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Mirrors > Home > ILE Home > Th. List > mul02d | Unicode version |
Description: Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
mul01d.1 |
Ref | Expression |
---|---|
mul02d |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul01d.1 | . 2 | |
2 | mul02 8262 | . 2 | |
3 | 1, 2 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1335 wcel 2128 (class class class)co 5824 cc 7730 cc0 7732 cmul 7737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-setind 4496 ax-resscn 7824 ax-1cn 7825 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-distr 7836 ax-i2m1 7837 ax-0id 7840 ax-rnegex 7841 ax-cnre 7843 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-sub 8048 |
This theorem is referenced by: mulneg1 8270 mulap0r 8490 mulap0 8528 un0mulcl 9124 mul2lt0rgt0 9667 mul2lt0np 9670 lincmb01cmp 9907 iccf1o 9908 bcval5 10637 hashxp 10700 remul2 10773 immul2 10780 fsumconst 11351 binomlem 11380 fprodeq0 11514 fprodeq0g 11535 efne0 11575 dvds0 11701 mulmoddvds 11754 mulgcd 11899 bezoutr1 11916 lcmgcd 11954 qnumgt0 12072 dvmptcmulcn 13083 dvef 13088 sin0pilem1 13102 sinhalfpip 13141 sinhalfpim 13142 coshalfpip 13143 coshalfpim 13144 |
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