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Theorem colline 25262
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 763 . . . . . . . 8 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝐺 ∈ TarskiG)
6 colline.1 . . . . . . . . 9 (𝜑𝑋𝑃)
76ad4antr 763 . . . . . . . 8 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑋𝑃)
8 simplr 787 . . . . . . . 8 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑥𝑃)
9 simpr 475 . . . . . . . 8 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑋𝑥)
101, 2, 3, 5, 7, 8, 9tgelrnln 25243 . . . . . . 7 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → (𝑋𝐿𝑥) ∈ ran 𝐿)
111, 2, 3, 5, 7, 8, 9tglinerflx1 25246 . . . . . . 7 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑋 ∈ (𝑋𝐿𝑥))
12 simp-4r 802 . . . . . . . 8 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑌 = 𝑍)
13 simpllr 794 . . . . . . . . 9 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑋 = 𝑍)
1413, 11eqeltrrd 2688 . . . . . . . 8 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑍 ∈ (𝑋𝐿𝑥))
1512, 14eqeltrd 2687 . . . . . . 7 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑌 ∈ (𝑋𝐿𝑥))
16 eleq2 2676 . . . . . . . . 9 (𝑎 = (𝑋𝐿𝑥) → (𝑋𝑎𝑋 ∈ (𝑋𝐿𝑥)))
17 eleq2 2676 . . . . . . . . 9 (𝑎 = (𝑋𝐿𝑥) → (𝑌𝑎𝑌 ∈ (𝑋𝐿𝑥)))
18 eleq2 2676 . . . . . . . . 9 (𝑎 = (𝑋𝐿𝑥) → (𝑍𝑎𝑍 ∈ (𝑋𝐿𝑥)))
1916, 17, 183anbi123d 1390 . . . . . . . 8 (𝑎 = (𝑋𝐿𝑥) → ((𝑋𝑎𝑌𝑎𝑍𝑎) ↔ (𝑋 ∈ (𝑋𝐿𝑥) ∧ 𝑌 ∈ (𝑋𝐿𝑥) ∧ 𝑍 ∈ (𝑋𝐿𝑥))))
2019rspcev 3281 . . . . . . 7 (((𝑋𝐿𝑥) ∈ ran 𝐿 ∧ (𝑋 ∈ (𝑋𝐿𝑥) ∧ 𝑌 ∈ (𝑋𝐿𝑥) ∧ 𝑍 ∈ (𝑋𝐿𝑥))) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
2110, 11, 15, 14, 20syl13anc 1319 . . . . . 6 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
22 eqid 2609 . . . . . . . 8 (dist‘𝐺) = (dist‘𝐺)
23 colline.4 . . . . . . . 8 (𝜑 → 2 ≤ (#‘𝑃))
241, 22, 2, 4, 23, 6tglowdim1i 25113 . . . . . . 7 (𝜑 → ∃𝑥𝑃 𝑋𝑥)
2524ad2antrr 757 . . . . . 6 (((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) → ∃𝑥𝑃 𝑋𝑥)
2621, 25r19.29a 3059 . . . . 5 (((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
274ad2antrr 757 . . . . . . 7 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝐺 ∈ TarskiG)
286ad2antrr 757 . . . . . . 7 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑋𝑃)
29 colline.3 . . . . . . . 8 (𝜑𝑍𝑃)
3029ad2antrr 757 . . . . . . 7 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑍𝑃)
31 simpr 475 . . . . . . 7 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑋𝑍)
321, 2, 3, 27, 28, 30, 31tgelrnln 25243 . . . . . 6 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → (𝑋𝐿𝑍) ∈ ran 𝐿)
331, 2, 3, 27, 28, 30, 31tglinerflx1 25246 . . . . . 6 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑋 ∈ (𝑋𝐿𝑍))
34 simplr 787 . . . . . . 7 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑌 = 𝑍)
351, 2, 3, 27, 28, 30, 31tglinerflx2 25247 . . . . . . 7 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑍 ∈ (𝑋𝐿𝑍))
3634, 35eqeltrd 2687 . . . . . 6 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑌 ∈ (𝑋𝐿𝑍))
37 eleq2 2676 . . . . . . . 8 (𝑎 = (𝑋𝐿𝑍) → (𝑋𝑎𝑋 ∈ (𝑋𝐿𝑍)))
38 eleq2 2676 . . . . . . . 8 (𝑎 = (𝑋𝐿𝑍) → (𝑌𝑎𝑌 ∈ (𝑋𝐿𝑍)))
39 eleq2 2676 . . . . . . . 8 (𝑎 = (𝑋𝐿𝑍) → (𝑍𝑎𝑍 ∈ (𝑋𝐿𝑍)))
4037, 38, 393anbi123d 1390 . . . . . . 7 (𝑎 = (𝑋𝐿𝑍) → ((𝑋𝑎𝑌𝑎𝑍𝑎) ↔ (𝑋 ∈ (𝑋𝐿𝑍) ∧ 𝑌 ∈ (𝑋𝐿𝑍) ∧ 𝑍 ∈ (𝑋𝐿𝑍))))
4140rspcev 3281 . . . . . 6 (((𝑋𝐿𝑍) ∈ ran 𝐿 ∧ (𝑋 ∈ (𝑋𝐿𝑍) ∧ 𝑌 ∈ (𝑋𝐿𝑍) ∧ 𝑍 ∈ (𝑋𝐿𝑍))) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
4232, 33, 36, 35, 41syl13anc 1319 . . . . 5 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
4326, 42pm2.61dane 2868 . . . 4 ((𝜑𝑌 = 𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
4443adantlr 746 . . 3 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 = 𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
45 simpll 785 . . . . 5 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → 𝜑)
46 simpr 475 . . . . . . 7 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → 𝑌𝑍)
4746neneqd 2786 . . . . . 6 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → ¬ 𝑌 = 𝑍)
48 simplr 787 . . . . . 6 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
