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Theorem 3dvds 16277
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 12594 . . 3 3 ∈ β„€
21a1i 11 . 2 ((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) β†’ 3 ∈ β„€)
3 fzfid 13939 . . 3 ((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) β†’ (0...𝑁) ∈ Fin)
4 ffvelcdm 7074 . . . . 5 ((𝐹:(0...𝑁)βŸΆβ„€ ∧ π‘˜ ∈ (0...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ β„€)
54adantll 711 . . . 4 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ β„€)
6 10nn 12692 . . . . . 6 10 ∈ β„•
76nnzi 12585 . . . . 5 10 ∈ β„€
8 elfznn0 13595 . . . . . 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 14043 . . . . 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 12671 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ ((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) ∈ β„€)
133, 12fsumzcl 15683 . 2 ((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) β†’ Ξ£π‘˜ ∈ (0...𝑁)((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) ∈ β„€)
143, 5fsumzcl 15683 . 2 ((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) β†’ Ξ£π‘˜ ∈ (0...𝑁)(πΉβ€˜π‘˜) ∈ β„€)
1512, 5zsubcld 12670 . . . 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 11165 . . . . . . . . . . . 12 1 ∈ β„‚
176nncni 12221 . . . . . . . . . . . 12 10 ∈ β„‚
1816, 17negsubdi2i 11545 . . . . . . . . . . 11 -(1 βˆ’ 10) = (10 βˆ’ 1)
19 9p1e10 12678 . . . . . . . . . . . . 13 (9 + 1) = 10
2019eqcomi 2733 . . . . . . . . . . . 12 10 = (9 + 1)
2120oveq1i 7412 . . . . . . . . . . 11 (10 βˆ’ 1) = ((9 + 1) βˆ’ 1)
22 9cn 12311 . . . . . . . . . . . 12 9 ∈ β„‚
2322, 16pncan3oi 11475 . . . . . . . . . . 11 ((9 + 1) βˆ’ 1) = 9
2418, 21, 233eqtri 2756 . . . . . . . . . 10 -(1 βˆ’ 10) = 9
25 3t3e9 12378 . . . . . . . . . 10 (3 Β· 3) = 9
2624, 25eqtr4i 2755 . . . . . . . . 9 -(1 βˆ’ 10) = (3 Β· 3)
2717a1i 11 . . . . . . . . . . . . . . 15 (π‘˜ ∈ β„•0 β†’ 10 ∈ β„‚)
28 1re 11213 . . . . . . . . . . . . . . . . 17 1 ∈ ℝ
29 1lt10 12815 . . . . . . . . . . . . . . . . 17 1 < 10
3028, 29gtneii 11325 . . . . . . . . . . . . . . . 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 15815 . . . . . . . . . . . . . 14 (π‘˜ ∈ β„•0 β†’ Σ𝑗 ∈ (0...(π‘˜ βˆ’ 1))(10↑𝑗) = ((1 βˆ’ (10β†‘π‘˜)) / (1 βˆ’ 10)))
34 fzfid 13939 . . . . . . . . . . . . . . 15 (π‘˜ ∈ β„•0 β†’ (0...(π‘˜ βˆ’ 1)) ∈ Fin)
35 elfznn0 13595 . . . . . . . . . . . . . . . . 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 14043 . . . . . . . . . . . . . . . 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 15683 . . . . . . . . . . . . . 14 (π‘˜ ∈ β„•0 β†’ Σ𝑗 ∈ (0...(π‘˜ βˆ’ 1))(10↑𝑗) ∈ β„€)
4033, 39eqeltrrd 2826 . . . . . . . . . . . . 13 (π‘˜ ∈ β„•0 β†’ ((1 βˆ’ (10β†‘π‘˜)) / (1 βˆ’ 10)) ∈ β„€)
41 1z 12591 . . . . . . . . . . . . . . 15 1 ∈ β„€
42 zsubcl 12603 . . . . . . . . . . . . . . 15 ((1 ∈ β„€ ∧ 10 ∈ β„€) β†’ (1 βˆ’ 10) ∈ β„€)
4341, 7, 42mp2an 689 . . . . . . . . . . . . . 14 (1 βˆ’ 10) ∈ β„€
4428, 29ltneii 11326 . . . . . . . . . . . . . . 15 1 β‰  10
4516, 17subeq0i 11539 . . . . . . . . . . . . . . . 16 ((1 βˆ’ 10) = 0 ↔ 1 = 10)
4645necon3bii 2985 . . . . . . . . . . . . . . 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 12603 . . . . . . . . . . . . . . 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 16203 . . . . . . . . . . . . . 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 1461 . . . . . . . . . . . . 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 12666 . . . . . . . . . . . . 13 (π‘˜ ∈ β„•0 β†’ (10β†‘π‘˜) ∈ β„‚)
55 negsubdi2 11518 . . . . . . . . . . . . 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 5167 . . . . . . . . . . 11 (π‘˜ ∈ β„•0 β†’ (1 βˆ’ 10) βˆ₯ -((10β†‘π‘˜) βˆ’ 1))
58 peano2zm 12604 . . . . . . . . . . . . 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 16220 . . . . . . . . . . . 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 16219 . . . . . . . . . . 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 5175 . . . . . . . 8 (π‘˜ ∈ β„•0 β†’ (3 Β· 3) βˆ₯ ((10β†‘π‘˜) βˆ’ 1))
67 muldvds1 16227 . . . . . . . . 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 1461 . . . . . . . 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 16244 . . . . . . 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 1462 . . . . . 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 12666 . . . . . 6 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
7611zcnd 12666 . . . . . 6 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ (10β†‘π‘˜) ∈ β„‚)
7775, 76muls1d 11673 . . . . 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 5165 . . . 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 16254 . . 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 12666 . . . 4 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ ((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) ∈ β„‚)
813, 80, 75fsumsub 15736 . . 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 5165 . 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 16247 . 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 1370 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 1533   ∈ wcel 2098   β‰  wne 2932   class class class wbr 5139  βŸΆwf 6530  β€˜cfv 6534  (class class class)co 7402  β„‚cc 11105  0cc0 11107  1c1 11108   + caddc 11110   Β· cmul 11112   βˆ’ cmin 11443  -cneg 11444   / cdiv 11870  3c3 12267  9c9 12273  β„•0cn0 12471  β„€cz 12557  cdc 12676  ...cfz 13485  β†‘cexp 14028  Ξ£csu 15634   βˆ₯ cdvds 16200
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-inf2 9633  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-se 5623  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-isom 6543  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-sup 9434  df-oi 9502  df-card 9931  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12472  df-z 12558  df-dec 12677  df-uz 12822  df-rp 12976  df-fz 13486  df-fzo 13629  df-seq 13968  df-exp 14029  df-hash 14292  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-clim 15434  df-sum 15635  df-dvds 16201
This theorem is referenced by: (None)
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