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Theorem 3dvds 16308
Description: A rule for divisibility by 3 of a number written in base 10. This is Metamath 100 proof #85. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 17-Jan-2015.) (Revised by AV, 8-Sep-2021.)
Assertion
Ref Expression
3dvds ((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) β†’ (3 βˆ₯ Ξ£π‘˜ ∈ (0...𝑁)((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) ↔ 3 βˆ₯ Ξ£π‘˜ ∈ (0...𝑁)(πΉβ€˜π‘˜)))
Distinct variable groups:   π‘˜,𝐹   π‘˜,𝑁
Proof of Theorem 3dvds
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 3z 12626 . . 3 3 ∈ β„€
21a1i 11 . 2 ((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) β†’ 3 ∈ β„€)
3 fzfid 13971 . . 3 ((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) β†’ (0...𝑁) ∈ Fin)
4 ffvelcdm 7091 . . . . 5 ((𝐹:(0...𝑁)βŸΆβ„€ ∧ π‘˜ ∈ (0...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ β„€)
54adantll 713 . . . 4 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ β„€)
6 10nn 12724 . . . . . 6 10 ∈ β„•
76nnzi 12617 . . . . 5 10 ∈ β„€
8 elfznn0 13627 . . . . . 6 (π‘˜ ∈ (0...𝑁) β†’ π‘˜ ∈ β„•0)
98adantl 481 . . . . 5 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ π‘˜ ∈ β„•0)
10 zexpcl 14074 . . . . 5 ((10 ∈ β„€ ∧ π‘˜ ∈ β„•0) β†’ (10β†‘π‘˜) ∈ β„€)
117, 9, 10sylancr 586 . . . 4 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ (10β†‘π‘˜) ∈ β„€)
125, 11zmulcld 12703 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ ((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) ∈ β„€)
133, 12fsumzcl 15714 . 2 ((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) β†’ Ξ£π‘˜ ∈ (0...𝑁)((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) ∈ β„€)
143, 5fsumzcl 15714 . 2 ((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) β†’ Ξ£π‘˜ ∈ (0...𝑁)(πΉβ€˜π‘˜) ∈ β„€)
1512, 5zsubcld 12702 . . . 4 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ (((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) βˆ’ (πΉβ€˜π‘˜)) ∈ β„€)
16 ax-1cn 11197 . . . . . . . . . . . 12 1 ∈ β„‚
176nncni 12253 . . . . . . . . . . . 12 10 ∈ β„‚
1816, 17negsubdi2i 11577 . . . . . . . . . . 11 -(1 βˆ’ 10) = (10 βˆ’ 1)
19 9p1e10 12710 . . . . . . . . . . . . 13 (9 + 1) = 10
2019eqcomi 2737 . . . . . . . . . . . 12 10 = (9 + 1)
2120oveq1i 7430 . . . . . . . . . . 11 (10 βˆ’ 1) = ((9 + 1) βˆ’ 1)
22 9cn 12343 . . . . . . . . . . . 12 9 ∈ β„‚
2322, 16pncan3oi 11507 . . . . . . . . . . 11 ((9 + 1) βˆ’ 1) = 9
2418, 21, 233eqtri 2760 . . . . . . . . . 10 -(1 βˆ’ 10) = 9
25 3t3e9 12410 . . . . . . . . . 10 (3 Β· 3) = 9
2624, 25eqtr4i 2759 . . . . . . . . 9 -(1 βˆ’ 10) = (3 Β· 3)
2717a1i 11 . . . . . . . . . . . . . . 15 (π‘˜ ∈ β„•0 β†’ 10 ∈ β„‚)
