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Mirrors > Home > ILE Home > Th. List > mul01d | GIF version |
Description: Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
mul01d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
mul01d | ⊢ (𝜑 → (𝐴 · 0) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul01d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | mul01 8278 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐴 · 0) = 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 (class class class)co 5836 ℂcc 7742 0cc0 7744 · cmul 7749 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-setind 4508 ax-resscn 7836 ax-1cn 7837 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-sub 8062 |
This theorem is referenced by: mulap0r 8504 diveqap0 8569 div0ap 8589 mulle0r 8830 un0mulcl 9139 modqid 10274 addmodlteq 10323 expmul 10490 bcval5 10665 fsummulc2 11375 geolim 11438 fprodeq0 11544 0dvds 11737 gcdaddm 11902 bezoutlema 11917 bezoutlemb 11918 lcmgcd 11989 mulgcddvds 12005 cncongr2 12015 prmdiv 12146 pcaddlem 12249 qexpz 12261 dvcnp2cntop 13210 sin0pilem1 13249 sin0pilem2 13250 sinmpi 13283 cosmpi 13284 sinppi 13285 cosppi 13286 trilpolemclim 13756 trilpolemisumle 13758 trilpolemeq1 13760 nconstwlpolem0 13782 |
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