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| Mirrors > Home > MPE Home > Th. List > mullid | Structured version Visualization version GIF version | ||
| Description: Identity law for multiplication. See mulrid 11259 for commuted version. (Contributed by NM, 8-Oct-1999.) |
| Ref | Expression |
|---|---|
| mullid | ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-1cn 11213 | . . 3 ⊢ 1 ∈ ℂ | |
| 2 | mulcom 11241 | . . 3 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 · 𝐴) = (𝐴 · 1)) | |
| 3 | 1, 2 | mpan 690 | . 2 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = (𝐴 · 1)) |
| 4 | mulrid 11259 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴) | |
| 5 | 3, 4 | eqtrd 2777 | 1 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 1c1 11156 · cmul 11160 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-mulcom 11219 ax-mulass 11221 ax-distr 11222 ax-1rid 11225 ax-cnre 11228 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: mullidi 11266 mullidd 11279 muladd11 11431 1p1times 11432 mul02lem1 11437 cnegex2 11443 mulm1 11704 div1 11957 subdivcomb2 11963 recdiv 11973 divdiv2 11979 conjmul 11984 ser1const 14099 expp1 14109 recan 15375 arisum 15896 geo2sum 15909 prodrblem 15965 prodmolem2a 15970 risefac1 16069 fallfac1 16070 bpoly3 16094 bpoly4 16095 sinhval 16190 coshval 16191 demoivreALT 16237 gcdadd 16563 gcdid 16564 cncrng 21401 cncrngOLD 21402 cnfld1 21406 cnfld1OLD 21407 blcvx 24819 icccvx 24981 cnlmod 25173 coeidp 26303 dgrid 26304 quartlem1 26900 asinsinlem 26934 asinsin 26935 atantan 26966 musumsum 27235 brbtwn2 28920 axsegconlem1 28932 ax5seglem1 28943 ax5seglem2 28944 ax5seglem4 28947 ax5seglem5 28948 axeuclid 28978 axcontlem2 28980 axcontlem4 28982 cncvcOLD 30602 dvcosax 45941 |
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