|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > 1p0e1 | Structured version Visualization version GIF version | ||
| Description: 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.) | 
| Ref | Expression | 
|---|---|
| 1p0e1 | ⊢ (1 + 0) = 1 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ax-1cn 11214 | . 2 ⊢ 1 ∈ ℂ | |
| 2 | 1 | addridi 11449 | 1 ⊢ (1 + 0) = 1 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 (class class class)co 7432 0cc0 11156 1c1 11157 + 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: xov1plusxeqvd 13539 bernneq 14269 bcpasc 14361 relexpaddg 15093 4sqlem19 17002 1259lem1 17169 2503lem2 17176 pzriprng1ALT 21508 ef2pi 26520 dvsqrt 26785 dvcnsqrt 26787 loglesqrt 26805 efrlim 27013 efrlimOLD 27014 basellem7 27131 1sgm2ppw 27245 addsqnreup 27488 chpchtlim 27524 axlowdimlem16 28973 vc0 30594 ballotlemic 34510 hgt750lemd 34664 divcnvlin 35734 faclim 35747 poimirlem16 37644 poimirlem31 37659 12gcd5e1 42005 3exp7 42055 sticksstones7 42154 sticksstones12a 42159 sticksstones12 42160 metakunt29 42235 3cubeslem1 42700 pell1qr1 42887 pell1qrgaplem 42889 rmxy0 42940 binomcxplemnotnn0 44380 clim1fr1 45621 dvxpaek 45960 itgiccshift 46000 itgperiod 46001 wallispi2lem2 46092 | 
| Copyright terms: Public domain | W3C validator |