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Theorem iscgra1 25901
Description: A special version of iscgra 25900 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 25900 . 2 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑦𝑃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)))
129ad3antrrr 768 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐸𝑃)
136ad3antrrr 768 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐵𝑃)
145ad3antrrr 768 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐴𝑃)
154ad3antrrr 768 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐺 ∈ TarskiG)
168ad3antrrr 768 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐷𝑃)
17 iscgra1.m . . . . . . . 8 = (dist‘𝐺)
18 simpllr 817 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑦𝑃)
19 simpr2 1236 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑦(𝐾𝐸)𝐷)
201, 2, 3, 18, 16, 12, 15, 19hlne2 25700 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐷𝐸)
21 iscgra1.2 . . . . . . . . . . . 12 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
2221ad3antrrr 768 . . . . . . . . . . 11 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝐴 𝐵) = (𝐷 𝐸))
2322eqcomd 2766 . . . . . . . . . 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 25577 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐴𝐵)
2524necomd 2987 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐵𝐴)
261, 2, 3, 16, 12, 12, 15, 20hlid 25703 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐷(𝐾𝐸)𝐷)
27 eqid 2760 . . . . . . . . . . 11 (cgrG‘𝐺) = (cgrG‘𝐺)
287ad3antrrr 768 . . . . . . . . . . 11 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐶𝑃)
29 simplr 809 . . . . . . . . . . 11 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑥𝑃)
30 simpr1 1234 . . . . . . . . . . 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 25614 . . . . . . . . . 10 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝐴 𝐵) = (𝑦 𝐸))
3231eqcomd 2766 . . . . . . . . 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 25574 . . . . . . . 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 25574 . . . . . . . 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 25711 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑦 = 𝐷)
36 simpr3 1238 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑥(𝐾𝐸)𝐹)
3735, 30, 36jca32 559 . . . . . 6 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
38 simprrl 823 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩)
39 id 22 . . . . . . . . 9 (𝑦 = 𝐷𝑦 = 𝐷)
4039ad2antrl 766 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝑦 = 𝐷)
418ad3antrrr 768 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐷𝑃)
429ad3antrrr 768 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐸𝑃)
434ad3antrrr 768 . . . . . . . . 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 25577 . . . . . . . . . 10 (𝜑𝐷𝐸)
4645ad3antrrr 768 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐷𝐸)
471, 2, 3, 41, 41, 42, 43, 46hlid 25703 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐷(𝐾𝐸)𝐷)
4840, 47eqbrtrd 4826 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝑦(𝐾𝐸)𝐷)
49 simprrr 824 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝑥(𝐾𝐸)𝐹)
5038, 48, 493jca 1123 . . . . . 6 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹))
5137, 50impbida 913 . . . . 5 (((𝜑𝑦𝑃) ∧ 𝑥𝑃) → ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹) ↔ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))))
5251rexbidva 3187 . . . 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 3230 . . . 4 (∃𝑥𝑃 (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)) ↔ (𝑦 = 𝐷 ∧ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
5452, 53syl6bb 276 . . 3 ((𝜑𝑦𝑃) → (∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹) ↔ (𝑦 = 𝐷 ∧ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))))
5554rexbidva 3187 . 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 2761 . . . . . . . 8 (𝑦 = 𝐷𝐸 = 𝐸)
57 eqidd 2761 . . . . . . . 8 (𝑦 = 𝐷𝑥 = 𝑥)
5839, 56, 57s3eqd 13809 . . . . . . 7 (𝑦 = 𝐷 → ⟨“𝑦𝐸𝑥”⟩ = ⟨“𝐷𝐸𝑥”⟩)
5958breq2d 4816 . . . . . 6 (𝑦 = 𝐷 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩))
6059anbi1d 743 . . . . 5 (𝑦 = 𝐷 → ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹) ↔ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
6160rexbidv 3190 . . . 4 (𝑦 = 𝐷 → (∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹) ↔ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
6261ceqsrexv 3475 . . 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 294 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wrex 3051   class class class wbr 4804  cfv 6049  (class class class)co 6813  ⟨“cs3 13787  Basecbs 16059  distcds 16152  TarskiGcstrkg 25528  Itvcitv 25534  cgrGccgrg 25604  hlGchlg 25694  cgrAccgra 25898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-card 8955  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-xnn0 11556  df-z 11570  df-uz 11880  df-fz 12520  df-fzo 12660  df-hash 13312  df-word 13485  df-concat 13487  df-s1 13488  df-s2 13793  df-s3 13794  df-trkgc 25546  df-trkgb 25547  df-trkgcb 25548  df-trkg 25551  df-cgrg 25605  df-hlg 25695  df-cgra 25899
This theorem is referenced by:  acopyeu  25924
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