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Theorem colline 25900
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 725 . . . . . . . 8 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝐺 ∈ TarskiG)
6 colline.1 . . . . . . . . 9 (𝜑𝑋𝑃)
76ad4antr 725 . . . . . . . 8 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑋𝑃)
8 simplr 786 . . . . . . . 8 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑥𝑃)
9 simpr 478 . . . . . . . 8 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑋𝑥)
101, 2, 3, 5, 7, 8, 9tgelrnln 25881 . . . . . . 7 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → (𝑋𝐿𝑥) ∈ ran 𝐿)
111, 2, 3, 5, 7, 8, 9tglinerflx1 25884 . . . . . . 7 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑋 ∈ (𝑋𝐿𝑥))
12 simp-4r 804 . . . . . . . 8 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑌 = 𝑍)
13 simpllr 794 . . . . . . . . 9 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑋 = 𝑍)
1413, 11eqeltrrd 2879 . . . . . . . 8 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑍 ∈ (𝑋𝐿𝑥))
1512, 14eqeltrd 2878 . . . . . . 7 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → 𝑌 ∈ (𝑋𝐿𝑥))
16 eleq2 2867 . . . . . . . . 9 (𝑎 = (𝑋𝐿𝑥) → (𝑋𝑎𝑋 ∈ (𝑋𝐿𝑥)))
17 eleq2 2867 . . . . . . . . 9 (𝑎 = (𝑋𝐿𝑥) → (𝑌𝑎𝑌 ∈ (𝑋𝐿𝑥)))
18 eleq2 2867 . . . . . . . . 9 (𝑎 = (𝑋𝐿𝑥) → (𝑍𝑎𝑍 ∈ (𝑋𝐿𝑥)))
1916, 17, 183anbi123d 1561 . . . . . . . 8 (𝑎 = (𝑋𝐿𝑥) → ((𝑋𝑎𝑌𝑎𝑍𝑎) ↔ (𝑋 ∈ (𝑋𝐿𝑥) ∧ 𝑌 ∈ (𝑋𝐿𝑥) ∧ 𝑍 ∈ (𝑋𝐿𝑥))))
2019rspcev 3497 . . . . . . 7 (((𝑋𝐿𝑥) ∈ ran 𝐿 ∧ (𝑋 ∈ (𝑋𝐿𝑥) ∧ 𝑌 ∈ (𝑋𝐿𝑥) ∧ 𝑍 ∈ (𝑋𝐿𝑥))) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
2110, 11, 15, 14, 20syl13anc 1492 . . . . . 6 (((((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥𝑃) ∧ 𝑋𝑥) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
22 eqid 2799 . . . . . . . 8 (dist‘𝐺) = (dist‘𝐺)
23 colline.4 . . . . . . . 8 (𝜑 → 2 ≤ (♯‘𝑃))
241, 22, 2, 4, 23, 6tglowdim1i 25752 . . . . . . 7 (𝜑 → ∃𝑥𝑃 𝑋𝑥)
2524ad2antrr 718 . . . . . 6 (((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) → ∃𝑥𝑃 𝑋𝑥)
2621, 25r19.29a 3259 . . . . 5 (((𝜑𝑌 = 𝑍) ∧ 𝑋 = 𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
274ad2antrr 718 . . . . . . 7 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝐺 ∈ TarskiG)
286ad2antrr 718 . . . . . . 7 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑋𝑃)
29 colline.3 . . . . . . . 8 (𝜑𝑍𝑃)
3029ad2antrr 718 . . . . . . 7 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑍𝑃)
31 simpr 478 . . . . . . 7 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑋𝑍)
321, 2, 3, 27, 28, 30, 31tgelrnln 25881 . . . . . 6 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → (𝑋𝐿𝑍) ∈ ran 𝐿)
331, 2, 3, 27, 28, 30, 31tglinerflx1 25884 . . . . . 6 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑋 ∈ (𝑋𝐿𝑍))
34 simplr 786 . . . . . . 7 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑌 = 𝑍)
