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Mirrors > Home > MPE Home > Th. List > remul2 | Structured version Visualization version GIF version |
Description: Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.) |
Ref | Expression |
---|---|
remul2 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (𝐴 · (ℜ‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 10619 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | remul 14480 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℜ‘𝐵)) − ((ℑ‘𝐴) · (ℑ‘𝐵)))) | |
3 | 1, 2 | sylan 582 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℜ‘𝐵)) − ((ℑ‘𝐴) · (ℑ‘𝐵)))) |
4 | rere 14473 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (ℜ‘𝐴) = 𝐴) | |
5 | 4 | adantr 483 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘𝐴) = 𝐴) |
6 | 5 | oveq1d 7163 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → ((ℜ‘𝐴) · (ℜ‘𝐵)) = (𝐴 · (ℜ‘𝐵))) |
7 | reim0 14469 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (ℑ‘𝐴) = 0) | |
8 | 7 | oveq1d 7163 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((ℑ‘𝐴) · (ℑ‘𝐵)) = (0 · (ℑ‘𝐵))) |
9 | imcl 14462 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (ℑ‘𝐵) ∈ ℝ) | |
10 | 9 | recnd 10661 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (ℑ‘𝐵) ∈ ℂ) |
11 | 10 | mul02d 10830 | . . . 4 ⊢ (𝐵 ∈ ℂ → (0 · (ℑ‘𝐵)) = 0) |
12 | 8, 11 | sylan9eq 2874 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → ((ℑ‘𝐴) · (ℑ‘𝐵)) = 0) |
13 | 6, 12 | oveq12d 7166 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (((ℜ‘𝐴) · (ℜ‘𝐵)) − ((ℑ‘𝐴) · (ℑ‘𝐵))) = ((𝐴 · (ℜ‘𝐵)) − 0)) |
14 | recl 14461 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (ℜ‘𝐵) ∈ ℝ) | |
15 | 14 | recnd 10661 | . . . 4 ⊢ (𝐵 ∈ ℂ → (ℜ‘𝐵) ∈ ℂ) |
16 | mulcl 10613 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐵) ∈ ℂ) → (𝐴 · (ℜ‘𝐵)) ∈ ℂ) | |
17 | 1, 15, 16 | syl2an 597 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (𝐴 · (ℜ‘𝐵)) ∈ ℂ) |
18 | 17 | subid1d 10978 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → ((𝐴 · (ℜ‘𝐵)) − 0) = (𝐴 · (ℜ‘𝐵))) |
19 | 3, 13, 18 | 3eqtrd 2858 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (𝐴 · (ℜ‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ‘cfv 6348 (class class class)co 7148 ℂcc 10527 ℝcr 10528 0cc0 10529 · cmul 10534 − cmin 10862 ℜcre 14448 ℑcim 14449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-2 11692 df-cj 14450 df-re 14451 df-im 14452 |
This theorem is referenced by: rediv 14482 remul2d 14578 abscxp 25267 asinsin 25462 |
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