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| Mirrors > Home > MPE Home > Th. List > addlidi | Structured version Visualization version GIF version | ||
| Description: 0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.) |
| Ref | Expression |
|---|---|
| mul.1 | ⊢ 𝐴 ∈ ℂ |
| Ref | Expression |
|---|---|
| addlidi | ⊢ (0 + 𝐴) = 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mul.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
| 2 | addlid 11444 | . 2 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (0 + 𝐴) = 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 0cc0 11155 + caddc 11158 |
| 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: ine0 11698 muleqadd 11907 inelr 12256 nnne0 12300 0p1e1 12388 num0h 12745 nummul1c 12782 decrmac 12791 fz0tp 13668 fzo0to3tp 13791 cats1fvn 14897 rei 15195 imi 15196 ef01bndlem 16220 5ndvds3 16450 gcdaddmlem 16561 dec5dvds2 17103 2exp11 17127 2exp16 17128 43prm 17159 83prm 17160 139prm 17161 163prm 17162 317prm 17163 631prm 17164 1259lem1 17168 1259lem2 17169 1259lem3 17170 1259lem4 17171 1259lem5 17172 2503lem1 17174 2503lem2 17175 2503lem3 17176 2503prm 17177 4001lem1 17178 4001lem2 17179 4001lem3 17180 4001prm 17182 frgpnabllem1 19891 pcoass 25057 dvradcnv 26464 efhalfpi 26513 sinq34lt0t 26551 efifo 26589 logm1 26631 argimgt0 26654 ang180lem4 26855 1cubr 26885 asin1 26937 atanlogsublem 26958 dvatan 26978 log2ublem3 26991 log2ub 26992 basellem9 27132 cht2 27215 log2sumbnd 27588 ax5seglem7 28950 ex-fac 30470 dp20h 32861 dpmul4 32896 hgt750lem2 34667 12gcd5e1 42004 3exp7 42054 3lexlogpow5ineq1 42055 3lexlogpow5ineq5 42061 aks4d1p1 42077 posbezout 42101 sqn5i 42320 decpmul 42323 sqdeccom12 42324 sq3deccom12 42325 ex-decpmul 42340 fltnltalem 42672 dirkertrigeqlem1 46113 dirkertrigeqlem3 46115 fourierdlem103 46224 sqwvfoura 46243 sqwvfourb 46244 fouriersw 46246 fmtno5lem1 47540 fmtno5lem2 47541 fmtno5lem4 47543 fmtno4prmfac 47559 fmtno5faclem2 47567 fmtno5faclem3 47568 fmtno5fac 47569 139prmALT 47583 127prm 47586 2exp340mod341 47720 nfermltl8rev 47729 ackval1012 48611 ackval2012 48612 ackval3012 48613 |
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