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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 10640 | . 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) | |
2 | recn 10629 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
3 | ax-icn 10598 | . . . . . . 7 ⊢ i ∈ ℂ | |
4 | recn 10629 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
5 | mulcl 10623 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝑦 ∈ ℂ) → (i · 𝑦) ∈ ℂ) | |
6 | 3, 4, 5 | sylancr 589 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → (i · 𝑦) ∈ ℂ) |
7 | ax-1cn 10597 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
8 | adddir 10634 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑥 + (i · 𝑦)) · 1) = ((𝑥 · 1) + ((i · 𝑦) · 1))) | |
9 | 7, 8 | mp3an3 1446 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ) → ((𝑥 + (i · 𝑦)) · 1) = ((𝑥 · 1) + ((i · 𝑦) · 1))) |
10 | 2, 6, 9 | syl2an 597 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 + (i · 𝑦)) · 1) = ((𝑥 · 1) + ((i · 𝑦) · 1))) |
11 | ax-1rid 10609 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → (𝑥 · 1) = 𝑥) | |
12 | mulass 10627 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 1 ∈ ℂ) → ((i · 𝑦) · 1) = (i · (𝑦 · 1))) | |
13 | 3, 7, 12 | mp3an13 1448 | . . . . . . . 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 10609 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (𝑦 · 1) = 𝑦) | |
16 | 15 | oveq2d 7174 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (i · (𝑦 · 1)) = (i · 𝑦)) |
17 | 14, 16 | eqtrd 2858 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ((i · 𝑦) · 1) = (i · 𝑦)) |
18 | 11, 17 | oveqan12d 7177 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 · 1) + ((i · 𝑦) · 1)) = (𝑥 + (i · 𝑦))) |
19 | 10, 18 | eqtrd 2858 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 + (i · 𝑦)) · 1) = (𝑥 + (i · 𝑦))) |
20 | oveq1 7165 | . . . . 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 2839 | . . . 4 ⊢ (𝐴 = (𝑥 + (i · 𝑦)) → ((𝐴 · 1) = 𝐴 ↔ ((𝑥 + (i · 𝑦)) · 1) = (𝑥 + (i · 𝑦)))) |
23 | 19, 22 | syl5ibrcom 249 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐴 = (𝑥 + (i · 𝑦)) → (𝐴 · 1) = 𝐴)) |
24 | 23 | rexlimivv 3294 | . 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 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 (class class class)co 7158 ℂcc 10537 ℝcr 10538 1c1 10540 ici 10541 + caddc 10542 · cmul 10544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-mulcl 10601 ax-mulcom 10603 ax-mulass 10605 ax-distr 10606 ax-1rid 10609 ax-cnre 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 |
This theorem is referenced by: mulid2 10642 mulid1i 10647 mulid1d 10660 muleqadd 11286 divdiv1 11353 conjmul 11359 expmul 13477 binom21 13583 binom2sub1 13585 sq01 13589 bernneq 13593 hashiun 15179 fprodcvg 15286 prodmolem2a 15290 efexp 15456 cncrng 20568 cnfld1 20572 0dgr 24837 ecxp 25258 dvcxp1 25323 dvcncxp1 25326 efrlim 25549 lgsdilem2 25911 axcontlem7 26758 ipasslem2 28611 addltmulALT 30225 0dp2dp 30587 zrhnm 31212 2even 44211 |
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