . . . . 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 778 . . . . . . . . . . . 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 729 . . . . . . . . . . 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 26588 . . . . . . . . . 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 780 . . . . . . . . . . 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 729 . . . . . . . . . 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 26589 . . . . . . . . 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 1192 . . . . . . . . 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 1193 . . . . . . . . 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 26292 . . . . . . . . . . . 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 2827 . . . . . . . . . . 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 26252 . . . . . . . . . 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 729 . . . . . . . . . 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 2856 . . . . . . . . 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 26293 . . . . . . . . . 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) = (𝐸(dist‘𝐺)𝑣))
6021ad3antrrr 729 . . . . . . . . . 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))
6159, 60eqtr3d 2858 . . . . . . . . 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))
621, 16, 34, 22, 36, 28, 38, 32, 39, 33, 51, 26, 2, 29, 52, 53, 58, 61cgracgr 26590 . . . . . . . 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))
631, 2, 16, 22, 26, 29, 32, 33, 62tgcgrcomlr 26252 . . . . . . 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))
6431, 63eqtrd 2856 . . . . . 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))
65 cgracom.1 . . . . . . . 8 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
661, 16, 34, 4, 6, 8, 10, 35, 27, 37iscgra 26581 . . . . . . . 8 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑢𝑃𝑣𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹)))
6765, 66mpbid 235 . . . . . . 7 (𝜑 → ∃𝑢𝑃𝑣𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑢𝐸𝑣”⟩ ∧ 𝑢(𝐾𝐸)𝐷𝑣(𝐾𝐸)𝐹))
6867ad3antrrr 729 . . . . . 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))
6964, 68r19.29vva 3321 . . . . 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))
701, 2, 3, 5, 7, 9, 11, 12, 14, 15, 19, 21, 69trgcgr 26288 . . . 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”⟩)
7170, 46, 493jca 1125 . . 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))
721, 16, 34, 4, 35, 27, 37, 42, 13, 40, 44cgrane3 26586 . . . . . 6 (𝜑𝑈𝐻)
7372necomd 3062 . . . . 5 (𝜑𝐻𝑈)
741, 16, 34, 4, 6, 8, 10, 35, 27, 37, 65cgrane1 26584 . . . . . 6 (𝜑𝐴𝐵)
7574necomd 3062 . . . . 5 (𝜑𝐵𝐴)
761, 16, 34, 13, 8, 6, 4, 42, 2, 73, 75hlcgrex 26388 . . . 4 (𝜑 → ∃𝑥𝑃 (𝑥(𝐾𝑈)𝐻 ∧ (𝑈(dist‘𝐺)𝑥) = (𝐵(dist‘𝐺)𝐴)))
771, 16, 34, 4, 35, 27, 37, 42, 13, 40, 44cgrane4 26587 . . . . . 6 (𝜑𝑈𝐽)
7877necomd 3062 . . . . 5 (𝜑𝐽𝑈)
791, 16, 34, 4, 6, 8, 10, 35, 27, 37, 65cgrane2 26585 . . . . 5 (𝜑𝐵𝐶)
801, 16, 34, 13, 8, 10, 4, 40, 2, 78, 79hlcgrex 26388 . . . 4 (𝜑 → ∃𝑦𝑃 (𝑦(𝐾𝑈)𝐽 ∧ (𝑈(dist‘𝐺)𝑦) = (𝐵(dist‘𝐺)𝐶)))
81 reeanv 3352 . . . 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))))
8276, 80, 81sylanbrc 586 . . 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))))
8371, 82reximddv2 3264 . 2 (𝜑 → ∃𝑥𝑃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝑈𝑦”⟩ ∧ 𝑥(𝐾𝑈)𝐻𝑦(𝐾𝑈)𝐽))
841, 16, 34, 4, 6, 8, 10, 42, 13, 40iscgra 26581 . 2 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩ ↔ ∃𝑥𝑃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝑈𝑦”⟩ ∧ 𝑥(𝐾𝑈)𝐻𝑦(𝐾𝑈)𝐽)))
8583, 84mpbird 260 1 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wrex 3127   class class class wbr 5039  cfv 6328  (class class class)co 7130  ⟨“cs3 14183  Basecbs 16461  distcds 16552  TarskiGcstrkg 26202  Itvcitv 26208  cgrGccgrg 26282  hlGchlg 26372  cgrAccgra 26579
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-map 8383  df-pm 8384  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-dju 9306  df-card 9344  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-2 11678  df-3 11679  df-n0 11876  df-xnn0 11946  df-z 11960  df-uz 12222  df-fz 12876  df-fzo 13017  df-hash 13675  df-word 13846  df-concat 13902  df-s1 13929  df-s2 14189  df-s3 14190  df-trkgc 26220  df-trkgb 26221  df-trkgcb 26222  df-trkg 26225  df-cgrg 26283  df-leg 26355  df-hlg 26373  df-cgra 26580
This theorem is referenced by:  cgraswaplr  26597  sacgr  26603  oacgr  26604  tgasa1  26630
  Copyright terms: Public domain W3C validator