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Mirrors > Home > MPE Home > Th. List > addid1i | 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 |
---|---|
addid1i | ⊢ (𝐴 + 0) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | addid1 10820 | . 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 1537 ∈ wcel 2114 (class class class)co 7156 ℂcc 10535 0cc0 10537 + caddc 10540 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 |
This theorem is referenced by: 1p0e1 11762 9p1e10 12101 num0u 12110 numnncl2 12122 decrmanc 12156 decaddi 12159 decaddci 12160 decmul1 12163 decmulnc 12166 fsumrelem 15162 bpoly4 15413 demoivreALT 15554 decexp2 16411 decsplit0 16417 37prm 16454 43prm 16455 139prm 16457 163prm 16458 317prm 16459 631prm 16460 1259lem2 16465 1259lem3 16466 1259lem4 16467 1259lem5 16468 2503lem1 16470 2503lem2 16471 2503lem3 16472 4001lem1 16474 4001lem2 16475 4001lem3 16476 4001lem4 16477 sinhalfpilem 25049 efipi 25059 asin1 25472 log2ublem3 25526 log2ub 25527 emcllem6 25578 lgam1 25641 ip2i 28605 pythi 28627 normlem6 28892 normpythi 28919 normpari 28931 pjneli 29500 dp20u 30554 1mhdrd 30592 ballotth 31795 hgt750lemd 31919 hgt750lem2 31923 dirkertrigeqlem3 42405 fourierdlem103 42514 fourierdlem104 42515 fouriersw 42536 257prm 43743 fmtno4nprmfac193 43756 fmtno5faclem3 43763 fmtno5fac 43764 139prmALT 43779 127prm 43783 m11nprm 43786 |
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