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Mirrors > Home > MPE Home > Th. List > itg1sub | Structured version Visualization version GIF version |
Description: The integral of a difference of two simple functions. (Contributed by Mario Carneiro, 6-Aug-2014.) |
Ref | Expression |
---|---|
itg1sub | ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → (∫1‘(𝐹 ∘𝑓 − 𝐺)) = ((∫1‘𝐹) − (∫1‘𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 474 | . . . 4 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → 𝐹 ∈ dom ∫1) | |
2 | simpr 479 | . . . . 5 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → 𝐺 ∈ dom ∫1) | |
3 | neg1rr 11317 | . . . . . 6 ⊢ -1 ∈ ℝ | |
4 | 3 | a1i 11 | . . . . 5 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → -1 ∈ ℝ) |
5 | 2, 4 | i1fmulc 23669 | . . . 4 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → ((ℝ × {-1}) ∘𝑓 · 𝐺) ∈ dom ∫1) |
6 | 1, 5 | itg1add 23667 | . . 3 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → (∫1‘(𝐹 ∘𝑓 + ((ℝ × {-1}) ∘𝑓 · 𝐺))) = ((∫1‘𝐹) + (∫1‘((ℝ × {-1}) ∘𝑓 · 𝐺)))) |
7 | 2, 4 | itg1mulc 23670 | . . . . 5 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → (∫1‘((ℝ × {-1}) ∘𝑓 · 𝐺)) = (-1 · (∫1‘𝐺))) |
8 | itg1cl 23651 | . . . . . . . 8 ⊢ (𝐺 ∈ dom ∫1 → (∫1‘𝐺) ∈ ℝ) | |
9 | 8 | recnd 10260 | . . . . . . 7 ⊢ (𝐺 ∈ dom ∫1 → (∫1‘𝐺) ∈ ℂ) |
10 | 2, 9 | syl 17 | . . . . . 6 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → (∫1‘𝐺) ∈ ℂ) |
11 | 10 | mulm1d 10674 | . . . . 5 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → (-1 · (∫1‘𝐺)) = -(∫1‘𝐺)) |
12 | 7, 11 | eqtrd 2794 | . . . 4 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → (∫1‘((ℝ × {-1}) ∘𝑓 · 𝐺)) = -(∫1‘𝐺)) |
13 | 12 | oveq2d 6829 | . . 3 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → ((∫1‘𝐹) + (∫1‘((ℝ × {-1}) ∘𝑓 · 𝐺))) = ((∫1‘𝐹) + -(∫1‘𝐺))) |
14 | 6, 13 | eqtrd 2794 | . 2 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → (∫1‘(𝐹 ∘𝑓 + ((ℝ × {-1}) ∘𝑓 · 𝐺))) = ((∫1‘𝐹) + -(∫1‘𝐺))) |
15 | i1ff 23642 | . . . . 5 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ) | |
16 | ax-resscn 10185 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
17 | fss 6217 | . . . . 5 ⊢ ((𝐹:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:ℝ⟶ℂ) | |
18 | 15, 16, 17 | sylancl 697 | . . . 4 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℂ) |
19 | i1ff 23642 | . . . . 5 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ) | |
20 | fss 6217 | . . . . 5 ⊢ ((𝐺:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐺:ℝ⟶ℂ) | |
21 | 19, 16, 20 | sylancl 697 | . . . 4 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℂ) |
22 | reex 10219 | . . . . 5 ⊢ ℝ ∈ V | |
23 | ofnegsub 11210 | . . . . 5 ⊢ ((ℝ ∈ V ∧ 𝐹:ℝ⟶ℂ ∧ 𝐺:ℝ⟶ℂ) → (𝐹 ∘𝑓 + ((ℝ × {-1}) ∘𝑓 · 𝐺)) = (𝐹 ∘𝑓 − 𝐺)) | |
24 | 22, 23 | mp3an1 1560 | . . . 4 ⊢ ((𝐹:ℝ⟶ℂ ∧ 𝐺:ℝ⟶ℂ) → (𝐹 ∘𝑓 + ((ℝ × {-1}) ∘𝑓 · 𝐺)) = (𝐹 ∘𝑓 − 𝐺)) |
25 | 18, 21, 24 | syl2an 495 | . . 3 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → (𝐹 ∘𝑓 + ((ℝ × {-1}) ∘𝑓 · 𝐺)) = (𝐹 ∘𝑓 − 𝐺)) |
26 | 25 | fveq2d 6356 | . 2 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → (∫1‘(𝐹 ∘𝑓 + ((ℝ × {-1}) ∘𝑓 · 𝐺))) = (∫1‘(𝐹 ∘𝑓 − 𝐺))) |
27 | itg1cl 23651 | . . . 4 ⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) ∈ ℝ) | |
28 | 27 | recnd 10260 | . . 3 ⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) ∈ ℂ) |
29 | negsub 10521 | . . 3 ⊢ (((∫1‘𝐹) ∈ ℂ ∧ (∫1‘𝐺) ∈ ℂ) → ((∫1‘𝐹) + -(∫1‘𝐺)) = ((∫1‘𝐹) − (∫1‘𝐺))) | |
30 | 28, 9, 29 | syl2an 495 | . 2 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → ((∫1‘𝐹) + -(∫1‘𝐺)) = ((∫1‘𝐹) − (∫1‘𝐺))) |
31 | 14, 26, 30 | 3eqtr3d 2802 | 1 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → (∫1‘(𝐹 ∘𝑓 − 𝐺)) = ((∫1‘𝐹) − (∫1‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ⊆ wss 3715 {csn 4321 × cxp 5264 dom cdm 5266 ⟶wf 6045 ‘cfv 6049 (class class class)co 6813 ∘𝑓 cof 7060 ℂcc 10126 ℝcr 10127 1c1 10129 + caddc 10131 · cmul 10133 − cmin 10458 -cneg 10459 ∫1citg1 23583 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-inf2 8711 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 ax-addf 10207 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-disj 4773 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-2o 7730 df-oadd 7733 df-er 7911 df-map 8025 df-pm 8026 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-sup 8513 df-inf 8514 df-oi 8580 df-card 8955 df-cda 9182 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-n0 11485 df-z 11570 df-uz 11880 df-q 11982 df-rp 12026 df-xadd 12140 df-ioo 12372 df-ico 12374 df-icc 12375 df-fz 12520 df-fzo 12660 df-fl 12787 df-seq 12996 df-exp 13055 df-hash 13312 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-clim 14418 df-sum 14616 df-xmet 19941 df-met 19942 df-ovol 23433 df-vol 23434 df-mbf 23587 df-itg1 23588 |
This theorem is referenced by: itg1lea 23678 itgitg1 23774 itg2addnclem 33774 |
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