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Theorem 3dvds 16270
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 12591 . . 3 3 ∈ β„€
21a1i 11 . 2 ((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) β†’ 3 ∈ β„€)
3 fzfid 13934 . . 3 ((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) β†’ (0...𝑁) ∈ Fin)
4 ffvelcdm 7080 . . . . 5 ((𝐹:(0...𝑁)βŸΆβ„€ ∧ π‘˜ ∈ (0...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ β„€)
54adantll 712 . . . 4 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ β„€)
6 10nn 12689 . . . . . 6 10 ∈ β„•
76nnzi 12582 . . . . 5 10 ∈ β„€
8 elfznn0 13590 . . . . . 6 (π‘˜ ∈ (0...𝑁) β†’ π‘˜ ∈ β„•0)
98adantl 482 . . . . 5 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ π‘˜ ∈ β„•0)
10 zexpcl 14038 . . . . 5 ((10 ∈ β„€ ∧ π‘˜ ∈ β„•0) β†’ (10β†‘π‘˜) ∈ β„€)
117, 9, 10sylancr 587 . . . 4 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ (10β†‘π‘˜) ∈ β„€)
125, 11zmulcld 12668 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ ((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) ∈ β„€)
133, 12fsumzcl 15677 . 2 ((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) β†’ Ξ£π‘˜ ∈ (0...𝑁)((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) ∈ β„€)
143, 5fsumzcl 15677 . 2 ((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) β†’ Ξ£π‘˜ ∈ (0...𝑁)(πΉβ€˜π‘˜) ∈ β„€)
1512, 5zsubcld 12667 . . . 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 11164 . . . . . . . . . . . 12 1 ∈ β„‚
176nncni 12218 . . . . . . . . . . . 12 10 ∈ β„‚
1816, 17negsubdi2i 11542 . . . . . . . . . . 11 -(1 βˆ’ 10) = (10 βˆ’ 1)
19 9p1e10 12675 . . . . . . . . . . . . 13 (9 + 1) = 10
2019eqcomi 2741 . . . . . . . . . . . 12 10 = (9 + 1)
2120oveq1i 7415 . . . . . . . . . . 11 (10 βˆ’ 1) = ((9 + 1) βˆ’ 1)
22 9cn 12308 . . . . . . . . . . . 12 9 ∈ β„‚
2322, 16pncan3oi 11472 . . . . . . . . . . 11 ((9 + 1) βˆ’ 1) = 9
2418, 21, 233eqtri 2764 . . . . . . . . . 10 -(1 βˆ’ 10) = 9
25 3t3e9 12375 . . . . . . . . . 10 (3 Β· 3) = 9
2624, 25eqtr4i 2763 . . . . . . . . 9 -(1 βˆ’ 10) = (3 Β· 3)
2717a1i 11 . . . . . . . . . . . . . . 15 (π‘˜ ∈ β„•0 β†’ 10 ∈ β„‚)
28 1re 11210 . . . . . . . . . . . . . . . . 17 1 ∈ ℝ
29 1lt10 12812 . . . . . . . . . . . . . . . . 17 1 < 10
3028, 29gtneii 11322 . . . . . . . . . . . . . . . 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 15809 . . . . . . . . . . . . . 14 (π‘˜ ∈ β„•0 β†’ Σ𝑗 ∈ (0...(π‘˜ βˆ’ 1))(10↑𝑗) = ((1 βˆ’ (10β†‘π‘˜)) / (1 βˆ’ 10)))
34 fzfid 13934 . . . . . . . . . . . . . . 15 (π‘˜ ∈ β„•0 β†’ (0...(π‘˜ βˆ’ 1)) ∈ Fin)
35 elfznn0 13590 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (0...(π‘˜ βˆ’ 1)) β†’ 𝑗 ∈ β„•0)
3635adantl 482 . . . . . . . . . . . . . . . 16 ((π‘˜ ∈ β„•0 ∧ 𝑗 ∈ (0...(π‘˜ βˆ’ 1))) β†’ 𝑗 ∈ β„•0)
37 zexpcl 14038 . . . . . . . . . . . . . . . 16 ((10 ∈ β„€ ∧ 𝑗 ∈ β„•0) β†’ (10↑𝑗) ∈ β„€)
387, 36, 37sylancr 587 . . . . . . . . . . . . . . 15 ((π‘˜ ∈ β„•0 ∧ 𝑗 ∈ (0...(π‘˜ βˆ’ 1))) β†’ (10↑𝑗) ∈ β„€)
3934, 38fsumzcl 15677 . . . . . . . . . . . . . 14 (π‘˜ ∈ β„•0 β†’ Σ𝑗 ∈ (0...(π‘˜ βˆ’ 1))(10↑𝑗) ∈ β„€)
4033, 39eqeltrrd 2834 . . . . . . . . . . . . 13 (π‘˜ ∈ β„•0 β†’ ((1 βˆ’ (10β†‘π‘˜)) / (1 βˆ’ 10)) ∈ β„€)
41 1z 12588 . . . . . . . . . . . . . . 15 1 ∈ β„€
42 zsubcl 12600 . . . . . . . . . . . . . . 15 ((1 ∈ β„€ ∧ 10 ∈ β„€) β†’ (1 βˆ’ 10) ∈ β„€)
4341, 7, 42mp2an 690 . . . . . . . . . . . . . 14 (1 βˆ’ 10) ∈ β„€
4428, 29ltneii 11323 . . . . . . . . . . . . . . 15 1 β‰  10
4516, 17subeq0i 11536 . . . . . . . . . . . . . . . 16 ((1 βˆ’ 10) = 0 ↔ 1 = 10)