49 orel2 396 . . . . . 6 𝑌 = 𝑍 → ((𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍) → 𝑋 ∈ (𝑌𝐿𝑍)))
5047, 48, 49sylc 62 . . . . 5 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → 𝑋 ∈ (𝑌𝐿𝑍))
514ad2antrr 757 . . . . . 6 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → 𝐺 ∈ TarskiG)
52 colline.2 . . . . . . 7 (𝜑𝑌𝑃)
5352ad2antrr 757 . . . . . 6 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → 𝑌𝑃)
5429ad2antrr 757 . . . . . 6 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → 𝑍𝑃)
55 simpr 475 . . . . . 6 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → 𝑌𝑍)
561, 2, 3, 51, 53, 54, 55tgelrnln 25243 . . . . 5 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → (𝑌𝐿𝑍) ∈ ran 𝐿)
5745, 50, 46, 56syl21anc 1316 . . . 4 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → (𝑌𝐿𝑍) ∈ ran 𝐿)
581, 2, 3, 51, 53, 54, 55tglinerflx1 25246 . . . . 5 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → 𝑌 ∈ (𝑌𝐿𝑍))
5945, 50, 46, 58syl21anc 1316 . . . 4 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → 𝑌 ∈ (𝑌𝐿𝑍))
601, 2, 3, 51, 53, 54, 55tglinerflx2 25247 . . . . 5 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → 𝑍 ∈ (𝑌𝐿𝑍))
6145, 50, 46, 60syl21anc 1316 . . . 4 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → 𝑍 ∈ (𝑌𝐿𝑍))
62 eleq2 2676 . . . . . 6 (𝑎 = (𝑌𝐿𝑍) → (𝑋𝑎𝑋 ∈ (𝑌𝐿𝑍)))
63 eleq2 2676 . . . . . 6 (𝑎 = (𝑌𝐿𝑍) → (𝑌𝑎𝑌 ∈ (𝑌𝐿𝑍)))
64 eleq2 2676 . . . . . 6 (𝑎 = (𝑌𝐿𝑍) → (𝑍𝑎𝑍 ∈ (𝑌𝐿𝑍)))
6562, 63, 643anbi123d 1390 . . . . 5 (𝑎 = (𝑌𝐿𝑍) → ((𝑋𝑎𝑌𝑎𝑍𝑎) ↔ (𝑋 ∈ (𝑌𝐿𝑍) ∧ 𝑌 ∈ (𝑌𝐿𝑍) ∧ 𝑍 ∈ (𝑌𝐿𝑍))))
6665rspcev 3281 . . . 4 (((𝑌𝐿𝑍) ∈ ran 𝐿 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∧ 𝑌 ∈ (𝑌𝐿𝑍) ∧ 𝑍 ∈ (𝑌𝐿𝑍))) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
6757, 50, 59, 61, 66syl13anc 1319 . . 3 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
6844, 67pm2.61dane 2868 . 2 ((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
69 df-ne 2781 . . . . . 6 (𝑌𝑍 ↔ ¬ 𝑌 = 𝑍)
70 simplr1 1095 . . . . . . . 8 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑋𝑎)
714ad3antrrr 761 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝐺 ∈ TarskiG)
7252ad3antrrr 761 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑌𝑃)
7329ad3antrrr 761 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑍𝑃)
74 simpr 475 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑌𝑍)
75 simpllr 794 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑎 ∈ ran 𝐿)
76 simplr2 1096 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑌𝑎)
77 simplr3 1097 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑍𝑎)
781, 2, 3, 71, 72, 73, 74, 74, 75, 76, 77tglinethru 25249 . . . . . . . 8 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑎 = (𝑌𝐿𝑍))
7970, 78eleqtrd 2689 . . . . . . 7 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑋 ∈ (𝑌𝐿𝑍))
8079ex 448 . . . . . 6 (((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) → (𝑌𝑍𝑋 ∈ (𝑌𝐿𝑍)))
8169, 80syl5bir 231 . . . . 5 (((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) → (¬ 𝑌 = 𝑍𝑋 ∈ (𝑌𝐿𝑍)))
8281orrd 391 . . . 4 (((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) → (𝑌 = 𝑍𝑋 ∈ (𝑌𝐿𝑍)))
8382orcomd 401 . . 3 (((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
8483r19.29an 3058 . 2 ((𝜑 ∧ ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎)) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
8568, 84impbida 872 1 (𝜑 → ((𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍) ↔ ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wrex 2896   class class class wbr 4577  ran crn 5029  cfv 5790  (class class class)co 6527  cle 9931  2c2 10917  #chash 12934  Basecbs 15641  distcds 15723  TarskiGcstrkg 25046  Itvcitv 25052  LineGclng 25053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290  df-hash 12935  df-word 13100  df-concat 13102  df-s1 13103  df-s2 13390  df-s3 13391  df-trkgc 25064  df-trkgb 25065  df-trkgcb 25066  df-trkg 25069  df-cgrg 25124
This theorem is referenced by: (None)
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