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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 10638 | . 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) | |
2 | 0cn 10633 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
3 | recn 10627 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
4 | ax-icn 10596 | . . . . . . . 8 ⊢ i ∈ ℂ | |
5 | recn 10627 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
6 | mulcl 10621 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝑦 ∈ ℂ) → (i · 𝑦) ∈ ℂ) | |
7 | 4, 5, 6 | sylancr 589 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (i · 𝑦) ∈ ℂ) |
8 | adddi 10626 | . . . . . . 7 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ) → (0 · (𝑥 + (i · 𝑦))) = ((0 · 𝑥) + (0 · (i · 𝑦)))) | |
9 | 2, 3, 7, 8 | mp3an3an 1463 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 · (𝑥 + (i · 𝑦))) = ((0 · 𝑥) + (0 · (i · 𝑦)))) |
10 | mul02lem2 10817 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → (0 · 𝑥) = 0) | |
11 | mul12 10805 | . . . . . . . . 9 ⊢ ((0 ∈ ℂ ∧ i ∈ ℂ ∧ 𝑦 ∈ ℂ) → (0 · (i · 𝑦)) = (i · (0 · 𝑦))) | |
12 | 2, 4, 5, 11 | mp3an12i 1461 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (0 · (i · 𝑦)) = (i · (0 · 𝑦))) |
13 | mul02lem2 10817 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (0 · 𝑦) = 0) | |
14 | 13 | oveq2d 7172 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (i · (0 · 𝑦)) = (i · 0)) |
15 | 12, 14 | eqtrd 2856 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (0 · (i · 𝑦)) = (i · 0)) |
16 | 10, 15 | oveqan12d 7175 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((0 · 𝑥) + (0 · (i · 𝑦))) = (0 + (i · 0))) |
17 | 9, 16 | eqtrd 2856 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 · (𝑥 + (i · 𝑦))) = (0 + (i · 0))) |
18 | cnre 10638 | . . . . . . . 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 7164 | . . . . . . . . . 10 ⊢ (0 = (𝑥 + (i · 𝑦)) → (0 · 0) = (0 · (𝑥 + (i · 𝑦)))) | |
21 | 20 | eqeq1d 2823 | . . . . . . . . 9 ⊢ (0 = (𝑥 + (i · 𝑦)) → ((0 · 0) = (0 + (i · 0)) ↔ (0 · (𝑥 + (i · 𝑦))) = (0 + (i · 0)))) |
22 | 17, 21 | syl5ibrcom 249 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 = (𝑥 + (i · 𝑦)) → (0 · 0) = (0 + (i · 0)))) |
23 | 22 | rexlimivv 3292 | . . . . . . 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 10643 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
26 | mul02lem2 10817 | . . . . . . 7 ⊢ (0 ∈ ℝ → (0 · 0) = 0) | |
27 | 25, 26 | ax-mp 5 | . . . . . 6 ⊢ (0 · 0) = 0 |
28 | 24, 27 | eqtr3i 2846 | . . . . 5 ⊢ (0 + (i · 0)) = 0 |
29 | 17, 28 | syl6eq 2872 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 · (𝑥 + (i · 𝑦))) = 0) |
30 | oveq2 7164 | . . . . 5 ⊢ (𝐴 = (𝑥 + (i · 𝑦)) → (0 · 𝐴) = (0 · (𝑥 + (i · 𝑦)))) | |
31 | 30 | eqeq1d 2823 | . . . 4 ⊢ (𝐴 = (𝑥 + (i · 𝑦)) → ((0 · 𝐴) = 0 ↔ (0 · (𝑥 + (i · 𝑦))) = 0)) |
32 | 29, 31 | syl5ibrcom 249 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐴 = (𝑥 + (i · 𝑦)) → (0 · 𝐴) = 0)) |
33 | 32 | rexlimivv 3292 | . 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 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 ici 10539 + caddc 10540 · cmul 10542 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 |
This theorem is referenced by: mul01 10819 cnegex2 10822 mul02i 10829 mul02d 10838 bcval5 13679 fsumconst 15145 demoivreALT 15554 nnnn0modprm0 16143 cnfldmulg 20577 itg2mulc 24348 dvcmulf 24542 coe0 24846 plymul0or 24870 sineq0 25109 jensen 25566 musumsum 25769 lgsne0 25911 brbtwn2 26691 ax5seglem4 26718 axeuclidlem 26748 axeuclid 26749 axcontlem2 26751 axcontlem4 26753 eulerpartlemb 31626 expgrowth 40687 dvcosax 42231 |
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