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Theorem colline 25344
 Description: Three points are colinear iff there is a line through all three of them. Theorem 6.23 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 28-May-2019.)
Hypotheses
Ref Expression
tglineintmo.p 𝑃 = (Base‘𝐺)
tglineintmo.i 𝐼 = (Itv‘𝐺)
tglineintmo.l 𝐿 = (LineG‘𝐺)
tglineintmo.g (𝜑𝐺 ∈ TarskiG)
colline.1 (𝜑𝑋𝑃)
colline.2 (𝜑𝑌𝑃)
colline.3 (𝜑𝑍𝑃)
colline.4 (𝜑 → 2 ≤ (#‘𝑃))
Assertion
Ref Expression
colline (𝜑 → ((𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍) ↔ ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎)))
Distinct variable groups:   𝐿,𝑎   𝑋,𝑎   𝑌,𝑎   𝑍,𝑎   𝜑,𝑎
Allowed substitution hints:   𝑃(𝑎)   𝐺(𝑎)   𝐼(𝑎)

Proof of Theorem colline
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 tglineintmo.p . . . . . . . 8 𝑃 = (Base‘𝐺)
2 tglineintmo.i . . . . . . . 8 𝐼 = (Itv‘𝐺)
3 tglineintmo.l . . . . . . . 8 𝐿 = (LineG‘𝐺)
4 tglineintmo.g . . . . . . . . 9 (𝜑𝐺 ∈ TarskiG)
54ad4antr 764 . . . . . . . 8 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝐺 ∈ TarskiG)
6 colline.1 . . . . . . . . 9 (𝜑𝑋𝑃)
76ad4antr 764 . . . . . . . 8 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑋𝑃)
8 simplr 788 . . . . . . . 8 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑥𝑃)
9 simpr 476 . . . . . . . 8 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑋𝑥)
101, 2, 3, 5, 7, 8, 9tgelrnln 25325 . . . . . . 7 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → (𝑋𝐿𝑥) ∈ ran 𝐿)
111, 2, 3, 5, 7, 8, 9tglinerflx1 25328 . . . . . . 7 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑋 ∈ (𝑋𝐿𝑥))
12 simp-4r 803 . . . . . . . 8 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑌 = 𝑍)
13 simpllr 795 . . . . . . . . 9 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑋 = 𝑍)
1413, 11eqeltrrd 2689 . . . . . . . 8 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑍 ∈ (𝑋𝐿𝑥))
1512, 14eqeltrd 2688 . . . . . . 7 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑌 ∈ (𝑋𝐿𝑥))
16 eleq2 2677 . . . . . . . . 9 (𝑎 = (𝑋𝐿𝑥) → (𝑋𝑎𝑋 ∈ (𝑋𝐿𝑥)))
17 eleq2 2677 . . . . . . . . 9 (𝑎 = (𝑋𝐿𝑥) → (𝑌𝑎𝑌 ∈ (𝑋𝐿𝑥)))
18 eleq2 2677 . . . . . . . . 9 (𝑎 = (𝑋𝐿𝑥) → (𝑍𝑎𝑍 ∈ (𝑋𝐿𝑥)))
1916, 17, 183anbi123d 1391 . . . . . . . 8 (𝑎 = (𝑋𝐿𝑥) → ((𝑋𝑎𝑌𝑎𝑍𝑎) ↔ (𝑋 ∈ (𝑋𝐿𝑥) ∧ 𝑌 ∈ (𝑋𝐿𝑥) ∧ 𝑍 ∈ (𝑋𝐿𝑥))))
2019rspcev 3282 . . . . . . 7 (((𝑋𝐿𝑥) ∈ ran 𝐿 ∧ (𝑋 ∈ (𝑋𝐿𝑥) ∧ 𝑌 ∈ (𝑋𝐿𝑥) ∧ 𝑍 ∈ (𝑋𝐿𝑥))) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
2110, 11, 15, 14, 20syl13anc 1320 . . . . . 6 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
22 eqid 2610 . . . . . . . 8 (dist‘𝐺) = (dist‘𝐺)
23 colline.4 . . . . . . . 8 (𝜑 → 2 ≤ (#‘𝑃))
241, 22, 2, 4, 23, 6tglowdim1i 25196 . . . . . . 7 (𝜑 → ∃𝑥𝑃 𝑋𝑥)
2524ad2antrr 758 . . . . . 6 (((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) → ∃𝑥𝑃 𝑋𝑥)
2621, 25r19.29a 3060 . . . . 5 (((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
