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Mirrors > Home > ILE Home > Th. List > mulid2i | GIF version |
Description: Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) |
Ref | Expression |
---|---|
axi.1 | ⊢ 𝐴 ∈ ℂ |
Ref | Expression |
---|---|
mulid2i | ⊢ (1 · 𝐴) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axi.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | mulid2 7876 | . 2 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (1 · 𝐴) = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = 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: halfpm6th 9053 div4p1lem1div2 9086 3halfnz 9261 sq10 10586 fac2 10605 efival 11629 ef01bndlem 11653 3dvdsdec 11756 3dvds2dec 11757 odd2np1lem 11763 m1expo 11791 m1exp1 11792 nno 11797 sin2pim 13145 cos2pim 13146 sincosq3sgn 13160 sincosq4sgn 13161 cosq23lt0 13165 tangtx 13170 sincosq1eq 13171 sincos4thpi 13172 sincos6thpi 13174 abssinper 13178 cosq34lt1 13182 ex-fl 13312 |
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