![]() |
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 12339 | . 2 ⊢ 2 ∈ ℂ | |
2 | ax-1cn 11216 | . 2 ⊢ 1 ∈ ℂ | |
3 | df-3 12328 | . 2 ⊢ 3 = (2 + 1) | |
4 | 1, 2, 3 | mvrraddi 11527 | 1 ⊢ (3 − 1) = 2 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 (class class class)co 7424 1c1 11159 − cmin 11494 2c2 12319 3c3 12320 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-ltxr 11303 df-sub 11496 df-2 12327 df-3 12328 |
This theorem is referenced by: halfpm6th 12485 ige3m2fz 13579 fzo13pr 13770 fzo0to3tp 13772 fldiv4p1lem1div2 13855 lsws3 14914 bpoly3 16060 rpnnen2lem3 16218 rpnnen2lem11 16226 3prm 16695 prmo3 17043 1cubrlem 26869 1cubr 26870 quart1 26884 log2cnv 26972 log2ublem3 26976 2lgslem3b 27426 2lgslem3d 27428 axlowdimlem16 28891 2pthd 29874 wlk2v2e 30090 ex-bc 30385 cyc3fv1 33015 cyc3fv2 33016 cyc3fv3 33017 fib4 34238 circlemethhgt 34489 cusgracyclt3v 34984 itg2addnclem3 37374 lcm3un 41714 aks4d1p1 41775 2np3bcnp1 41842 lhe4.4ex1a 44003 wallispilem4 45689 fmtnoge3 47102 fmtnoprmfac2lem1 47138 nnsum3primesle9 47366 |
Copyright terms: Public domain | W3C validator |