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| Description: Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) | 
| Ref | Expression | 
|---|---|
| mul.1 | ⊢ 𝐴 ∈ ℂ | 
| Ref | Expression | 
|---|---|
| mul02i | ⊢ (0 · 𝐴) = 0 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mul.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
| 2 | mul02 11439 | . 2 ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (0 · 𝐴) = 0 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 0cc0 11155 · cmul 11160 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 | 
| This theorem is referenced by: abs0 15324 odd2np1lem 16377 divalglem8 16437 11prm 17152 631prm 17164 1259lem1 17168 1259lem3 17170 1259lem4 17171 2503lem1 17174 2503lem2 17175 4001lem1 17178 4001lem2 17179 4001lem3 17180 4001prm 17182 pzriprnglem5 21496 pzriprnglem6 21497 pzriprng1ALT 21507 pcoass 25057 sin2pi 26517 abscxpbnd 26796 log2ub 26992 dchrmullid 27296 lgsdir2 27374 lgsdir 27376 ex-prmo 30478 siilem2 30871 nmophmi 32050 ccfldextdgrr 33722 hgt750lem2 34667 60gcd6e6 42005 3exp7 42054 3lexlogpow5ineq1 42055 3lexlogpow5ineq5 42061 aks4d1p1 42077 sqn5i 42320 sqdeccom12 42324 stoweidlem36 46051 fmtnofac1 47557 fmtno5faclem1 47566 fmtno5faclem2 47567 31prm 47584 2exp340mod341 47720 8exp8mod9 47723 nfermltl8rev 47729 line2ylem 48672 | 
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