351, 2, 3, 27, 28, 30, 31tglinerflx2 25885 . . . . . . 7 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑍 ∈ (𝑋𝐿𝑍))
3634, 35eqeltrd 2878 . . . . . 6 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → 𝑌 ∈ (𝑋𝐿𝑍))
37 eleq2 2867 . . . . . . . 8 (𝑎 = (𝑋𝐿𝑍) → (𝑋𝑎𝑋 ∈ (𝑋𝐿𝑍)))
38 eleq2 2867 . . . . . . . 8 (𝑎 = (𝑋𝐿𝑍) → (𝑌𝑎𝑌 ∈ (𝑋𝐿𝑍)))
39 eleq2 2867 . . . . . . . 8 (𝑎 = (𝑋𝐿𝑍) → (𝑍𝑎𝑍 ∈ (𝑋𝐿𝑍)))
4037, 38, 393anbi123d 1561 . . . . . . 7 (𝑎 = (𝑋𝐿𝑍) → ((𝑋𝑎𝑌𝑎𝑍𝑎) ↔ (𝑋 ∈ (𝑋𝐿𝑍) ∧ 𝑌 ∈ (𝑋𝐿𝑍) ∧ 𝑍 ∈ (𝑋𝐿𝑍))))
4140rspcev 3497 . . . . . 6 (((𝑋𝐿𝑍) ∈ ran 𝐿 ∧ (𝑋 ∈ (𝑋𝐿𝑍) ∧ 𝑌 ∈ (𝑋𝐿𝑍) ∧ 𝑍 ∈ (𝑋𝐿𝑍))) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
4232, 33, 36, 35, 41syl13anc 1492 . . . . 5 (((𝜑𝑌 = 𝑍) ∧ 𝑋𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
4326, 42pm2.61dane 3058 . . . 4 ((𝜑𝑌 = 𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
4443adantlr 707 . . 3 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 = 𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
45 simpll 784 . . . . 5 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → 𝜑)
46 simpr 478 . . . . . . 7 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → 𝑌𝑍)
4746neneqd 2976 . . . . . 6 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → ¬ 𝑌 = 𝑍)
48 simplr 786 . . . . . 6 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
49 orel2 915 . . . . . 6 𝑌 = 𝑍 → ((𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍) → 𝑋 ∈ (𝑌𝐿𝑍)))
5047, 48, 49sylc 65 . . . . 5 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → 𝑋 ∈ (𝑌𝐿𝑍))
514ad2antrr 718 . . . . . 6 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → 𝐺 ∈ TarskiG)
52 colline.2 . . . . . . 7 (𝜑𝑌𝑃)
5352ad2antrr 718 . . . . . 6 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → 𝑌𝑃)
5429ad2antrr 718 . . . . . 6 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → 𝑍𝑃)
55 simpr 478 . . . . . 6 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → 𝑌𝑍)
561, 2, 3, 51, 53, 54, 55tgelrnln 25881 . . . . 5 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → (𝑌𝐿𝑍) ∈ ran 𝐿)
5745, 50, 46, 56syl21anc 867 . . . 4 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → (𝑌𝐿𝑍) ∈ ran 𝐿)
581, 2, 3, 51, 53, 54, 55tglinerflx1 25884 . . . . 5 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → 𝑌 ∈ (𝑌𝐿𝑍))
5945, 50, 46, 58syl21anc 867 . . . 4 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → 𝑌 ∈ (𝑌𝐿𝑍))
601, 2, 3, 51, 53, 54, 55tglinerflx2 25885 . . . . 5 (((𝜑𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌𝑍) → 𝑍 ∈ (𝑌𝐿𝑍))
6145, 50, 46, 60syl21anc 867 . . . 4 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → 𝑍 ∈ (𝑌𝐿𝑍))
62 eleq2 2867 . . . . . 6 (𝑎 = (𝑌𝐿𝑍) → (𝑋𝑎𝑋 ∈ (𝑌𝐿𝑍)))
63 eleq2 2867 . . . . . 6 (𝑎 = (𝑌𝐿𝑍) → (𝑌𝑎𝑌 ∈ (𝑌𝐿𝑍)))
64 eleq2 2867 . . . . . 6 (𝑎 = (𝑌𝐿𝑍) → (𝑍𝑎𝑍 ∈ (𝑌𝐿𝑍)))
6562, 63, 643anbi123d 1561 . . . . 5 (𝑎 = (𝑌𝐿𝑍) → ((𝑋𝑎𝑌𝑎𝑍𝑎) ↔ (𝑋 ∈ (𝑌𝐿𝑍) ∧ 𝑌 ∈ (𝑌𝐿𝑍) ∧ 𝑍 ∈ (𝑌𝐿𝑍))))
