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Mirrors > Home > ILE Home > Th. List > mulid2d | GIF version |
Description: Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
addcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
mulid2d | ⊢ (𝜑 → (1 · 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | mulid2 7547 | . 2 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (1 · 𝐴) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1290 ∈ wcel 1439 (class class class)co 5666 ℂcc 7409 1c1 7412 · cmul 7416 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-resscn 7498 ax-1cn 7499 ax-icn 7501 ax-addcl 7502 ax-mulcl 7504 ax-mulcom 7507 ax-mulass 7509 ax-distr 7510 ax-1rid 7513 ax-cnre 7517 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-iota 4993 df-fv 5036 df-ov 5669 |
This theorem is referenced by: adddirp1d 7575 mulsubfacd 7957 mulcanapd 8191 receuap 8199 divdivdivap 8241 divcanap5 8242 ltrec 8405 recp1lt1 8421 nndivtr 8525 xp1d2m1eqxm1d2 8729 gtndiv 8902 lincmb01cmp 9481 iccf1o 9482 modqfrac 9805 qnegmod 9837 addmodid 9840 m1expcl2 10038 expgt1 10054 ltexp2a 10068 leexp2a 10069 binom3 10132 faclbnd 10210 facavg 10215 ibcval5 10232 cvg1nlemcau 10478 resqrexlemover 10504 resqrexlemcalc2 10509 absimle 10578 maxabslemlub 10701 binom1p 10940 binom1dif 10942 efcllemp 11009 ef01bndlem 11108 efieq1re 11122 eirraplem 11125 iddvds 11148 gcdaddm 11314 rpmulgcd 11354 prmind2 11441 phiprm 11538 hashgcdlem 11542 qdencn 12187 |
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