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Mirrors > Home > ILE Home > Th. List > mul02d | 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 |
---|---|
mul02d | ⊢ (𝜑 → (0 · 𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul01d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | mul02 8369 | . 2 ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (0 · 𝐴) = 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 (class class class)co 5892 ℂcc 7834 0cc0 7836 · cmul 7841 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-setind 4551 ax-resscn 7928 ax-1cn 7929 ax-icn 7931 ax-addcl 7932 ax-addrcl 7933 ax-mulcl 7934 ax-addcom 7936 ax-mulcom 7937 ax-addass 7938 ax-distr 7940 ax-i2m1 7941 ax-0id 7944 ax-rnegex 7945 ax-cnre 7947 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-iota 5193 df-fun 5234 df-fv 5240 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-sub 8155 |
This theorem is referenced by: mulneg1 8377 mulap0r 8597 mulap0 8636 un0mulcl 9235 mul2lt0rgt0 9785 mul2lt0np 9788 lincmb01cmp 10028 iccf1o 10029 bcval5 10770 hashxp 10833 remul2 10909 immul2 10916 fsumconst 11489 binomlem 11518 fprodeq0 11652 fprodeq0g 11673 efne0 11713 dvds0 11840 mulmoddvds 11896 mulgcd 12044 bezoutr1 12061 lcmgcd 12105 qnumgt0 12225 pcexp 12336 mulgnn0ass 13091 dvmptcmulcn 14620 dvef 14625 sin0pilem1 14639 sinhalfpip 14678 sinhalfpim 14679 coshalfpip 14680 coshalfpim 14681 lgsdir2 14872 lgsdir 14874 lgsdirnn0 14886 lgsdinn0 14887 |
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