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Mirrors > Home > MPE Home > Th. List > mulid1 | Structured version Visualization version GIF version |
Description: The number 1 is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) |
Ref | Expression |
---|---|
mulid1 | ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnre 10626 | . 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) | |
2 | recn 10615 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
3 | ax-icn 10584 | . . . . . . 7 ⊢ i ∈ ℂ | |
4 | recn 10615 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
5 | mulcl 10609 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝑦 ∈ ℂ) → (i · 𝑦) ∈ ℂ) | |
6 | 3, 4, 5 | sylancr 587 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → (i · 𝑦) ∈ ℂ) |
7 | ax-1cn 10583 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
8 | adddir 10620 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑥 + (i · 𝑦)) · 1) = ((𝑥 · 1) + ((i · 𝑦) · 1))) | |
9 | 7, 8 | mp3an3 1441 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ) → ((𝑥 + (i · 𝑦)) · 1) = ((𝑥 · 1) + ((i · 𝑦) · 1))) |
10 | 2, 6, 9 | syl2an 595 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 + (i · 𝑦)) · 1) = ((𝑥 · 1) + ((i · 𝑦) · 1))) |
11 | ax-1rid 10595 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → (𝑥 · 1) = 𝑥) | |
12 | mulass 10613 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 1 ∈ ℂ) → ((i · 𝑦) · 1) = (i · (𝑦 · 1))) | |
13 | 3, 7, 12 | mp3an13 1443 | . . . . . . . 8 ⊢ (𝑦 ∈ ℂ → ((i · 𝑦) · 1) = (i · (𝑦 · 1))) |
14 | 4, 13 | syl 17 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → ((i · 𝑦) · 1) = (i · (𝑦 · 1))) |
15 | ax-1rid 10595 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (𝑦 · 1) = 𝑦) | |
16 | 15 | oveq2d 7161 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (i · (𝑦 · 1)) = (i · 𝑦)) |
17 | 14, 16 | eqtrd 2853 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ((i · 𝑦) · 1) = (i · 𝑦)) |
18 | 11, 17 | oveqan12d 7164 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 · 1) + ((i · 𝑦) · 1)) = (𝑥 + (i · 𝑦))) |
19 | 10, 18 | eqtrd 2853 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 + (i · 𝑦)) · 1) = (𝑥 + (i · 𝑦))) |
20 | oveq1 7152 | . . . . 5 ⊢ (𝐴 = (𝑥 + (i · 𝑦)) → (𝐴 · 1) = ((𝑥 + (i · 𝑦)) · 1)) | |
21 | id 22 | . . . . 5 ⊢ (𝐴 = (𝑥 + (i · 𝑦)) → 𝐴 = (𝑥 + (i · 𝑦))) | |
22 | 20, 21 | eqeq12d 2834 | . . . 4 ⊢ (𝐴 = (𝑥 + (i · 𝑦)) → ((𝐴 · 1) = 𝐴 ↔ ((𝑥 + (i · 𝑦)) · 1) = (𝑥 + (i · 𝑦)))) |
23 | 19, 22 | syl5ibrcom 248 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐴 = (𝑥 + (i · 𝑦)) → (𝐴 · 1) = 𝐴)) |
24 | 23 | rexlimivv 3289 | . 2 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)) → (𝐴 · 1) = 𝐴) |
25 | 1, 24 | syl 17 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 (class class class)co 7145 ℂcc 10523 ℝcr 10524 1c1 10526 ici 10527 + caddc 10528 · cmul 10530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-mulcl 10587 ax-mulcom 10589 ax-mulass 10591 ax-distr 10592 ax-1rid 10595 ax-cnre 10598 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 |
This theorem is referenced by: mulid2 10628 mulid1i 10633 mulid1d 10646 muleqadd 11272 divdiv1 11339 conjmul 11345 expmul 13462 binom21 13568 binom2sub1 13570 sq01 13574 bernneq 13578 hashiun 15165 fprodcvg 15272 prodmolem2a 15276 efexp 15442 cncrng 20494 cnfld1 20498 0dgr 24762 ecxp 25183 dvcxp1 25248 dvcncxp1 25251 efrlim 25474 lgsdilem2 25836 axcontlem7 26683 ipasslem2 28536 addltmulALT 30150 0dp2dp 30512 zrhnm 31109 2even 44132 |
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