Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > itgre | Structured version Visualization version GIF version |
Description: Real part of an integral. (Contributed by Mario Carneiro, 14-Aug-2014.) |
Ref | Expression |
---|---|
itgcnval.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
itgcnval.2 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) |
Ref | Expression |
---|---|
itgre | ⊢ (𝜑 → (ℜ‘∫𝐴𝐵 d𝑥) = ∫𝐴(ℜ‘𝐵) d𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itgcnval.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
2 | itgcnval.2 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) | |
3 | 1, 2 | itgcnval 24710 | . . 3 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥))) |
4 | 3 | fveq2d 6730 | . 2 ⊢ (𝜑 → (ℜ‘∫𝐴𝐵 d𝑥) = (ℜ‘(∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)))) |
5 | iblmbf 24678 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) | |
6 | 2, 5 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) |
7 | 6, 1 | mbfmptcl 24546 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
8 | 7 | recld 14770 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℝ) |
9 | 7 | iblcn 24709 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿1))) |
10 | 2, 9 | mpbid 235 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿1)) |
11 | 10 | simpld 498 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1) |
12 | 8, 11 | itgrecl 24708 | . . 3 ⊢ (𝜑 → ∫𝐴(ℜ‘𝐵) d𝑥 ∈ ℝ) |
13 | 7 | imcld 14771 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℝ) |
14 | 10 | simprd 499 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿1) |
15 | 13, 14 | itgrecl 24708 | . . 3 ⊢ (𝜑 → ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℝ) |
16 | 12, 15 | crred 14807 | . 2 ⊢ (𝜑 → (ℜ‘(∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥))) = ∫𝐴(ℜ‘𝐵) d𝑥) |
17 | 4, 16 | eqtrd 2778 | 1 ⊢ (𝜑 → (ℜ‘∫𝐴𝐵 d𝑥) = ∫𝐴(ℜ‘𝐵) d𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 ↦ cmpt 5144 ‘cfv 6389 (class class class)co 7222 ici 10744 + caddc 10745 · cmul 10747 ℜcre 14673 ℑcim 14674 MblFncmbf 24524 𝐿1cibl 24527 ∫citg 24528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5188 ax-sep 5201 ax-nul 5208 ax-pow 5267 ax-pr 5331 ax-un 7532 ax-inf2 9269 ax-cnex 10798 ax-resscn 10799 ax-1cn 10800 ax-icn 10801 ax-addcl 10802 ax-addrcl 10803 ax-mulcl 10804 ax-mulrcl 10805 ax-mulcom 10806 ax-addass 10807 ax-mulass 10808 ax-distr 10809 ax-i2m1 10810 ax-1ne0 10811 ax-1rid 10812 ax-rnegex 10813 ax-rrecex 10814 ax-cnre 10815 ax-pre-lttri 10816 ax-pre-lttrn 10817 ax-pre-ltadd 10818 ax-pre-mulgt0 10819 ax-pre-sup 10820 ax-addf 10821 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3417 df-sbc 3704 df-csb 3821 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-pss 3894 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-tp 4555 df-op 4557 df-uni 4829 df-int 4869 df-iun 4915 df-disj 5028 df-br 5063 df-opab 5125 df-mpt 5145 df-tr 5171 df-id 5464 df-eprel 5469 df-po 5477 df-so 5478 df-fr 5518 df-se 5519 df-we 5520 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-res 5572 df-ima 5573 df-pred 6169 df-ord 6225 df-on 6226 df-lim 6227 df-suc 6228 df-iota 6347 df-fun 6391 df-fn 6392 df-f 6393 df-f1 6394 df-fo 6395 df-f1o 6396 df-fv 6397 df-isom 6398 df-riota 7179 df-ov 7225 df-oprab 7226 df-mpo 7227 df-of 7478 df-ofr 7479 df-om 7654 df-1st 7770 df-2nd 7771 df-wrecs 8056 df-recs 8117 df-rdg 8155 df-1o 8211 df-2o 8212 df-er 8400 df-map 8519 df-pm 8520 df-en 8636 df-dom 8637 df-sdom 8638 df-fin 8639 df-sup 9071 df-inf 9072 df-oi 9139 df-dju 9530 df-card 9568 df-pnf 10882 df-mnf 10883 df-xr 10884 df-ltxr 10885 df-le 10886 df-sub 11077 df-neg 11078 df-div 11503 df-nn 11844 df-2 11906 df-3 11907 df-4 11908 df-n0 12104 df-z 12190 df-uz 12452 df-q 12558 df-rp 12600 df-xadd 12718 df-ioo 12952 df-ico 12954 df-icc 12955 df-fz 13109 df-fzo 13252 df-fl 13380 df-mod 13456 df-seq 13588 df-exp 13649 df-hash 13910 df-cj 14675 df-re 14676 df-im 14677 df-sqrt 14811 df-abs 14812 df-clim 15062 df-sum 15263 df-xmet 20369 df-met 20370 df-ovol 24374 df-vol 24375 df-mbf 24529 df-itg1 24530 df-itg2 24531 df-ibl 24532 df-itg 24533 df-0p 24580 |
This theorem is referenced by: itgabs 24745 itgabsnc 35596 |
Copyright terms: Public domain | W3C validator |