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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 7876 | . 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 1335 ∈ wcel 2128 (class class class)co 5824 ℂcc 7730 1c1 7733 · cmul 7737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-resscn 7824 ax-1cn 7825 ax-icn 7827 ax-addcl 7828 ax-mulcl 7830 ax-mulcom 7833 ax-mulass 7835 ax-distr 7836 ax-1rid 7839 ax-cnre 7843 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-iota 5135 df-fv 5178 df-ov 5827 |
This theorem is referenced by: adddirp1d 7904 mulsubfacd 8293 mulcanapd 8535 receuap 8543 divdivdivap 8586 divcanap5 8587 subrecap 8711 ltrec 8754 recp1lt1 8770 nndivtr 8875 xp1d2m1eqxm1d2 9085 gtndiv 9259 lincmb01cmp 9907 iccf1o 9908 modqfrac 10236 qnegmod 10268 addmodid 10271 m1expcl2 10441 expgt1 10457 ltexp2a 10471 leexp2a 10472 binom3 10535 faclbnd 10615 facavg 10620 bcval5 10637 cvg1nlemcau 10884 resqrexlemover 10910 resqrexlemcalc2 10915 absimle 10984 maxabslemlub 11107 reccn2ap 11210 binom1p 11382 binom1dif 11384 fprodsplitdc 11493 fprodcl2lem 11502 efcllemp 11555 ef01bndlem 11653 efieq1re 11668 eirraplem 11673 iddvds 11699 gcdaddm 11867 rpmulgcd 11909 prmind2 11996 phiprm 12097 eulerthlemth 12106 fermltl 12108 hashgcdlem 12112 odzdvds 12119 dvexp 13075 dvef 13088 reeff1oleme 13093 sin0pilem1 13102 sinhalfpip 13141 sinhalfpim 13142 coshalfpip 13143 coshalfpim 13144 tangtx 13159 logdivlti 13202 binom4 13296 qdencn 13598 |
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