28 1re 11245 . . . . . . . . . . . . . . . . 17 1 ∈ ℝ
29 1lt10 12847 . . . . . . . . . . . . . . . . 17 1 < 10
3028, 29gtneii 11357 . . . . . . . . . . . . . . . 16 10 β‰  1
3130a1i 11 . . . . . . . . . . . . . . 15 (π‘˜ ∈ β„•0 β†’ 10 β‰  1)
32 id 22 . . . . . . . . . . . . . . 15 (π‘˜ ∈ β„•0 β†’ π‘˜ ∈ β„•0)
3327, 31, 32geoser 15846 . . . . . . . . . . . . . 14 (π‘˜ ∈ β„•0 β†’ Σ𝑗 ∈ (0...(π‘˜ βˆ’ 1))(10↑𝑗) = ((1 βˆ’ (10β†‘π‘˜)) / (1 βˆ’ 10)))
34 fzfid 13971 . . . . . . . . . . . . . . 15 (π‘˜ ∈ β„•0 β†’ (0...(π‘˜ βˆ’ 1)) ∈ Fin)
35 elfznn0 13627 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (0...(π‘˜ βˆ’ 1)) β†’ 𝑗 ∈ β„•0)
3635adantl 481 . . . . . . . . . . . . . . . 16 ((π‘˜ ∈ β„•0 ∧ 𝑗 ∈ (0...(π‘˜ βˆ’ 1))) β†’ 𝑗 ∈ β„•0)
37 zexpcl 14074 . . . . . . . . . . . . . . . 16 ((10 ∈ β„€ ∧ 𝑗 ∈ β„•0) β†’ (10↑𝑗) ∈ β„€)
387, 36, 37sylancr 586 . . . . . . . . . . . . . . 15 ((π‘˜ ∈ β„•0 ∧ 𝑗 ∈ (0...(π‘˜ βˆ’ 1))) β†’ (10↑𝑗) ∈ β„€)
3934, 38fsumzcl 15714 . . . . . . . . . . . . . 14 (π‘˜ ∈ β„•0 β†’ Σ𝑗 ∈ (0...(π‘˜ βˆ’ 1))(10↑𝑗) ∈ β„€)
4033, 39eqeltrrd 2830 . . . . . . . . . . . . 13 (π‘˜ ∈ β„•0 β†’ ((1 βˆ’ (10β†‘π‘˜)) / (1 βˆ’ 10)) ∈ β„€)
41 1z 12623 . . . . . . . . . . . . . . 15 1 ∈ β„€
42 zsubcl 12635 . . . . . . . . . . . . . . 15 ((1 ∈ β„€ ∧ 10 ∈ β„€) β†’ (1 βˆ’ 10) ∈ β„€)
4341, 7, 42mp2an 691 . . . . . . . . . . . . . 14 (1 βˆ’ 10) ∈ β„€
4428, 29ltneii 11358 . . . . . . . . . . . . . . 15 1 β‰  10
4516, 17subeq0i 11571 . . . . . . . . . . . . . . . 16 ((1 βˆ’ 10) = 0 ↔ 1 = 10)
4645necon3bii 2990 . . . . . . . . . . . . . . 15 ((1 βˆ’ 10) β‰  0 ↔ 1 β‰  10)
4744, 46mpbir 230 . . . . . . . . . . . . . 14 (1 βˆ’ 10) β‰  0
487, 32, 10sylancr 586 . . . . . . . . . . . . . . 15 (π‘˜ ∈ β„•0 β†’ (10β†‘π‘˜) ∈ β„€)
49 zsubcl 12635 . . . . . . . . . . . . . . 15 ((1 ∈ β„€ ∧ (10β†‘π‘˜) ∈ β„€) β†’ (1 βˆ’ (10β†‘π‘˜)) ∈ β„€)
5041, 48, 49sylancr 586 . . . . . . . . . . . . . 14 (π‘˜ ∈ β„•0 β†’ (1 βˆ’ (10β†‘π‘˜)) ∈ β„€)
51 dvdsval2 16234 . . . . . . . . . . . . . 14 (((1 βˆ’ 10) ∈ β„€ ∧ (1 βˆ’ 10) β‰  0 ∧ (1 βˆ’ (10β†‘π‘˜)) ∈ β„€) β†’ ((1 βˆ’ 10) βˆ₯ (1 βˆ’ (10β†‘π‘˜)) ↔ ((1 βˆ’ (10β†‘π‘˜)) / (1 βˆ’ 10)) ∈ β„€))
5243, 47, 50, 51mp3an12i 1462 . . . . . . . . . . . . 13 (π‘˜ ∈ β„•0 β†’ ((1 βˆ’ 10) βˆ₯ (1 βˆ’ (10β†‘π‘˜)) ↔ ((1 βˆ’ (10β†‘π‘˜)) / (1 βˆ’ 10)) ∈ β„€))
5340, 52mpbird 257 . . . . . . . . . . . 12 (π‘˜ ∈ β„•0 β†’ (1 βˆ’ 10) βˆ₯ (1 βˆ’ (10β†‘π‘˜)))
5448zcnd 12698 . . . . . . . . . . . . 13 (π‘˜ ∈ β„•0 β†’ (10β†‘π‘˜) ∈ β„‚)
55 negsubdi2 11550 . . . . . . . . . . . . 13 (((10β†‘π‘˜) ∈ β„‚ ∧ 1 ∈ β„‚) β†’ -((10β†‘π‘˜) βˆ’ 1) = (1 βˆ’ (10β†‘π‘˜)))
