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Mirrors > Home > MPE Home > Th. List > mul01i | Structured version Visualization version GIF version |
Description: Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.) |
Ref | Expression |
---|---|
mul.1 | ⊢ 𝐴 ∈ ℂ |
Ref | Expression |
---|---|
mul01i | ⊢ (𝐴 · 0) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | mul01 10094 | . 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 1475 ∈ wcel 1977 (class class class)co 6549 ℂcc 9813 0cc0 9815 · cmul 9820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 |
This theorem is referenced by: ine0 10344 msqge0 10428 recextlem2 10537 eqneg 10624 crne0 10890 2t0e0 11060 it0e0 11131 num0h 11385 decmul1 11461 decmul1OLD 11462 discr 12863 sin4lt0 14764 demoivreALT 14770 gcdaddmlem 15083 bezout 15098 139prm 15669 317prm 15671 631prm 15672 1259lem4 15679 2503lem1 15682 2503lem2 15683 4001lem1 15686 4001lem2 15687 4001lem3 15688 4001lem4 15689 odadd1 18074 minveclem7 23014 itg1addlem4 23272 aalioulem3 23893 dcubic 24373 log2ublem3 24475 basellem7 24613 basellem9 24615 lgsdir2 24855 selberg2lem 25039 logdivbnd 25045 pntrsumo1 25054 pntrlog2bndlem5 25070 axpaschlem 25620 axlowdimlem6 25627 nmblolbii 27038 siilem1 27090 minvecolem7 27123 eigorthi 28080 nmbdoplbi 28267 nmcoplbi 28271 nmbdfnlbi 28292 nmcfnlbi 28295 nmopcoi 28338 subfacval2 30423 areacirc 32675 139prmALT 40049 |
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