274ad2antrr 758 . . . . . . 7 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝐺 ∈ TarskiG)
286ad2antrr 758 . . . . . . 7 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑋𝑃)
29 colline.3 . . . . . . . 8 (𝜑𝑍𝑃)
3029ad2antrr 758 . . . . . . 7 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑍𝑃)
31 simpr 476 . . . . . . 7 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑋𝑍)
321, 2, 3, 27, 28, 30, 31tgelrnln 25325 . . . . . 6 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → (𝑋𝐿𝑍) ∈ ran 𝐿)
331, 2, 3, 27, 28, 30, 31tglinerflx1 25328 . . . . . 6 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑋 ∈ (𝑋𝐿𝑍))
34 simplr 788 . . . . . . 7 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑌 = 𝑍)
351, 2, 3, 27, 28, 30, 31tglinerflx2 25329 . . . . . . 7 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑍 ∈ (𝑋𝐿𝑍))
3634, 35eqeltrd 2688 . . . . . 6 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑌 ∈ (𝑋𝐿𝑍))
37 eleq2 2677 . . . . . . . 8 (𝑎 = (𝑋𝐿𝑍) → (𝑋𝑎𝑋 ∈ (𝑋𝐿𝑍)))
38 eleq2 2677 . . . . . . . 8 (𝑎 = (𝑋𝐿𝑍) → (𝑌𝑎𝑌 ∈ (𝑋𝐿𝑍)))
39 eleq2 2677 . . . . . . . 8 (𝑎 = (𝑋𝐿𝑍) → (𝑍𝑎𝑍 ∈ (𝑋𝐿𝑍)))
4037, 38, 393anbi123d 1391 . . . . . . 7 (𝑎 = (𝑋𝐿𝑍) → ((𝑋𝑎𝑌𝑎𝑍𝑎) ↔ (𝑋 ∈ (𝑋𝐿𝑍) ∧ 𝑌 ∈ (𝑋𝐿𝑍) ∧ 𝑍 ∈ (𝑋𝐿𝑍))))
4140rspcev 3282 . . . . . 6 (((𝑋𝐿𝑍) ∈ ran 𝐿 ∧ (𝑋 ∈ (𝑋𝐿𝑍) ∧ 𝑌 ∈ (𝑋𝐿𝑍) ∧ 𝑍 ∈ (𝑋𝐿𝑍))) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
4232, 33, 36, 35, 41syl13anc 1320 . . . . 5 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
4326, 42pm2.61dane 2869 . . . 4 ((𝜑𝑌 = 𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
4443adantlr 747 . . 3 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 = 𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
45 simpll 786 . . . . 5 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → 𝜑)
46 simpr 476 . . . . . . 7 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → 𝑌𝑍)
4746neneqd 2787 . . . . . 6 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → ¬ 𝑌 = 𝑍)
48 simplr 788 . . . . . 6 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
49 orel2 397 . . . . . 6 𝑌 = 𝑍 → ((𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍) → 𝑋 ∈ (𝑌𝐿𝑍)))
5047, 48, 49sylc 63 . . . . 5 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → 𝑋 ∈ (𝑌𝐿𝑍))
514ad2antrr 758 . . . . . 6 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → 𝐺 ∈ TarskiG)
52 colline.2 . . . . . . 7 (𝜑𝑌𝑃)
5352ad2antrr 758 . . . . . 6 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → 𝑌𝑃)
5429ad2antrr 758 . . . . . 6 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → 𝑍𝑃)
55 simpr 476 . . . . . 6 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → 𝑌𝑍)
561, 2, 3, 51, 53, 54, 55tgelrnln 25325 . . . . 5 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → (𝑌𝐿𝑍) ∈ ran 𝐿)
5745, 50, 46, 56syl21anc 1317 . . . 4 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → (𝑌𝐿𝑍) ∈ ran 𝐿)
581, 2, 3, 51, 53, 54, 55tglinerflx1 25328 . . . . 5 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → 𝑌 ∈ (𝑌𝐿𝑍))