5654, 16, 55sylancl 585 . . . . . . . . . . . 12 (π‘˜ ∈ β„•0 β†’ -((10β†‘π‘˜) βˆ’ 1) = (1 βˆ’ (10β†‘π‘˜)))
5753, 56breqtrrd 5176 . . . . . . . . . . 11 (π‘˜ ∈ β„•0 β†’ (1 βˆ’ 10) βˆ₯ -((10β†‘π‘˜) βˆ’ 1))
58 peano2zm 12636 . . . . . . . . . . . . 13 ((10β†‘π‘˜) ∈ β„€ β†’ ((10β†‘π‘˜) βˆ’ 1) ∈ β„€)
5948, 58syl 17 . . . . . . . . . . . 12 (π‘˜ ∈ β„•0 β†’ ((10β†‘π‘˜) βˆ’ 1) ∈ β„€)
60 dvdsnegb 16251 . . . . . . . . . . . 12 (((1 βˆ’ 10) ∈ β„€ ∧ ((10β†‘π‘˜) βˆ’ 1) ∈ β„€) β†’ ((1 βˆ’ 10) βˆ₯ ((10β†‘π‘˜) βˆ’ 1) ↔ (1 βˆ’ 10) βˆ₯ -((10β†‘π‘˜) βˆ’ 1)))
6143, 59, 60sylancr 586 . . . . . . . . . . 11 (π‘˜ ∈ β„•0 β†’ ((1 βˆ’ 10) βˆ₯ ((10β†‘π‘˜) βˆ’ 1) ↔ (1 βˆ’ 10) βˆ₯ -((10β†‘π‘˜) βˆ’ 1)))
6257, 61mpbird 257 . . . . . . . . . 10 (π‘˜ ∈ β„•0 β†’ (1 βˆ’ 10) βˆ₯ ((10β†‘π‘˜) βˆ’ 1))
63 negdvdsb 16250 . . . . . . . . . . 11 (((1 βˆ’ 10) ∈ β„€ ∧ ((10β†‘π‘˜) βˆ’ 1) ∈ β„€) β†’ ((1 βˆ’ 10) βˆ₯ ((10β†‘π‘˜) βˆ’ 1) ↔ -(1 βˆ’ 10) βˆ₯ ((10β†‘π‘˜) βˆ’ 1)))
6443, 59, 63sylancr 586 . . . . . . . . . 10 (π‘˜ ∈ β„•0 β†’ ((1 βˆ’ 10) βˆ₯ ((10β†‘π‘˜) βˆ’ 1) ↔ -(1 βˆ’ 10) βˆ₯ ((10β†‘π‘˜) βˆ’ 1)))
6562, 64mpbid 231 . . . . . . . . 9 (π‘˜ ∈ β„•0 β†’ -(1 βˆ’ 10) βˆ₯ ((10β†‘π‘˜) βˆ’ 1))
6626, 65eqbrtrrid 5184 . . . . . . . 8 (π‘˜ ∈ β„•0 β†’ (3 Β· 3) βˆ₯ ((10β†‘π‘˜) βˆ’ 1))
67 muldvds1 16258 . . . . . . . . 9 ((3 ∈ β„€ ∧ 3 ∈ β„€ ∧ ((10β†‘π‘˜) βˆ’ 1) ∈ β„€) β†’ ((3 Β· 3) βˆ₯ ((10β†‘π‘˜) βˆ’ 1) β†’ 3 βˆ₯ ((10β†‘π‘˜) βˆ’ 1)))
681, 1, 59, 67mp3an12i 1462 . . . . . . . 8 (π‘˜ ∈ β„•0 β†’ ((3 Β· 3) βˆ₯ ((10β†‘π‘˜) βˆ’ 1) β†’ 3 βˆ₯ ((10β†‘π‘˜) βˆ’ 1)))
6966, 68mpd 15 . . . . . . 7 (π‘˜ ∈ β„•0 β†’ 3 βˆ₯ ((10β†‘π‘˜) βˆ’ 1))
709, 69syl 17 . . . . . 6 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ 3 βˆ₯ ((10β†‘π‘˜) βˆ’ 1))
7111, 58syl 17 . . . . . . 7 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ ((10β†‘π‘˜) βˆ’ 1) ∈ β„€)
72 dvdsmultr2 16275 . . . . . . 7 ((3 ∈ β„€ ∧ (πΉβ€˜π‘˜) ∈ β„€ ∧ ((10β†‘π‘˜) βˆ’ 1) ∈ β„€) β†’ (3 βˆ₯ ((10β†‘π‘˜) βˆ’ 1) β†’ 3 βˆ₯ ((πΉβ€˜π‘˜) Β· ((10β†‘π‘˜) βˆ’ 1))))
731, 5, 71, 72mp3an2i 1463 . . . . . 6 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ (3 βˆ₯ ((10β†‘π‘˜) βˆ’ 1) β†’ 3 βˆ₯ ((πΉβ€˜π‘˜) Β· ((10β†‘π‘˜) βˆ’ 1))))
7470, 73mpd 15 . . . . 5 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ 3 βˆ₯ ((πΉβ€˜π‘˜) Β· ((10β†‘π‘˜) βˆ’ 1)))
755zcnd 12698 . . . . . 6 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
7611zcnd 12698 . . . . . 6 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ (10β†‘π‘˜) ∈ β„‚)
7775, 76muls1d 11705 . . . . 5 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ ((πΉβ€˜π‘˜) Β· ((10β†‘π‘˜) βˆ’ 1)) = (((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) βˆ’ (πΉβ€˜π‘˜)))
