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Mirrors > Home > ILE Home > Th. List > ex-dvds | GIF version |
Description: Example for df-dvds 11790: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.) |
Ref | Expression |
---|---|
ex-dvds | ⊢ 3 ∥ 6 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z 9279 | . . 3 ⊢ 2 ∈ ℤ | |
2 | 3z 9280 | . . 3 ⊢ 3 ∈ ℤ | |
3 | 6nn 9082 | . . . 4 ⊢ 6 ∈ ℕ | |
4 | 3 | nnzi 9272 | . . 3 ⊢ 6 ∈ ℤ |
5 | 1, 2, 4 | 3pm3.2i 1175 | . 2 ⊢ (2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 6 ∈ ℤ) |
6 | 3cn 8992 | . . . 4 ⊢ 3 ∈ ℂ | |
7 | 6 | 2timesi 9047 | . . 3 ⊢ (2 · 3) = (3 + 3) |
8 | 3p3e6 9059 | . . 3 ⊢ (3 + 3) = 6 | |
9 | 7, 8 | eqtri 2198 | . 2 ⊢ (2 · 3) = 6 |
10 | dvds0lem 11803 | . 2 ⊢ (((2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 6 ∈ ℤ) ∧ (2 · 3) = 6) → 3 ∥ 6) | |
11 | 5, 9, 10 | mp2an 426 | 1 ⊢ 3 ∥ 6 |
Colors of variables: wff set class |
Syntax hints: ∧ w3a 978 = wceq 1353 ∈ wcel 2148 class class class wbr 4003 (class class class)co 5874 + caddc 7813 · cmul 7815 2c2 8968 3c3 8969 6c6 8972 ℤcz 9251 ∥ cdvds 11789 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-inn 8918 df-2 8976 df-3 8977 df-4 8978 df-5 8979 df-6 8980 df-z 9252 df-dvds 11790 |
This theorem is referenced by: (None) |
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