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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 8142 | . 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 1331 ∈ wcel 1480 (class class class)co 5767 ℂcc 7611 0cc0 7613 · cmul 7618 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-setind 4447 ax-resscn 7705 ax-1cn 7706 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-sub 7928 |
This theorem is referenced by: mulneg1 8150 mulap0r 8370 mulap0 8408 un0mulcl 9004 mul2lt0rgt0 9540 mul2lt0np 9543 lincmb01cmp 9779 iccf1o 9780 bcval5 10502 hashxp 10565 remul2 10638 immul2 10645 fsumconst 11216 binomlem 11245 efne0 11373 dvds0 11497 mulmoddvds 11550 mulgcd 11693 bezoutr1 11710 lcmgcd 11748 qnumgt0 11865 dvmptcmulcn 12841 dvef 12845 sin0pilem1 12851 sinhalfpip 12890 sinhalfpim 12891 coshalfpip 12892 coshalfpim 12893 |
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