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Theorem iscgra1 25636
Description: A special version of iscgra 25635 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 25635 . 2 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑦𝑃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)))
129ad3antrrr 765 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐸𝑃)
136ad3antrrr 765 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐵𝑃)
145ad3antrrr 765 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐴𝑃)
154ad3antrrr 765 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐺 ∈ TarskiG)
168ad3antrrr 765 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐷𝑃)
17 iscgra1.m . . . . . . . 8 = (dist‘𝐺)
18 simpllr 798 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑦𝑃)
19 simpr2 1066 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑦(𝐾𝐸)𝐷)
201, 2, 3, 18, 16, 12, 15, 19hlne2 25435 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐷𝐸)
21 iscgra1.2 . . . . . . . . . . . 12 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
2221ad3antrrr 765 . . . . . . . . . . 11 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝐴 𝐵) = (𝐷 𝐸))
2322eqcomd 2627 . . . . . . . . . 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 25312 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐴𝐵)
2524necomd 2845 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐵𝐴)
261, 2, 3, 16, 12, 12, 15, 20hlid 25438 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐷(𝐾𝐸)𝐷)
27 eqid 2621 . . . . . . . . . . 11 (cgrG‘𝐺) = (cgrG‘𝐺)
287ad3antrrr 765 . . . . . . . . . . 11 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐶𝑃)
29 simplr 791 . . . . . . . . . . 11 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑥𝑃)
30 simpr1 1065 . . . . . . . . . . 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 25349 . . . . . . . . . 10 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝐴 𝐵) = (𝑦 𝐸))
3231eqcomd 2627 . . . . . . . . 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 25309 . . . . . . . 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 25309 . . . . . . . 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 25446 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑦 = 𝐷)
36 simpr3 1067 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑥(𝐾𝐸)𝐹)
3735, 30, 36jca32 557 . . . . . 6 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
38 simprrl 803 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩)
39 id 22 . . . . . . . . 9 (𝑦 = 𝐷𝑦 = 𝐷)
4039ad2antrl 763 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝑦 = 𝐷)
418ad3antrrr 765 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐷𝑃)
429ad3antrrr 765 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐸𝑃)
434ad3antrrr 765 . . . . . . . . 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 25312 . . . . . . . . . 10 (𝜑𝐷𝐸)
4645ad3antrrr 765 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐷𝐸)
471, 2, 3, 41, 41, 42, 43, 46hlid 25438 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐷(𝐾𝐸)𝐷)
4840, 47eqbrtrd 4645 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝑦(𝐾𝐸)𝐷)
49 simprrr 804 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝑥(𝐾𝐸)𝐹)
5038, 48, 493jca 1240 . . . . . 6 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹))
5137, 50impbida 876 . . . . 5 (((𝜑𝑦𝑃) ∧ 𝑥𝑃) → ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹) ↔ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))))
5251rexbidva 3044 . . . 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 3086 . . . 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 3044 . 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 2622 . . . . . . . 8 (𝑦 = 𝐷𝐸 = 𝐸)
57 eqidd 2622 . . . . . . . 8 (𝑦 = 𝐷𝑥 = 𝑥)
5839, 56, 57s3eqd 13562 . . . . . . 7 (𝑦 = 𝐷 → ⟨“𝑦𝐸𝑥”⟩ = ⟨“𝐷𝐸𝑥”⟩)
5958breq2d 4635 . . . . . 6 (𝑦 = 𝐷 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩))
6059anbi1d 740 . . . . 5 (𝑦 = 𝐷 → ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹) ↔ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
6160rexbidv 3047 . . . 4 (𝑦 = 𝐷 → (∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹) ↔ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
6261ceqsrexv 3324 . . 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2909   class class class wbr 4623  cfv 5857  (class class class)co 6615  ⟨“cs3 13540  Basecbs 15800  distcds 15890  TarskiGcstrkg 25263  Itvcitv 25269  cgrGccgrg 25339  hlGchlg 25429  cgrAccgra 25633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-3 11040  df-n0 11253  df-xnn0 11324  df-z 11338  df-uz 11648  df-fz 12285  df-fzo 12423  df-hash 13074  df-word 13254  df-concat 13256  df-s1 13257  df-s2 13546  df-s3 13547  df-trkgc 25281  df-trkgb 25282  df-trkgcb 25283  df-trkg 25286  df-cgrg 25340  df-hlg 25430  df-cgra 25634
This theorem is referenced by:  acopyeu  25659
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