mul01i - Metamath Proof Explorer

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Theorem mul01i 11434

Description: Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)

**Hypothesis**
Ref	Expression
mul.1	⊢ 𝐴 ∈ ℂ

**Assertion**
Ref	Expression
mul01i	⊢ (𝐴 · 0) = 0

**Proof of Theorem mul01i**
Step	Hyp	Ref	Expression
1		mul.1	. 2 ⊢ 𝐴 ∈ ℂ
2		mul01 11423	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
3	1, 2	ax-mp 5	1 ⊢ (𝐴 · 0) = 0

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 ∈ wcel 2099 (class class class)co 7420 ℂcc 11136 0cc0 11138 · cmul 11143

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-ltxr 11283

This theorem is referenced by: ine0 11679 msqge0 11765 recextlem2 11875 eqneg 11964 crne0 12235 2t0e0 12411 it0e0 12464 num0h 12719 discr 14234 sin4lt0 16171 demoivreALT 16177 gcdaddmlem 16498 bezout 16518 139prm 17092 317prm 17094 631prm 17095 1259lem4 17102 2503lem1 17105 2503lem2 17106 4001lem1 17109 4001lem2 17110 4001lem3 17111 4001lem4 17112 odadd1 19802 minveclem7 25362 itg1addlem4 25627 itg1addlem4OLD 25628 aalioulem3 26268 dcubic 26777 log2ublem3 26879 basellem7 27018 basellem9 27020 lgsdir2 27262 selberg2lem 27482 logdivbnd 27488 pntrsumo1 27497 pntrlog2bndlem5 27513 axpaschlem 28750 axlowdimlem6 28757 nmblolbii 30608 siilem1 30660 minvecolem7 30692 eigorthi 31646 nmbdoplbi 31833 nmcoplbi 31837 nmbdfnlbi 31858 nmcfnlbi 31861 nmopcoi 31904 itgexpif 34238 hgt750lem2 34284 subfacval2 34797 areacirc 37186 60lcm7e420 41481 3lexlogpow5ineq1 41525 sqn5i 41859 139prmALT 46936

	
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