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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 11402 | . 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 2105 (class class class)co 7412 ℂcc 11112 0cc0 11114 + caddc 11117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11255 df-mnf 11256 df-ltxr 11258 |
This theorem is referenced by: ine0 11654 muleqadd 11863 inelr 12207 nnne0 12251 0p1e1 12339 num0h 12694 nummul1c 12731 decrmac 12740 fz0tp 13607 fz0to4untppr 13609 fzo0to3tp 13723 cats1fvn 14814 rei 15108 imi 15109 ef01bndlem 16132 gcdaddmlem 16470 dec5dvds2 17003 2exp11 17028 2exp16 17029 43prm 17060 83prm 17061 139prm 17062 163prm 17063 317prm 17064 631prm 17065 1259lem1 17069 1259lem2 17070 1259lem3 17071 1259lem4 17072 1259lem5 17073 2503lem1 17075 2503lem2 17076 2503lem3 17077 2503prm 17078 4001lem1 17079 4001lem2 17080 4001lem3 17081 4001prm 17083 frgpnabllem1 19783 pcoass 24772 dvradcnv 26170 efhalfpi 26218 sinq34lt0t 26256 efifo 26293 logm1 26334 argimgt0 26357 ang180lem4 26554 1cubr 26584 asin1 26636 atanlogsublem 26657 dvatan 26677 log2ublem3 26690 log2ub 26691 basellem9 26830 cht2 26913 log2sumbnd 27284 ax5seglem7 28461 ex-fac 29972 dp20h 32313 dpmul4 32348 hgt750lem2 33963 12gcd5e1 41175 3exp7 41225 3lexlogpow5ineq1 41226 3lexlogpow5ineq5 41232 aks4d1p1 41248 sqn5i 41500 decpmul 41503 sqdeccom12 41504 sq3deccom12 41505 ex-decpmul 41509 fltnltalem 41707 dirkertrigeqlem1 45113 dirkertrigeqlem3 45115 fourierdlem103 45224 sqwvfoura 45243 sqwvfourb 45244 fouriersw 45246 fmtno5lem1 46520 fmtno5lem2 46521 fmtno5lem4 46523 fmtno4prmfac 46539 fmtno5faclem2 46547 fmtno5faclem3 46548 fmtno5fac 46549 139prmALT 46563 127prm 46566 2exp340mod341 46700 nfermltl8rev 46709 ackval1012 47464 ackval2012 47465 ackval3012 47466 |
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