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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 |
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Ref | Expression |
---|---|
mul02d |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul01d.1 |
. 2
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2 | mul02 8340 |
. 2
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3 | 1, 2 | syl 14 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
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-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-setind 4535 ax-resscn 7900 ax-1cn 7901 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-mulcom 7909 ax-addass 7910 ax-distr 7912 ax-i2m1 7913 ax-0id 7916 ax-rnegex 7917 ax-cnre 7919 |
This theorem depends on definitions: df-bi 117 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-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-br 4003 df-opab 4064 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-iota 5177 df-fun 5217 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-sub 8126 |
This theorem is referenced by: mulneg1 8348 mulap0r 8568 mulap0 8607 un0mulcl 9206 mul2lt0rgt0 9756 mul2lt0np 9759 lincmb01cmp 9999 iccf1o 10000 bcval5 10736 hashxp 10799 remul2 10875 immul2 10882 fsumconst 11455 binomlem 11484 fprodeq0 11618 fprodeq0g 11639 efne0 11679 dvds0 11806 mulmoddvds 11861 mulgcd 12009 bezoutr1 12026 lcmgcd 12070 qnumgt0 12190 pcexp 12301 mulgnn0ass 12950 dvmptcmulcn 14054 dvef 14059 sin0pilem1 14073 sinhalfpip 14112 sinhalfpim 14113 coshalfpip 14114 coshalfpim 14115 lgsdir2 14305 lgsdir 14307 lgsdirnn0 14319 lgsdinn0 14320 |
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