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Theorem iscgra1 25502
 Description: A special version of iscgra 25501 where one distance is known to be equal. In this case, angle congruence can be written with only one quantifier. (Contributed by Thierry Arnoux, 9-Aug-2020.)
Hypotheses
Ref Expression
iscgra.p 𝑃 = (Base‘𝐺)
iscgra.i 𝐼 = (Itv‘𝐺)
iscgra.k 𝐾 = (hlG‘𝐺)
iscgra.g (𝜑𝐺 ∈ TarskiG)
iscgra.a (𝜑𝐴𝑃)
iscgra.b (𝜑𝐵𝑃)
iscgra.c (𝜑𝐶𝑃)
iscgra.d (𝜑𝐷𝑃)
iscgra.e (𝜑𝐸𝑃)
iscgra.f (𝜑𝐹𝑃)
iscgra1.m = (dist‘𝐺)
iscgra1.1 (𝜑𝐴𝐵)
iscgra1.2 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
Assertion
Ref Expression
iscgra1 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷   𝑥,𝐸   𝑥,𝐹   𝑥,𝐾   𝜑,𝑥   𝑥,𝐺   𝑥,𝐼   𝑥,𝑃
Allowed substitution hint:   (𝑥)

Proof of Theorem iscgra1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 iscgra.p . . 3 𝑃 = (Base‘𝐺)
2 iscgra.i . . 3 𝐼 = (Itv‘𝐺)
3 iscgra.k . . 3 𝐾 = (hlG‘𝐺)
4 iscgra.g . . 3 (𝜑𝐺 ∈ TarskiG)
5 iscgra.a . . 3 (𝜑𝐴𝑃)
6 iscgra.b . . 3 (𝜑𝐵𝑃)
7 iscgra.c . . 3 (𝜑𝐶𝑃)
8 iscgra.d . . 3 (𝜑𝐷𝑃)
9 iscgra.e . . 3 (𝜑𝐸𝑃)
10 iscgra.f . . 3 (𝜑𝐹𝑃)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10iscgra 25501 . 2 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑦𝑃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)))
129ad3antrrr 762 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐸𝑃)
136ad3antrrr 762 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐵𝑃)
145ad3antrrr 762 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐴𝑃)
154ad3antrrr 762 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐺 ∈ TarskiG)
168ad3antrrr 762 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐷𝑃)
17 iscgra1.m . . . . . . . 8 = (dist‘𝐺)
18 simpllr 795 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑦𝑃)
19 simpr2 1061 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑦(𝐾𝐸)𝐷)
201, 2, 3, 18, 16, 12, 15, 19hlne2 25301 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐷𝐸)
21 iscgra1.2 . . . . . . . . . . . 12 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
2221ad3antrrr 762 . . . . . . . . . . 11 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝐴 𝐵) = (𝐷 𝐸))
2322eqcomd 2616 . . . . . . . . . 10 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝐷 𝐸) = (𝐴 𝐵))
241, 17, 2, 15, 16, 12, 14, 13, 23, 20tgcgrneq 25178 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐴𝐵)
2524necomd 2837 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐵𝐴)
261, 2, 3, 16, 12, 12, 15, 20hlid 25304 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐷(𝐾𝐸)𝐷)
27 eqid 2610 . . . . . . . . . . 11 (cgrG‘𝐺) = (cgrG‘𝐺)
287ad3antrrr 762 . . . . . . . . . . 11 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐶𝑃)
29 simplr 788 . . . . . . . . . . 11 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑥𝑃)
30 simpr1 1060 . . . . . . . . . . 11 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩)
311, 17, 2, 27, 15, 14, 13, 28, 18, 12, 29, 30cgr3simp1 25215 . . . . . . . . . 10 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝐴 𝐵) = (𝑦 𝐸))
3231eqcomd 2616 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝑦 𝐸) = (𝐴 𝐵))
331, 17, 2, 15, 18, 12, 14, 13, 32tgcgrcomlr 25175 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝐸 𝑦) = (𝐵 𝐴))
341, 17, 2, 15, 16, 12, 14, 13, 23tgcgrcomlr 25175 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝐸 𝐷) = (𝐵 𝐴))
