Mathbox for Brendan Leahy |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > itgmulc2nclem1 | Structured version Visualization version GIF version |
Description: Lemma for itgmulc2nc 35500; cf. itgmulc2lem1 24596. (Contributed by Brendan Leahy, 17-Nov-2017.) |
Ref | Expression |
---|---|
itgmulc2nc.1 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
itgmulc2nc.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
itgmulc2nc.3 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) |
itgmulc2nc.m | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn) |
itgmulc2nc.4 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
itgmulc2nc.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
itgmulc2nc.6 | ⊢ (𝜑 → 0 ≤ 𝐶) |
itgmulc2nc.7 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵) |
Ref | Expression |
---|---|
itgmulc2nclem1 | ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itgmulc2nc.5 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
2 | itgmulc2nc.7 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵) | |
3 | elrege0 12940 | . . . . . . . 8 ⊢ (𝐵 ∈ (0[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) | |
4 | 1, 2, 3 | sylanbrc 586 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) |
5 | 0e0icopnf 12944 | . . . . . . . 8 ⊢ 0 ∈ (0[,)+∞) | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞)) |
7 | 4, 6 | ifclda 4459 | . . . . . 6 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ (0[,)+∞)) |
8 | 7 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ (0[,)+∞)) |
9 | 8 | fmpttd 6901 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)):ℝ⟶(0[,)+∞)) |
10 | itgmulc2nc.3 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) | |
11 | 1, 2 | iblpos 24557 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) ∈ ℝ))) |
12 | 10, 11 | mpbid 235 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) ∈ ℝ)) |
13 | 12 | simprd 499 | . . . 4 ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) ∈ ℝ) |
14 | itgmulc2nc.4 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
15 | itgmulc2nc.6 | . . . . 5 ⊢ (𝜑 → 0 ≤ 𝐶) | |
16 | elrege0 12940 | . . . . 5 ⊢ (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) | |
17 | 14, 15, 16 | sylanbrc 586 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) |
18 | 9, 13, 17 | itg2mulc 24512 | . . 3 ⊢ (𝜑 → (∫2‘((ℝ × {𝐶}) ∘f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)))) = (𝐶 · (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))))) |
19 | reex 10718 | . . . . . . 7 ⊢ ℝ ∈ V | |
20 | 19 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ∈ V) |
21 | itgmulc2nc.1 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
22 | 21 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐶 ∈ ℂ) |
23 | fconstmpt 5595 | . . . . . . 7 ⊢ (ℝ × {𝐶}) = (𝑥 ∈ ℝ ↦ 𝐶) | |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (ℝ × {𝐶}) = (𝑥 ∈ ℝ ↦ 𝐶)) |
25 | eqidd 2740 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) | |
26 | 20, 22, 8, 24, 25 | offval2 7456 | . . . . 5 ⊢ (𝜑 → ((ℝ × {𝐶}) ∘f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) = (𝑥 ∈ ℝ ↦ (𝐶 · if(𝑥 ∈ 𝐴, 𝐵, 0)))) |
27 | ovif2 7278 | . . . . . . 7 ⊢ (𝐶 · if(𝑥 ∈ 𝐴, 𝐵, 0)) = if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), (𝐶 · 0)) | |
28 | 21 | mul01d 10929 | . . . . . . . . 9 ⊢ (𝜑 → (𝐶 · 0) = 0) |
29 | 28 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐶 · 0) = 0) |
30 | 29 | ifeq2d 4444 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), (𝐶 · 0)) = if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0)) |
31 | 27, 30 | syl5eq 2786 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐶 · if(𝑥 ∈ 𝐴, 𝐵, 0)) = if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0)) |
32 | 31 | mpteq2dva 5135 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ (𝐶 · if(𝑥 ∈ 𝐴, 𝐵, 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0))) |
33 | 26, 32 | eqtrd 2774 | . . . 4 ⊢ (𝜑 → ((ℝ × {𝐶}) ∘f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0))) |
34 | 33 | fveq2d 6690 | . . 3 ⊢ (𝜑 → (∫2‘((ℝ × {𝐶}) ∘f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)))) = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0)))) |
35 | 18, 34 | eqtr3d 2776 | . 2 ⊢ (𝜑 → (𝐶 · (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)))) = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0)))) |
36 | 1, 10, 2 | itgposval 24560 | . . 3 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)))) |
37 | 36 | oveq2d 7198 | . 2 ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = (𝐶 · (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))))) |
38 | 14 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ) |
39 | 38, 1 | remulcld 10761 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℝ) |
40 | itgmulc2nc.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
41 | itgmulc2nc.m | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn) | |
42 | 21, 40, 10, 41 | iblmulc2nc 35497 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ 𝐿1) |
43 | 15 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐶) |
44 | 38, 1, 43, 2 | mulge0d 11307 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (𝐶 · 𝐵)) |
45 | 39, 42, 44 | itgposval 24560 | . 2 ⊢ (𝜑 → ∫𝐴(𝐶 · 𝐵) d𝑥 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0)))) |
46 | 35, 37, 45 | 3eqtr4d 2784 | 1 ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Vcvv 3400 ifcif 4424 {csn 4526 class class class wbr 5040 ↦ cmpt 5120 × cxp 5533 ‘cfv 6349 (class class class)co 7182 ∘f cof 7435 ℂcc 10625 ℝcr 10626 0cc0 10627 · cmul 10632 +∞cpnf 10762 ≤ cle 10766 [,)cico 12835 MblFncmbf 24378 ∫2citg2 24380 𝐿1cibl 24381 ∫citg 24382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-inf2 9189 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-pre-sup 10705 ax-addf 10706 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-disj 5006 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-of 7437 df-ofr 7438 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-2o 8144 df-er 8332 df-map 8451 df-pm 8452 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-fi 8960 df-sup 8991 df-inf 8992 df-oi 9059 df-dju 9415 df-card 9453 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-n0 11989 df-z 12075 df-uz 12337 df-q 12443 df-rp 12485 df-xneg 12602 df-xadd 12603 df-xmul 12604 df-ioo 12837 df-ico 12839 df-icc 12840 df-fz 12994 df-fzo 13137 df-fl 13265 df-mod 13341 df-seq 13473 df-exp 13534 df-hash 13795 df-cj 14560 df-re 14561 df-im 14562 df-sqrt 14696 df-abs 14697 df-clim 14947 df-sum 15148 df-rest 16811 df-topgen 16832 df-psmet 20221 df-xmet 20222 df-met 20223 df-bl 20224 df-mopn 20225 df-top 21657 df-topon 21674 df-bases 21709 df-cmp 22150 df-ovol 24228 df-vol 24229 df-mbf 24383 df-itg1 24384 df-itg2 24385 df-ibl 24386 df-itg 24387 df-0p 24434 |
This theorem is referenced by: itgmulc2nclem2 35499 |
Copyright terms: Public domain | W3C validator |