5945, 50, 46, 58syl21anc 1317 . . . 4 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → 𝑌 ∈ (𝑌𝐿𝑍))
601, 2, 3, 51, 53, 54, 55tglinerflx2 25329 . . . . 5 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → 𝑍 ∈ (𝑌𝐿𝑍))
6145, 50, 46, 60syl21anc 1317 . . . 4 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → 𝑍 ∈ (𝑌𝐿𝑍))
62 eleq2 2677 . . . . . 6 (𝑎 = (𝑌𝐿𝑍) → (𝑋𝑎𝑋 ∈ (𝑌𝐿𝑍)))
63 eleq2 2677 . . . . . 6 (𝑎 = (𝑌𝐿𝑍) → (𝑌𝑎𝑌 ∈ (𝑌𝐿𝑍)))
64 eleq2 2677 . . . . . 6 (𝑎 = (𝑌𝐿𝑍) → (𝑍𝑎𝑍 ∈ (𝑌𝐿𝑍)))
6562, 63, 643anbi123d 1391 . . . . 5 (𝑎 = (𝑌𝐿𝑍) → ((𝑋𝑎𝑌𝑎𝑍𝑎) ↔ (𝑋 ∈ (𝑌𝐿𝑍) ∧ 𝑌 ∈ (𝑌𝐿𝑍) ∧ 𝑍 ∈ (𝑌𝐿𝑍))))
6665rspcev 3282 . . . 4 (((𝑌𝐿𝑍) ∈ ran 𝐿 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∧ 𝑌 ∈ (𝑌𝐿𝑍) ∧ 𝑍 ∈ (𝑌𝐿𝑍))) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
6757, 50, 59, 61, 66syl13anc 1320 . . 3 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
6844, 67pm2.61dane 2869 . 2 ((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
69 df-ne 2782 . . . . . 6 (𝑌𝑍 ↔ ¬ 𝑌 = 𝑍)
70 simplr1 1096 . . . . . . . 8 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑋𝑎)
714ad3antrrr 762 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝐺 ∈ TarskiG)
7252ad3antrrr 762 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑌𝑃)
7329ad3antrrr 762 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑍𝑃)
74 simpr 476 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑌𝑍)
75 simpllr 795 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑎 ∈ ran 𝐿)
76 simplr2 1097 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑌𝑎)
77 simplr3 1098 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑍𝑎)
781, 2, 3, 71, 72, 73, 74, 74, 75, 76, 77tglinethru 25331 . . . . . . . 8 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑎 = (𝑌𝐿𝑍))
7970, 78eleqtrd 2690 . . . . . . 7 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑋 ∈ (𝑌𝐿𝑍))
8079ex 449 . . . . . 6 (((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) → (𝑌𝑍𝑋 ∈ (𝑌𝐿𝑍)))
8169, 80syl5bir 232 . . . . 5 (((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) → (¬ 𝑌 = 𝑍𝑋 ∈ (𝑌𝐿𝑍)))
8281orrd 392 . . . 4 (((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) → (𝑌 = 𝑍𝑋 ∈ (𝑌𝐿𝑍)))
8382orcomd 402 . . 3 (((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
8483r19.29an 3059 . 2 ((𝜑 ∧ ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎)) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
8568, 84impbida 873 1 (𝜑 → ((𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍) ↔ ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   class class class wbr 4583  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ≤ cle 9954  2c2 10947  #chash 12979  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135  LineGclng 25136 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-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 This theorem is referenced by: (None)
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