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Mirrors > Home > MPE Home > Th. List > resub | Structured version Visualization version GIF version |
Description: Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.) |
Ref | Expression |
---|---|
resub | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 − 𝐵)) = ((ℜ‘𝐴) − (ℜ‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negcl 10875 | . . . 4 ⊢ (𝐵 ∈ ℂ → -𝐵 ∈ ℂ) | |
2 | readd 14475 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ -𝐵 ∈ ℂ) → (ℜ‘(𝐴 + -𝐵)) = ((ℜ‘𝐴) + (ℜ‘-𝐵))) | |
3 | 1, 2 | sylan2 592 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 + -𝐵)) = ((ℜ‘𝐴) + (ℜ‘-𝐵))) |
4 | reneg 14474 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (ℜ‘-𝐵) = -(ℜ‘𝐵)) | |
5 | 4 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘-𝐵) = -(ℜ‘𝐵)) |
6 | 5 | oveq2d 7161 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((ℜ‘𝐴) + (ℜ‘-𝐵)) = ((ℜ‘𝐴) + -(ℜ‘𝐵))) |
7 | 3, 6 | eqtrd 2856 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 + -𝐵)) = ((ℜ‘𝐴) + -(ℜ‘𝐵))) |
8 | negsub 10923 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) | |
9 | 8 | fveq2d 6668 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 + -𝐵)) = (ℜ‘(𝐴 − 𝐵))) |
10 | recl 14459 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | |
11 | 10 | recnd 10658 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
12 | recl 14459 | . . . 4 ⊢ (𝐵 ∈ ℂ → (ℜ‘𝐵) ∈ ℝ) | |
13 | 12 | recnd 10658 | . . 3 ⊢ (𝐵 ∈ ℂ → (ℜ‘𝐵) ∈ ℂ) |
14 | negsub 10923 | . . 3 ⊢ (((ℜ‘𝐴) ∈ ℂ ∧ (ℜ‘𝐵) ∈ ℂ) → ((ℜ‘𝐴) + -(ℜ‘𝐵)) = ((ℜ‘𝐴) − (ℜ‘𝐵))) | |
15 | 11, 13, 14 | syl2an 595 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((ℜ‘𝐴) + -(ℜ‘𝐵)) = ((ℜ‘𝐴) − (ℜ‘𝐵))) |
16 | 7, 9, 15 | 3eqtr3d 2864 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 − 𝐵)) = ((ℜ‘𝐴) − (ℜ‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6349 (class class class)co 7145 ℂcc 10524 + caddc 10529 − cmin 10859 -cneg 10860 ℜcre 14446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 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 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 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-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-2 11689 df-cj 14448 df-re 14449 df-im 14450 |
This theorem is referenced by: resubd 14565 recn2 14947 caucvgr 15022 tanregt0 25050 logcnlem4 25155 isosctrlem1 25323 acoscos 25398 acosbnd 25405 atanlogsublem 25420 isosctrlem1ALT 41148 |
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