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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 10815 | . 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 1531 ∈ wcel 2108 (class class class)co 7148 ℂcc 10527 0cc0 10529 + caddc 10532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7151 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-ltxr 10672 |
This theorem is referenced by: ine0 11067 muleqadd 11276 inelr 11620 nnne0 11663 0p1e1 11751 num0h 12102 nummul1c 12139 decrmac 12148 fz0tp 13000 fz0to4untppr 13002 fzo0to3tp 13115 cats1fvn 14212 rei 14507 imi 14508 ef01bndlem 15529 gcdaddmlem 15864 dec5dvds2 16393 2exp16 16416 43prm 16447 83prm 16448 139prm 16449 163prm 16450 317prm 16451 631prm 16452 1259lem1 16456 1259lem2 16457 1259lem3 16458 1259lem4 16459 1259lem5 16460 2503lem1 16462 2503lem2 16463 2503lem3 16464 2503prm 16465 4001lem1 16466 4001lem2 16467 4001lem3 16468 4001prm 16470 frgpnabllem1 18985 pcoass 23620 dvradcnv 25001 efhalfpi 25049 sinq34lt0t 25087 efifo 25123 logm1 25164 argimgt0 25187 ang180lem4 25382 1cubr 25412 asin1 25464 atanlogsublem 25485 dvatan 25505 log2ublem3 25518 log2ub 25519 basellem9 25658 cht2 25741 log2sumbnd 26112 ax5seglem7 26713 ex-fac 28222 dp20h 30548 dpmul4 30583 hgt750lem2 31916 sqn5i 39162 decpmul 39165 sqdeccom12 39166 sq3deccom12 39167 ex-decpmul 39169 fltnltalem 39265 dirkertrigeqlem1 42374 dirkertrigeqlem3 42376 fourierdlem103 42485 sqwvfoura 42504 sqwvfourb 42505 fouriersw 42507 fmtno5lem1 43706 fmtno5lem2 43707 fmtno5lem4 43709 fmtno4prmfac 43725 fmtno5faclem2 43733 fmtno5faclem3 43734 fmtno5fac 43735 139prmALT 43750 127prm 43754 2exp11 43756 2exp340mod341 43889 nfermltl8rev 43898 |
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