351, 2, 3, 12, 13, 14, 15, 16, 17, 20, 25, 18, 16, 19, 26, 33, 34hlcgreulem 25312 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑦 = 𝐷)
36 simpr3 1062 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑥(𝐾𝐸)𝐹)
3735, 30, 36jca32 556 . . . . . 6 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
38 simprrl 800 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩)
39 id 22 . . . . . . . . 9 (𝑦 = 𝐷𝑦 = 𝐷)
4039ad2antrl 760 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝑦 = 𝐷)
418ad3antrrr 762 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐷𝑃)
429ad3antrrr 762 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐸𝑃)
434ad3antrrr 762 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐺 ∈ TarskiG)
44 iscgra1.1 . . . . . . . . . . 11 (𝜑𝐴𝐵)
451, 17, 2, 4, 5, 6, 8, 9, 21, 44tgcgrneq 25178 . . . . . . . . . 10 (𝜑𝐷𝐸)
4645ad3antrrr 762 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐷𝐸)
471, 2, 3, 41, 41, 42, 43, 46hlid 25304 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐷(𝐾𝐸)𝐷)
4840, 47eqbrtrd 4605 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝑦(𝐾𝐸)𝐷)
49 simprrr 801 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝑥(𝐾𝐸)𝐹)
5038, 48, 493jca 1235 . . . . . 6 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹))
5137, 50impbida 873 . . . . 5 (((𝜑𝑦𝑃) ∧ 𝑥𝑃) → ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹) ↔ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))))
5251rexbidva 3031 . . . 4 ((𝜑𝑦𝑃) → (∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹) ↔ ∃𝑥𝑃 (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))))
53 r19.42v 3073 . . . 4 (∃𝑥𝑃 (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)) ↔ (𝑦 = 𝐷 ∧ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
5452, 53syl6bb 275 . . 3 ((𝜑𝑦𝑃) → (∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹) ↔ (𝑦 = 𝐷 ∧ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))))
5554rexbidva 3031 . 2 (𝜑 → (∃𝑦𝑃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹) ↔ ∃𝑦𝑃 (𝑦 = 𝐷 ∧ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))))
56 eqidd 2611 . . . . . . . 8 (𝑦 = 𝐷𝐸 = 𝐸)
57 eqidd 2611 . . . . . . . 8 (𝑦 = 𝐷𝑥 = 𝑥)
5839, 56, 57s3eqd 13460 . . . . . . 7 (𝑦 = 𝐷 → ⟨“𝑦𝐸𝑥”⟩ = ⟨“𝐷𝐸𝑥”⟩)
5958breq2d 4595 . . . . . 6 (𝑦 = 𝐷 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩))
6059anbi1d 737 . . . . 5 (𝑦 = 𝐷 → ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹) ↔ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
6160rexbidv 3034 . . . 4 (𝑦 = 𝐷 → (∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹) ↔ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
6261ceqsrexv 3306 . . 3 (𝐷𝑃 → (∃𝑦𝑃 (𝑦 = 𝐷 ∧ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)) ↔ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
638, 62syl 17 . 2 (𝜑 → (∃𝑦𝑃 (𝑦 = 𝐷 ∧ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)) ↔ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
6411, 55, 633bitrd 293 1 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ⟨“cs3 13438  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135  cgrGccgrg 25205  hlGchlg 25295  cgrAccgra 25499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445  df-trkgc 25147  df-trkgb 25148  df-trkgcb 25149  df-trkg 25152  df-cgrg 25206  df-hlg 25296  df-cgra 25500 This theorem is referenced by:  acopyeu  25525
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