![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 3m1e2 | Structured version Visualization version GIF version |
Description: 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.) (Proof shortened by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
3m1e2 | ⊢ (3 − 1) = 2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2cn 12284 | . 2 ⊢ 2 ∈ ℂ | |
2 | ax-1cn 11165 | . 2 ⊢ 1 ∈ ℂ | |
3 | df-3 12273 | . 2 ⊢ 3 = (2 + 1) | |
4 | 1, 2, 3 | mvrraddi 11474 | 1 ⊢ (3 − 1) = 2 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 (class class class)co 7406 1c1 11108 − cmin 11441 2c2 12264 3c3 12265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 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 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-ltxr 11250 df-sub 11443 df-2 12272 df-3 12273 |
This theorem is referenced by: halfpm6th 12430 ige3m2fz 13522 fzo13pr 13713 fzo0to3tp 13715 fldiv4p1lem1div2 13797 lsws3 14853 bpoly3 15999 rpnnen2lem3 16156 rpnnen2lem11 16164 3prm 16628 prmo3 16971 1cubrlem 26336 1cubr 26337 quart1 26351 log2cnv 26439 log2ublem3 26443 2lgslem3b 26890 2lgslem3d 26892 axlowdimlem16 28205 2pthd 29184 wlk2v2e 29400 ex-bc 29695 cyc3fv1 32284 cyc3fv2 32285 cyc3fv3 32286 fib4 33392 circlemethhgt 33644 cusgracyclt3v 34136 itg2addnclem3 36530 lcm3un 40869 aks4d1p1 40930 2np3bcnp1 40949 lhe4.4ex1a 43074 wallispilem4 44771 fmtnoge3 46185 fmtnoprmfac2lem1 46221 nnsum3primesle9 46449 |
Copyright terms: Public domain | W3C validator |