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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 10976 | . 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 1543 ∈ wcel 2112 (class class class)co 7191 ℂcc 10692 0cc0 10694 · cmul 10699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-ltxr 10837 |
This theorem is referenced by: ine0 11232 msqge0 11318 recextlem2 11428 eqneg 11517 crne0 11788 2t0e0 11964 it0e0 12017 num0h 12270 discr 13772 sin4lt0 15719 demoivreALT 15725 gcdaddmlem 16046 bezout 16066 139prm 16640 317prm 16642 631prm 16643 1259lem4 16650 2503lem1 16653 2503lem2 16654 4001lem1 16657 4001lem2 16658 4001lem3 16659 4001lem4 16660 odadd1 19187 minveclem7 24286 itg1addlem4 24550 itg1addlem4OLD 24551 aalioulem3 25181 dcubic 25683 log2ublem3 25785 basellem7 25923 basellem9 25925 lgsdir2 26165 selberg2lem 26385 logdivbnd 26391 pntrsumo1 26400 pntrlog2bndlem5 26416 axpaschlem 26985 axlowdimlem6 26992 nmblolbii 28834 siilem1 28886 minvecolem7 28918 eigorthi 29872 nmbdoplbi 30059 nmcoplbi 30063 nmbdfnlbi 30084 nmcfnlbi 30087 nmopcoi 30130 itgexpif 32252 hgt750lem2 32298 subfacval2 32816 areacirc 35556 60lcm7e420 39701 3lexlogpow5ineq1 39745 sqn5i 39961 139prmALT 44664 |
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