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| Mirrors > Home > MPE Home > Th. List > mul02 | Structured version Visualization version GIF version | ||
| Description: Multiplication by 0. Theorem I.6 of [Apostol] p. 18. Based on ideas by Eric Schmidt. (Contributed by NM, 10-Aug-1999.) (Revised by Scott Fenton, 3-Jan-2013.) |
| Ref | Expression |
|---|---|
| mul02 | ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cnre 11258 | . 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) | |
| 2 | 0cn 11253 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
| 3 | recn 11245 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
| 4 | ax-icn 11214 | . . . . . . . 8 ⊢ i ∈ ℂ | |
| 5 | recn 11245 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
| 6 | mulcl 11239 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝑦 ∈ ℂ) → (i · 𝑦) ∈ ℂ) | |
| 7 | 4, 5, 6 | sylancr 587 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (i · 𝑦) ∈ ℂ) |
| 8 | adddi 11244 | . . . . . . 7 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ) → (0 · (𝑥 + (i · 𝑦))) = ((0 · 𝑥) + (0 · (i · 𝑦)))) | |
| 9 | 2, 3, 7, 8 | mp3an3an 1469 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 · (𝑥 + (i · 𝑦))) = ((0 · 𝑥) + (0 · (i · 𝑦)))) |
| 10 | mul02lem2 11438 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → (0 · 𝑥) = 0) | |
| 11 | mul12 11426 | . . . . . . . . 9 ⊢ ((0 ∈ ℂ ∧ i ∈ ℂ ∧ 𝑦 ∈ ℂ) → (0 · (i · 𝑦)) = (i · (0 · 𝑦))) | |
| 12 | 2, 4, 5, 11 | mp3an12i 1467 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (0 · (i · 𝑦)) = (i · (0 · 𝑦))) |
| 13 | mul02lem2 11438 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (0 · 𝑦) = 0) | |
| 14 | 13 | oveq2d 7447 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (i · (0 · 𝑦)) = (i · 0)) |
| 15 | 12, 14 | eqtrd 2777 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (0 · (i · 𝑦)) = (i · 0)) |
| 16 | 10, 15 | oveqan12d 7450 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((0 · 𝑥) + (0 · (i · 𝑦))) = (0 + (i · 0))) |
| 17 | 9, 16 | eqtrd 2777 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 · (𝑥 + (i · 𝑦))) = (0 + (i · 0))) |
| 18 | cnre 11258 | . . . . . . . 8 ⊢ (0 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 0 = (𝑥 + (i · 𝑦))) | |
| 19 | 2, 18 | ax-mp 5 | . . . . . . 7 ⊢ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 0 = (𝑥 + (i · 𝑦)) |
| 20 | oveq2 7439 | . . . . . . . . . 10 ⊢ (0 = (𝑥 + (i · 𝑦)) → (0 · 0) = (0 · (𝑥 + (i · 𝑦)))) | |
| 21 | 20 | eqeq1d 2739 | . . . . . . . . 9 ⊢ (0 = (𝑥 + (i · 𝑦)) → ((0 · 0) = (0 + (i · 0)) ↔ (0 · (𝑥 + (i · 𝑦))) = (0 + (i · 0)))) |
| 22 | 17, 21 | syl5ibrcom 247 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 = (𝑥 + (i · 𝑦)) → (0 · 0) = (0 + (i · 0)))) |
| 23 | 22 | rexlimivv 3201 | . . . . . . 7 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 0 = (𝑥 + (i · 𝑦)) → (0 · 0) = (0 + (i · 0))) |
| 24 | 19, 23 | ax-mp 5 | . . . . . 6 ⊢ (0 · 0) = (0 + (i · 0)) |
| 25 | 0re 11263 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
| 26 | mul02lem2 11438 | . . . . . . 7 ⊢ (0 ∈ ℝ → (0 · 0) = 0) | |
| 27 | 25, 26 | ax-mp 5 | . . . . . 6 ⊢ (0 · 0) = 0 |
| 28 | 24, 27 | eqtr3i 2767 | . . . . 5 ⊢ (0 + (i · 0)) = 0 |
| 29 | 17, 28 | eqtrdi 2793 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 · (𝑥 + (i · 𝑦))) = 0) |
| 30 | oveq2 7439 | . . . . 5 ⊢ (𝐴 = (𝑥 + (i · 𝑦)) → (0 · 𝐴) = (0 · (𝑥 + (i · 𝑦)))) | |
| 31 | 30 | eqeq1d 2739 | . . . 4 ⊢ (𝐴 = (𝑥 + (i · 𝑦)) → ((0 · 𝐴) = 0 ↔ (0 · (𝑥 + (i · 𝑦))) = 0)) |
| 32 | 29, 31 | syl5ibrcom 247 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐴 = (𝑥 + (i · 𝑦)) → (0 · 𝐴) = 0)) |
| 33 | 32 | rexlimivv 3201 | . 2 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)) → (0 · 𝐴) = 0) |
| 34 | 1, 33 | syl 17 | 1 ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 ici 11157 + caddc 11158 · 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 |
| This theorem is referenced by: mul01 11440 cnegex2 11443 mul02i 11450 mul02d 11459 bcval5 14357 fsumconst 15826 demoivreALT 16237 nnnn0modprm0 16844 cnfldmulg 21416 itg2mulc 25782 dvcmulf 25982 coe0 26295 plymul0or 26322 sineq0 26566 jensen 27032 musumsum 27235 lgsne0 27379 brbtwn2 28920 ax5seglem4 28947 axeuclidlem 28977 axeuclid 28978 axcontlem2 28980 axcontlem4 28982 eulerpartlemb 34370 expgrowth 44354 dvcosax 45941 |
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