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Theorem iscgra1 26529
Description: A special version of iscgra 26528 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 26528 . 2 (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑦𝑃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)))
129ad3antrrr 726 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐸𝑃)
136ad3antrrr 726 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐵𝑃)
145ad3antrrr 726 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐴𝑃)
154ad3antrrr 726 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐺 ∈ TarskiG)
168ad3antrrr 726 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐷𝑃)
17 iscgra1.m . . . . . . . 8 = (dist‘𝐺)
18 simpllr 772 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑦𝑃)
19 simpr2 1189 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑦(𝐾𝐸)𝐷)
201, 2, 3, 18, 16, 12, 15, 19hlne2 26325 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐷𝐸)
21 iscgra1.1 . . . . . . . . . 10 (𝜑𝐴𝐵)
2221ad3antrrr 726 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐴𝐵)
2322necomd 3076 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐵𝐴)
241, 2, 3, 16, 12, 12, 15, 20hlid 26328 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐷(𝐾𝐸)𝐷)
25 eqid 2826 . . . . . . . . . . 11 (cgrG‘𝐺) = (cgrG‘𝐺)
267ad3antrrr 726 . . . . . . . . . . 11 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝐶𝑃)
27 simplr 765 . . . . . . . . . . 11 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑥𝑃)
28 simpr1 1188 . . . . . . . . . . 11 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩)
291, 17, 2, 25, 15, 14, 13, 26, 18, 12, 27, 28cgr3simp1 26239 . . . . . . . . . 10 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝐴 𝐵) = (𝑦 𝐸))
3029eqcomd 2832 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝑦 𝐸) = (𝐴 𝐵))
311, 17, 2, 15, 18, 12, 14, 13, 30tgcgrcomlr 26199 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝐸 𝑦) = (𝐵 𝐴))
32 iscgra1.2 . . . . . . . . . . 11 (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
3332ad3antrrr 726 . . . . . . . . . 10 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝐴 𝐵) = (𝐷 𝐸))
3433eqcomd 2832 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝐷 𝐸) = (𝐴 𝐵))
351, 17, 2, 15, 16, 12, 14, 13, 34tgcgrcomlr 26199 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝐸 𝐷) = (𝐵 𝐴))
361, 2, 3, 12, 13, 14, 15, 16, 17, 20, 23, 18, 16, 19, 24, 31, 35hlcgreulem 26336 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑦 = 𝐷)
37 simpr3 1190 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → 𝑥(𝐾𝐸)𝐹)
3836, 28, 37jca32 516 . . . . . 6 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹)) → (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
39 simprrl 777 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩)
40 simprl 767 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝑦 = 𝐷)
418ad3antrrr 726 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐷𝑃)
429ad3antrrr 726 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐸𝑃)
434ad3antrrr 726 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐺 ∈ TarskiG)
441, 17, 2, 4, 5, 6, 8, 9, 32, 21tgcgrneq 26202 . . . . . . . . . 10 (𝜑𝐷𝐸)
4544ad3antrrr 726 . . . . . . . . 9 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐷𝐸)
461, 2, 3, 41, 41, 42, 43, 45hlid 26328 . . . . . . . 8 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝐷(𝐾𝐸)𝐷)
4740, 46eqbrtrd 5085 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝑦(𝐾𝐸)𝐷)
48 simprrr 778 . . . . . . 7 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → 𝑥(𝐾𝐸)𝐹)
4939, 47, 483jca 1122 . . . . . 6 ((((𝜑𝑦𝑃) ∧ 𝑥𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))) → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹))
5038, 49impbida 797 . . . . 5 (((𝜑𝑦𝑃) ∧ 𝑥𝑃) → ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹) ↔ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))))
5150rexbidva 3301 . . . 4 ((𝜑𝑦𝑃) → (∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹) ↔ ∃𝑥𝑃 (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))))
52 r19.42v 3355 . . . 4 (∃𝑥𝑃 (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)) ↔ (𝑦 = 𝐷 ∧ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
5351, 52syl6bb 288 . . 3 ((𝜑𝑦𝑃) → (∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹) ↔ (𝑦 = 𝐷 ∧ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))))
5453rexbidva 3301 . 2 (𝜑 → (∃𝑦𝑃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾𝐸)𝐷𝑥(𝐾𝐸)𝐹) ↔ ∃𝑦𝑃 (𝑦 = 𝐷 ∧ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹))))
55 id 22 . . . . . . . 8 (𝑦 = 𝐷𝑦 = 𝐷)
56 eqidd 2827 . . . . . . . 8 (𝑦 = 𝐷𝐸 = 𝐸)
57 eqidd 2827 . . . . . . . 8 (𝑦 = 𝐷𝑥 = 𝑥)
5855, 56, 57s3eqd 14221 . . . . . . 7 (𝑦 = 𝐷 → ⟨“𝑦𝐸𝑥”⟩ = ⟨“𝐷𝐸𝑥”⟩)
5958breq2d 5075 . . . . . 6 (𝑦 = 𝐷 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩))
6059anbi1d 629 . . . . 5 (𝑦 = 𝐷 → ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹) ↔ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
6160rexbidv 3302 . . . 4 (𝑦 = 𝐷 → (∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹) ↔ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
6261ceqsrexv 3653 . . 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, 54, 633bitrd 306 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021  wrex 3144   class class class wbr 5063  cfv 6354  (class class class)co 7150  ⟨“cs3 14199  Basecbs 16478  distcds 16569  TarskiGcstrkg 26149  Itvcitv 26155  cgrGccgrg 26229  hlGchlg 26319  cgrAccgra 26526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8284  df-map 8403  df-pm 8404  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-dju 9324  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-xnn0 11962  df-z 11976  df-uz 12238  df-fz 12888  df-fzo 13029  df-hash 13686  df-word 13857  df-concat 13918  df-s1 13945  df-s2 14205  df-s3 14206  df-trkgc 26167  df-trkgb 26168  df-trkgcb 26169  df-trkg 26172  df-cgrg 26230  df-hlg 26320  df-cgra 26527
This theorem is referenced by:  acopyeu  26553
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