Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > addid2i | 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 |
---|---|
addid2i | ⊢ (0 + 𝐴) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | addid2 10219 | . 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 1483 ∈ wcel 1990 (class class class)co 6650 ℂcc 9934 0cc0 9936 + caddc 9939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 |
This theorem is referenced by: ine0 10465 muleqadd 10671 inelr 11010 0p1e1 11132 num0h 11509 nummul1c 11562 decrmac 11577 decmul1 11585 decmul1OLD 11586 fz0tp 12440 fz0to4untppr 12442 fzo0to3tp 12554 cats1fvn 13603 rei 13896 imi 13897 ef01bndlem 14914 gcdaddmlem 15245 dec5dvds2 15769 2exp16 15797 43prm 15829 83prm 15830 139prm 15831 163prm 15832 317prm 15833 631prm 15834 1259lem1 15838 1259lem2 15839 1259lem3 15840 1259lem4 15841 1259lem5 15842 2503lem1 15844 2503lem2 15845 2503lem3 15846 2503prm 15847 4001lem1 15848 4001lem2 15849 4001lem3 15850 4001prm 15852 frgpnabllem1 18276 pcoass 22824 dvradcnv 24175 efhalfpi 24223 sinq34lt0t 24261 efifo 24293 logm1 24335 argimgt0 24358 ang180lem4 24542 1cubr 24569 asin1 24621 atanlogsublem 24642 dvatan 24662 log2ublem3 24675 log2ub 24676 basellem9 24815 cht2 24898 log2sumbnd 25233 ax5seglem7 25815 ex-fac 27308 dp20h 29586 dpmul4 29622 hgt750lem2 30730 dirkertrigeqlem1 40315 dirkertrigeqlem3 40317 fourierdlem103 40426 sqwvfoura 40445 sqwvfourb 40446 fouriersw 40448 fmtno5lem1 41465 fmtno5lem2 41466 fmtno5lem4 41468 fmtno4prmfac 41484 fmtno5faclem2 41492 fmtno5faclem3 41493 fmtno5fac 41494 139prmALT 41511 127prm 41515 2exp11 41517 |
Copyright terms: Public domain | W3C validator |