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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 10627 | . 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) | |
2 | 0cn 10622 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
3 | recn 10616 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
4 | ax-icn 10585 | . . . . . . . 8 ⊢ i ∈ ℂ | |
5 | recn 10616 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
6 | mulcl 10610 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝑦 ∈ ℂ) → (i · 𝑦) ∈ ℂ) | |
7 | 4, 5, 6 | sylancr 590 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (i · 𝑦) ∈ ℂ) |
8 | adddi 10615 | . . . . . . 7 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ) → (0 · (𝑥 + (i · 𝑦))) = ((0 · 𝑥) + (0 · (i · 𝑦)))) | |
9 | 2, 3, 7, 8 | mp3an3an 1464 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 · (𝑥 + (i · 𝑦))) = ((0 · 𝑥) + (0 · (i · 𝑦)))) |
10 | mul02lem2 10806 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → (0 · 𝑥) = 0) | |
11 | mul12 10794 | . . . . . . . . 9 ⊢ ((0 ∈ ℂ ∧ i ∈ ℂ ∧ 𝑦 ∈ ℂ) → (0 · (i · 𝑦)) = (i · (0 · 𝑦))) | |
12 | 2, 4, 5, 11 | mp3an12i 1462 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (0 · (i · 𝑦)) = (i · (0 · 𝑦))) |
13 | mul02lem2 10806 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (0 · 𝑦) = 0) | |
14 | 13 | oveq2d 7151 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (i · (0 · 𝑦)) = (i · 0)) |
15 | 12, 14 | eqtrd 2833 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (0 · (i · 𝑦)) = (i · 0)) |
16 | 10, 15 | oveqan12d 7154 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((0 · 𝑥) + (0 · (i · 𝑦))) = (0 + (i · 0))) |
17 | 9, 16 | eqtrd 2833 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 · (𝑥 + (i · 𝑦))) = (0 + (i · 0))) |
18 | cnre 10627 | . . . . . . . 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 7143 | . . . . . . . . . 10 ⊢ (0 = (𝑥 + (i · 𝑦)) → (0 · 0) = (0 · (𝑥 + (i · 𝑦)))) | |
21 | 20 | eqeq1d 2800 | . . . . . . . . 9 ⊢ (0 = (𝑥 + (i · 𝑦)) → ((0 · 0) = (0 + (i · 0)) ↔ (0 · (𝑥 + (i · 𝑦))) = (0 + (i · 0)))) |
22 | 17, 21 | syl5ibrcom 250 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 = (𝑥 + (i · 𝑦)) → (0 · 0) = (0 + (i · 0)))) |
23 | 22 | rexlimivv 3251 | . . . . . . 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 10632 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
26 | mul02lem2 10806 | . . . . . . 7 ⊢ (0 ∈ ℝ → (0 · 0) = 0) | |
27 | 25, 26 | ax-mp 5 | . . . . . 6 ⊢ (0 · 0) = 0 |
28 | 24, 27 | eqtr3i 2823 | . . . . 5 ⊢ (0 + (i · 0)) = 0 |
29 | 17, 28 | eqtrdi 2849 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 · (𝑥 + (i · 𝑦))) = 0) |
30 | oveq2 7143 | . . . . 5 ⊢ (𝐴 = (𝑥 + (i · 𝑦)) → (0 · 𝐴) = (0 · (𝑥 + (i · 𝑦)))) | |
31 | 30 | eqeq1d 2800 | . . . 4 ⊢ (𝐴 = (𝑥 + (i · 𝑦)) → ((0 · 𝐴) = 0 ↔ (0 · (𝑥 + (i · 𝑦))) = 0)) |
32 | 29, 31 | syl5ibrcom 250 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐴 = (𝑥 + (i · 𝑦)) → (0 · 𝐴) = 0)) |
33 | 32 | rexlimivv 3251 | . 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 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 ici 10528 + caddc 10529 · cmul 10531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 |
This theorem is referenced by: mul01 10808 cnegex2 10811 mul02i 10818 mul02d 10827 bcval5 13674 fsumconst 15137 demoivreALT 15546 nnnn0modprm0 16133 cnfldmulg 20123 itg2mulc 24351 dvcmulf 24548 coe0 24853 plymul0or 24877 sineq0 25116 jensen 25574 musumsum 25777 lgsne0 25919 brbtwn2 26699 ax5seglem4 26726 axeuclidlem 26756 axeuclid 26757 axcontlem2 26759 axcontlem4 26761 eulerpartlemb 31736 expgrowth 41039 dvcosax 42568 |
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