6665rspcev 3497 . . . 4 (((𝑌𝐿𝑍) ∈ ran 𝐿 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∧ 𝑌 ∈ (𝑌𝐿𝑍) ∧ 𝑍 ∈ (𝑌𝐿𝑍))) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
6757, 50, 59, 61, 66syl13anc 1492 . . 3 (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
6844, 67pm2.61dane 3058 . 2 ((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) → ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎))
69 df-ne 2972 . . . . . 6 (𝑌𝑍 ↔ ¬ 𝑌 = 𝑍)
70 simplr1 1276 . . . . . . . 8 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑋𝑎)
714ad3antrrr 722 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝐺 ∈ TarskiG)
7252ad3antrrr 722 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑌𝑃)
7329ad3antrrr 722 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑍𝑃)
74 simpr 478 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑌𝑍)
75 simpllr 794 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑎 ∈ ran 𝐿)
76 simplr2 1278 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑌𝑎)
77 simplr3 1280 . . . . . . . . 9 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑍𝑎)
781, 2, 3, 71, 72, 73, 74, 74, 75, 76, 77tglinethru 25887 . . . . . . . 8 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑎 = (𝑌𝐿𝑍))
7970, 78eleqtrd 2880 . . . . . . 7 ((((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) ∧ 𝑌𝑍) → 𝑋 ∈ (𝑌𝐿𝑍))
8079ex 402 . . . . . 6 (((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) → (𝑌𝑍𝑋 ∈ (𝑌𝐿𝑍)))
8169, 80syl5bir 235 . . . . 5 (((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) → (¬ 𝑌 = 𝑍𝑋 ∈ (𝑌𝐿𝑍)))
8281orrd 890 . . . 4 (((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) → (𝑌 = 𝑍𝑋 ∈ (𝑌𝐿𝑍)))
8382orcomd 898 . . 3 (((𝜑𝑎 ∈ ran 𝐿) ∧ (𝑋𝑎𝑌𝑎𝑍𝑎)) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
8483r19.29an 3258 . 2 ((𝜑 ∧ ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎)) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
8568, 84impbida 836 1 (𝜑 → ((𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍) ↔ ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  wo 874  w3a 1108   = wceq 1653  wcel 2157  wne 2971  wrex 3090   class class class wbr 4843  ran crn 5313  cfv 6101  (class class class)co 6878  cle 10364  2c2 11368  chash 13370  Basecbs 16184  distcds 16276  TarskiGcstrkg 25681  Itvcitv 25687  LineGclng 25688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-pm 8098  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-card 9051  df-cda 9278  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-2 11376  df-3 11377  df-n0 11581  df-xnn0 11653  df-z 11667  df-uz 11931  df-fz 12581  df-fzo 12721  df-hash 13371  df-word 13535  df-concat 13591  df-s1 13616  df-s2 13933  df-s3 13934  df-trkgc 25699  df-trkgb 25700  df-trkgcb 25701  df-trkg 25704  df-cgrg 25762
This theorem is referenced by: (None)
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