4645necon3bii 2993 . . . . . . . . . . . . . . 15 ((1 βˆ’ 10) β‰  0 ↔ 1 β‰  10)
4744, 46mpbir 230 . . . . . . . . . . . . . 14 (1 βˆ’ 10) β‰  0
487, 32, 10sylancr 587 . . . . . . . . . . . . . . 15 (π‘˜ ∈ β„•0 β†’ (10β†‘π‘˜) ∈ β„€)
49 zsubcl 12600 . . . . . . . . . . . . . . 15 ((1 ∈ β„€ ∧ (10β†‘π‘˜) ∈ β„€) β†’ (1 βˆ’ (10β†‘π‘˜)) ∈ β„€)
5041, 48, 49sylancr 587 . . . . . . . . . . . . . 14 (π‘˜ ∈ β„•0 β†’ (1 βˆ’ (10β†‘π‘˜)) ∈ β„€)
51 dvdsval2 16196 . . . . . . . . . . . . . 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 1465 . . . . . . . . . . . . 13 (π‘˜ ∈ β„•0 β†’ ((1 βˆ’ 10) βˆ₯ (1 βˆ’ (10β†‘π‘˜)) ↔ ((1 βˆ’ (10β†‘π‘˜)) / (1 βˆ’ 10)) ∈ β„€))
5340, 52mpbird 256 . . . . . . . . . . . 12 (π‘˜ ∈ β„•0 β†’ (1 βˆ’ 10) βˆ₯ (1 βˆ’ (10β†‘π‘˜)))
5448zcnd 12663 . . . . . . . . . . . . 13 (π‘˜ ∈ β„•0 β†’ (10β†‘π‘˜) ∈ β„‚)
55 negsubdi2 11515 . . . . . . . . . . . . 13 (((10β†‘π‘˜) ∈ β„‚ ∧ 1 ∈ β„‚) β†’ -((10β†‘π‘˜) βˆ’ 1) = (1 βˆ’ (10β†‘π‘˜)))
5654, 16, 55sylancl 586 . . . . . . . . . . . 12 (π‘˜ ∈ β„•0 β†’ -((10β†‘π‘˜) βˆ’ 1) = (1 βˆ’ (10β†‘π‘˜)))
5753, 56breqtrrd 5175 . . . . . . . . . . 11 (π‘˜ ∈ β„•0 β†’ (1 βˆ’ 10) βˆ₯ -((10β†‘π‘˜) βˆ’ 1))
58 peano2zm 12601 . . . . . . . . . . . . 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 16213 . . . . . . . . . . . 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 587 . . . . . . . . . . 11 (π‘˜ ∈ β„•0 β†’ ((1 βˆ’ 10) βˆ₯ ((10β†‘π‘˜) βˆ’ 1) ↔ (1 βˆ’ 10) βˆ₯ -((10β†‘π‘˜) βˆ’ 1)))
6257, 61mpbird 256 . . . . . . . . . 10 (π‘˜ ∈ β„•0 β†’ (1 βˆ’ 10) βˆ₯ ((10β†‘π‘˜) βˆ’ 1))
63 negdvdsb 16212 . . . . . . . . . . 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 587 . . . . . . . . . 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 5183 . . . . . . . 8 (π‘˜ ∈ β„•0 β†’ (3 Β· 3) βˆ₯ ((10β†‘π‘˜) βˆ’ 1))
67 muldvds1 16220 . . . . . . . . 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 1465 . . . . . . . 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 16237 . . . . . . 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 1466 . . . . . 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 12663 . . . . . 6 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
7611zcnd 12663 . . . . . 6 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ (10β†‘π‘˜) ∈ β„‚)
7775, 76muls1d 11670 . . . . 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 5173 . . . 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 16247 . . 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 12663 . . . 4 (((𝑁 ∈ β„•0 ∧ 𝐹:(0...𝑁)βŸΆβ„€) ∧ π‘˜ ∈ (0...𝑁)) β†’ ((πΉβ€˜π‘˜) Β· (10β†‘π‘˜)) ∈ β„‚)
813, 80, 75fsumsub 15730 . . 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 5173 . 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 16240 . 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 1373 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 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   class class class wbr 5147  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405  β„‚cc 11104  0cc0 11106  1c1 11107   + caddc 11109   Β· cmul 11111   βˆ’ cmin 11440  -cneg 11441   / cdiv 11867  3c3 12264  9c9 12270  β„•0cn0 12468  β„€cz 12554  cdc 12673  ...cfz 13480  β†‘cexp 14023  Ξ£csu 15628   βˆ₯ cdvds 16193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-rp 12971  df-fz 13481  df-fzo 13624  df-seq 13963  df-exp 14024  df-hash 14287  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-sum 15629  df-dvds 16194
This theorem is referenced by: (None)
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