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| Mirrors > Home > MPE Home > Th. List > addridi | Structured version Visualization version GIF version | ||
| Description: 0 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.) | 
| Ref | Expression | 
|---|---|
| mul.1 | ⊢ 𝐴 ∈ ℂ | 
| Ref | Expression | 
|---|---|
| addridi | ⊢ (𝐴 + 0) = 𝐴 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mul.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
| 2 | addrid 11442 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 + 0) = 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 (class class class)co 7432 ℂcc 11154 0cc0 11156 + caddc 11159 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-ltxr 11301 | 
| This theorem is referenced by: 1p0e1 12391 9p1e10 12737 num0u 12746 numnncl2 12758 decrmanc 12792 decaddi 12795 decaddci 12796 decmul1 12799 decmulnc 12802 fsumrelem 15844 bpoly4 16096 demoivreALT 16238 decsplit0 17119 37prm 17159 43prm 17160 139prm 17162 163prm 17163 317prm 17164 631prm 17165 1259lem2 17170 1259lem3 17171 1259lem4 17172 1259lem5 17173 2503lem1 17175 2503lem2 17176 2503lem3 17177 4001lem1 17179 4001lem2 17180 4001lem3 17181 4001lem4 17182 sinhalfpilem 26506 efipi 26516 asin1 26938 log2ublem3 26992 log2ub 26993 emcllem6 27045 lgam1 27108 ip2i 30848 pythi 30870 normlem6 31135 normpythi 31162 normpari 31174 pjneli 31743 dp20u 32861 1mhdrd 32899 ballotth 34541 hgt750lemd 34664 hgt750lem2 34668 420gcd8e4 42008 60lcm7e420 42012 420lcm8e840 42013 3lexlogpow5ineq1 42056 3lexlogpow5ineq5 42062 dirkertrigeqlem3 46120 fourierdlem103 46229 fourierdlem104 46230 fouriersw 46251 257prm 47553 fmtno4nprmfac193 47566 fmtno5faclem3 47573 fmtno5fac 47574 139prmALT 47588 127prm 47591 m11nprm 47593 | 
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