7874, 77breqtrd 5174 . . . 4 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ 3 βˆ₯ (((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) βˆ’ (πΉβ€˜π‘˜)))
793, 2, 15, 78fsumdvds 16285 . . 3 ((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) β†’ 3 βˆ₯ Ξ£π‘˜ ∈ (0...𝑁)(((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) βˆ’ (πΉβ€˜π‘˜)))
8012zcnd 12698 . . . 4 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ ((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) ∈ β„‚)
813, 80, 75fsumsub 15767 . . 3 ((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) β†’ Ξ£π‘˜ ∈ (0...𝑁)(((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) βˆ’ (πΉβ€˜π‘˜)) = (Ξ£π‘˜ ∈ (0...𝑁)((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) βˆ’ Ξ£π‘˜ ∈ (0...𝑁)(πΉβ€˜π‘˜)))
8279, 81breqtrd 5174 . 2 ((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) β†’ 3 βˆ₯ (Ξ£π‘˜ ∈ (0...𝑁)((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) βˆ’ Ξ£π‘˜ ∈ (0...𝑁)(πΉβ€˜π‘˜)))
83 dvdssub2 16278 . 2 (((3 ∈ β„€ ∧ Ξ£π‘˜ ∈ (0...𝑁)((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) ∈ β„€ ∧ Ξ£π‘˜ ∈ (0...𝑁)(πΉβ€˜π‘˜) ∈ β„€) ∧ 3 βˆ₯ (Ξ£π‘˜ ∈ (0...𝑁)((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) βˆ’ Ξ£π‘˜ ∈ (0...𝑁)(πΉβ€˜π‘˜))) β†’ (3 βˆ₯ Ξ£π‘˜ ∈ (0...𝑁)((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) ↔ 3 βˆ₯ Ξ£π‘˜ ∈ (0...𝑁)(πΉβ€˜π‘˜)))
842, 13, 14, 82, 83syl31anc 1371 1 ((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) β†’ (3 βˆ₯ Ξ£π‘˜ ∈ (0...𝑁)((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) ↔ 3 βˆ₯ Ξ£π‘˜ ∈ (0...𝑁)(πΉβ€˜π‘˜)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099   β‰  wne 2937   class class class wbr 5148  βŸΆwf 6544  β€˜cfv 6548  (class class class)co 7420  β„‚cc 11137  0cc0 11139  1c1 11140   + caddc 11142   Β· cmul 11144   βˆ’ cmin 11475  -cneg 11476   / cdiv 11902  3c3 12299  9c9 12305  β„•0cn0 12503  β„€cz 12589  cdc 12708  ...cfz 13517  β†‘cexp 14059  Ξ£csu 15665   βˆ₯ cdvds 16231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-inf2 9665  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216  ax-pre-sup 11217
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9466  df-oi 9534  df-card 9963  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-div 11903  df-nn 12244  df-2 12306  df-3 12307  df-4 12308  df-5 12309  df-6 12310  df-7 12311  df-8 12312  df-9 12313  df-n0 12504  df-z 12590  df-dec 12709  df-uz 12854  df-rp 13008  df-fz 13518  df-fzo 13661  df-seq 14000  df-exp 14060  df-hash 14323  df-cj 15079  df-re 15080  df-im 15081  df-sqrt 15215  df-abs 15216  df-clim 15465  df-sum 15666  df-dvds 16232
This theorem is referenced by: (None)
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