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Mirrors > Home > MPE Home > Th. List > mulrid | 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 |
---|---|
mulrid | ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnre 11256 | . 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) | |
2 | recn 11243 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
3 | ax-icn 11212 | . . . . . . 7 ⊢ i ∈ ℂ | |
4 | recn 11243 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
5 | mulcl 11237 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝑦 ∈ ℂ) → (i · 𝑦) ∈ ℂ) | |
6 | 3, 4, 5 | sylancr 587 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → (i · 𝑦) ∈ ℂ) |
7 | ax-1cn 11211 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
8 | adddir 11250 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑥 + (i · 𝑦)) · 1) = ((𝑥 · 1) + ((i · 𝑦) · 1))) | |
9 | 7, 8 | mp3an3 1449 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ) → ((𝑥 + (i · 𝑦)) · 1) = ((𝑥 · 1) + ((i · 𝑦) · 1))) |
10 | 2, 6, 9 | syl2an 596 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 + (i · 𝑦)) · 1) = ((𝑥 · 1) + ((i · 𝑦) · 1))) |
11 | ax-1rid 11223 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → (𝑥 · 1) = 𝑥) | |
12 | mulass 11241 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 1 ∈ ℂ) → ((i · 𝑦) · 1) = (i · (𝑦 · 1))) | |
13 | 3, 7, 12 | mp3an13 1451 | . . . . . . . 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 11223 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (𝑦 · 1) = 𝑦) | |
16 | 15 | oveq2d 7447 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (i · (𝑦 · 1)) = (i · 𝑦)) |
17 | 14, 16 | eqtrd 2775 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ((i · 𝑦) · 1) = (i · 𝑦)) |
18 | 11, 17 | oveqan12d 7450 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 · 1) + ((i · 𝑦) · 1)) = (𝑥 + (i · 𝑦))) |
19 | 10, 18 | eqtrd 2775 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 + (i · 𝑦)) · 1) = (𝑥 + (i · 𝑦))) |
20 | oveq1 7438 | . . . . 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 2751 | . . . 4 ⊢ (𝐴 = (𝑥 + (i · 𝑦)) → ((𝐴 · 1) = 𝐴 ↔ ((𝑥 + (i · 𝑦)) · 1) = (𝑥 + (i · 𝑦)))) |
23 | 19, 22 | syl5ibrcom 247 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐴 = (𝑥 + (i · 𝑦)) → (𝐴 · 1) = 𝐴)) |
24 | 23 | rexlimivv 3199 | . 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 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 (class class class)co 7431 ℂcc 11151 ℝcr 11152 1c1 11154 ici 11155 + caddc 11156 · cmul 11158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-mulcl 11215 ax-mulcom 11217 ax-mulass 11219 ax-distr 11220 ax-1rid 11223 ax-cnre 11226 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 |
This theorem is referenced by: mullid 11258 mulridi 11263 mulridd 11276 muleqadd 11905 divdiv1 11976 conjmul 11982 expmul 14145 binom21 14255 binom2sub1 14257 sq01 14261 bernneq 14265 hashiun 15855 fprodcvg 15963 prodmolem2a 15967 efexp 16134 cncrng 21419 cncrngOLD 21420 cnfld1 21424 cnfld1OLD 21425 0dgr 26299 ecxp 26730 dvcxp1 26797 dvcncxp1 26800 efrlim 27027 efrlimOLD 27028 lgsdilem2 27392 axcontlem7 29000 ipasslem2 30861 addltmulALT 32475 0dp2dp 32876 zrhnm 33930 2even 48083 |
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