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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 7918 | . 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 1348 ∈ wcel 2141 (class class class)co 5853 ℂcc 7772 1c1 7775 · cmul 7779 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-resscn 7866 ax-1cn 7867 ax-icn 7869 ax-addcl 7870 ax-mulcl 7872 ax-mulcom 7875 ax-mulass 7877 ax-distr 7878 ax-1rid 7881 ax-cnre 7885 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-iota 5160 df-fv 5206 df-ov 5856 |
This theorem is referenced by: halfpm6th 9098 div4p1lem1div2 9131 3halfnz 9309 sq10 10646 fac2 10665 efival 11695 ef01bndlem 11719 3dvdsdec 11824 3dvds2dec 11825 odd2np1lem 11831 m1expo 11859 m1exp1 11860 nno 11865 sin2pim 13528 cos2pim 13529 sincosq3sgn 13543 sincosq4sgn 13544 cosq23lt0 13548 tangtx 13553 sincosq1eq 13554 sincos4thpi 13555 sincos6thpi 13557 abssinper 13561 cosq34lt1 13565 lgsdir2lem1 13723 lgsdir2lem4 13726 lgsdir2lem5 13727 ex-fl 13760 |
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