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 34952; cf. itgmulc2lem1 24424. (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 12834 | . . . . . . . 8 ⊢ (𝐵 ∈ (0[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) | |
4 | 1, 2, 3 | sylanbrc 585 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) |
5 | 0e0icopnf 12838 | . . . . . . . 8 ⊢ 0 ∈ (0[,)+∞) | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞)) |
7 | 4, 6 | ifclda 4499 | . . . . . 6 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ (0[,)+∞)) |
8 | 7 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ (0[,)+∞)) |
9 | 8 | fmpttd 6872 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)):ℝ⟶(0[,)+∞)) |
10 | itgmulc2nc.3 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) | |
11 | 1, 2 | iblpos 24385 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) ∈ ℝ))) |
12 | 10, 11 | mpbid 234 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) ∈ ℝ)) |
13 | 12 | simprd 498 | . . . 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 12834 | . . . . 5 ⊢ (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) | |
17 | 14, 15, 16 | sylanbrc 585 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) |
18 | 9, 13, 17 | itg2mulc 24340 | . . 3 ⊢ (𝜑 → (∫2‘((ℝ × {𝐶}) ∘f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)))) = (𝐶 · (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))))) |
19 | reex 10620 | . . . . . . 7 ⊢ ℝ ∈ V | |
20 | 19 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ∈ V) |
21 | itgmulc2nc.1 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
22 | 21 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐶 ∈ ℂ) |
23 | fconstmpt 5607 | . . . . . . 7 ⊢ (ℝ × {𝐶}) = (𝑥 ∈ ℝ ↦ 𝐶) | |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (ℝ × {𝐶}) = (𝑥 ∈ ℝ ↦ 𝐶)) |
25 | eqidd 2820 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) | |
26 | 20, 22, 8, 24, 25 | offval2 7418 | . . . . 5 ⊢ (𝜑 → ((ℝ × {𝐶}) ∘f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) = (𝑥 ∈ ℝ ↦ (𝐶 · if(𝑥 ∈ 𝐴, 𝐵, 0)))) |
27 | ovif2 7244 | . . . . . . 7 ⊢ (𝐶 · if(𝑥 ∈ 𝐴, 𝐵, 0)) = if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), (𝐶 · 0)) | |
28 | 21 | mul01d 10831 | . . . . . . . . 9 ⊢ (𝜑 → (𝐶 · 0) = 0) |
29 | 28 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐶 · 0) = 0) |
30 | 29 | ifeq2d 4484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), (𝐶 · 0)) = if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0)) |
31 | 27, 30 | syl5eq 2866 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐶 · if(𝑥 ∈ 𝐴, 𝐵, 0)) = if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0)) |
32 | 31 | mpteq2dva 5152 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ (𝐶 · if(𝑥 ∈ 𝐴, 𝐵, 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0))) |
33 | 26, 32 | eqtrd 2854 | . . . 4 ⊢ (𝜑 → ((ℝ × {𝐶}) ∘f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0))) |
34 | 33 | fveq2d 6667 | . . 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 2856 | . 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 24388 | . . 3 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)))) |
37 | 36 | oveq2d 7164 | . 2 ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = (𝐶 · (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))))) |
38 | 14 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ) |
39 | 38, 1 | remulcld 10663 | . . 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 34949 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ 𝐿1) |
43 | 15 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐶) |
44 | 38, 1, 43, 2 | mulge0d 11209 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (𝐶 · 𝐵)) |
45 | 39, 42, 44 | itgposval 24388 | . 2 ⊢ (𝜑 → ∫𝐴(𝐶 · 𝐵) d𝑥 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0)))) |
46 | 35, 37, 45 | 3eqtr4d 2864 | 1 ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 Vcvv 3493 ifcif 4465 {csn 4559 class class class wbr 5057 ↦ cmpt 5137 × cxp 5546 ‘cfv 6348 (class class class)co 7148 ∘f cof 7399 ℂcc 10527 ℝcr 10528 0cc0 10529 · cmul 10534 +∞cpnf 10664 ≤ cle 10668 [,)cico 12732 MblFncmbf 24207 ∫2citg2 24209 𝐿1cibl 24210 ∫citg 24211 |
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-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-inf2 9096 ax-cnex 10585 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 ax-pre-sup 10607 ax-addf 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 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-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-disj 5023 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 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-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 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-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-ofr 7402 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-pm 8401 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fi 8867 df-sup 8898 df-inf 8899 df-oi 8966 df-dju 9322 df-card 9360 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-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-n0 11890 df-z 11974 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-ioo 12734 df-ico 12736 df-icc 12737 df-fz 12885 df-fzo 13026 df-fl 13154 df-mod 13230 df-seq 13362 df-exp 13422 df-hash 13683 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-rest 16688 df-topgen 16709 df-psmet 20529 df-xmet 20530 df-met 20531 df-bl 20532 df-mopn 20533 df-top 21494 df-topon 21511 df-bases 21546 df-cmp 21987 df-ovol 24057 df-vol 24058 df-mbf 24212 df-itg1 24213 df-itg2 24214 df-ibl 24215 df-itg 24216 df-0p 24263 |
This theorem is referenced by: itgmulc2nclem2 34951 |
Copyright terms: Public domain | W3C validator |