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Theorem iscgra1 25993
Description: A special version of iscgra 25992 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 25992 . 2 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑦𝑃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)))
129ad3antrrr 721 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐸𝑃)
136ad3antrrr 721 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐵𝑃)
145ad3antrrr 721 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐴𝑃)
154ad3antrrr 721 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐺 ∈ TarskiG)
168ad3antrrr 721 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐷𝑃)
17 iscgra1.m . . . . . . . 8 = (dist‘𝐺)
18 simpllr 793 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑦𝑃)
19 simpr2 1250 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑦(𝐾𝐸)𝐷)
201, 2, 3, 18, 16, 12, 15, 19hlne2 25792 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐷𝐸)
21 iscgra1.2 . . . . . . . . . . . 12 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
2221ad3antrrr 721 . . . . . . . . . . 11 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝐴 𝐵) = (𝐷 𝐸))
2322eqcomd 2771 . . . . . . . . . 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 25669 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐴𝐵)
2524necomd 2992 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐵𝐴)
261, 2, 3, 16, 12, 12, 15, 20hlid 25795 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐷(𝐾𝐸)𝐷)
27 eqid 2765 . . . . . . . . . . 11 (cgrG‘𝐺) = (cgrG‘𝐺)
287ad3antrrr 721 . . . . . . . . . . 11 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐶𝑃)
29 simplr 785 . . . . . . . . . . 11 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑥𝑃)
30 simpr1 1248 . . . . . . . . . . 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 25706 . . . . . . . . . 10 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝐴 𝐵) = (𝑦 𝐸))
3231eqcomd 2771 . . . . . . . . 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 25666 . . . . . . . 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 25666 . . . . . . . 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 25803 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑦 = 𝐷)
36 simpr3 1252 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑥(𝐾𝐸)𝐹)
3735, 30, 36jca32 511 . . . . . 6 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
38 simprrl 799 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩)
39 id 22 . . . . . . . . 9 (𝑦 = 𝐷𝑦 = 𝐷)
4039ad2antrl 719 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝑦 = 𝐷)
418ad3antrrr 721 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐷𝑃)
429ad3antrrr 721 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐸𝑃)
434ad3antrrr 721 . . . . . . . . 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 25669 . . . . . . . . . 10 (𝜑𝐷𝐸)
4645ad3antrrr 721 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐷𝐸)
471, 2, 3, 41, 41, 42, 43, 46hlid 25795 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐷(𝐾𝐸)𝐷)
4840, 47eqbrtrd 4831 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝑦(𝐾𝐸)𝐷)
49 simprrr 800 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝑥(𝐾𝐸)𝐹)
5038, 48, 493jca 1158 . . . . . 6 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹))
5137, 50impbida 835 . . . . 5 (((𝜑𝑦𝑃) ∧ 𝑥𝑃) → ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹) ↔ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))))
5251rexbidva 3196 . . . 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 3239 . . . 4 (∃𝑥𝑃 (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)) ↔ (𝑦 = 𝐷 ∧ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
5452, 53syl6bb 278 . . 3 ((𝜑𝑦𝑃) → (∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹) ↔ (𝑦 = 𝐷 ∧ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))))
5554rexbidva 3196 . 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 2766 . . . . . . . 8 (𝑦 = 𝐷𝐸 = 𝐸)
57 eqidd 2766 . . . . . . . 8 (𝑦 = 𝐷𝑥 = 𝑥)
5839, 56, 57s3eqd 13893 . . . . . . 7 (𝑦 = 𝐷 → ⟨“𝑦𝐸𝑥”⟩ = ⟨“𝐷𝐸𝑥”⟩)
5958breq2d 4821 . . . . . 6 (𝑦 = 𝐷 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩))
6059anbi1d 623 . . . . 5 (𝑦 = 𝐷 → ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹) ↔ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
6160rexbidv 3199 . . . 4 (𝑦 = 𝐷 → (∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹) ↔ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
6261ceqsrexv 3489 . . 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 296 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wrex 3056   class class class wbr 4809  cfv 6068  (class class class)co 6842  ⟨“cs3 13871  Basecbs 16130  distcds 16223  TarskiGcstrkg 25620  Itvcitv 25626  cgrGccgrg 25696  hlGchlg 25786  cgrAccgra 25990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-xnn0 11611  df-z 11625  df-uz 11887  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13487  df-concat 13542  df-s1 13567  df-s2 13877  df-s3 13878  df-trkgc 25638  df-trkgb 25639  df-trkgcb 25640  df-trkg 25643  df-cgrg 25697  df-hlg 25787  df-cgra 25991
This theorem is referenced by:  acopyeu  26016
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