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Mirrors > Home > MPE Home > Th. List > itgmulc2lem1 | Structured version Visualization version GIF version |
Description: Lemma for itgmulc2 23600: positive real case. (Contributed by Mario Carneiro, 25-Aug-2014.) |
Ref | Expression |
---|---|
itgmulc2.1 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
itgmulc2.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
itgmulc2.3 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) |
itgmulc2.4 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
itgmulc2.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
itgmulc2.6 | ⊢ (𝜑 → 0 ≤ 𝐶) |
itgmulc2.7 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵) |
Ref | Expression |
---|---|
itgmulc2lem1 | ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itgmulc2.5 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
2 | itgmulc2.7 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵) | |
3 | elrege0 12278 | . . . . . . . 8 ⊢ (𝐵 ∈ (0[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) | |
4 | 1, 2, 3 | sylanbrc 698 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) |
5 | 0e0icopnf 12282 | . . . . . . . 8 ⊢ 0 ∈ (0[,)+∞) | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞)) |
7 | 4, 6 | ifclda 4120 | . . . . . 6 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ (0[,)+∞)) |
8 | 7 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ (0[,)+∞)) |
9 | eqid 2622 | . . . . 5 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) | |
10 | 8, 9 | fmptd 6385 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)):ℝ⟶(0[,)+∞)) |
11 | itgmulc2.3 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) | |
12 | 1, 2 | iblpos 23559 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) ∈ ℝ))) |
13 | 11, 12 | mpbid 222 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) ∈ ℝ)) |
14 | 13 | simprd 479 | . . . 4 ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) ∈ ℝ) |
15 | itgmulc2.4 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
16 | itgmulc2.6 | . . . . 5 ⊢ (𝜑 → 0 ≤ 𝐶) | |
17 | elrege0 12278 | . . . . 5 ⊢ (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) | |
18 | 15, 16, 17 | sylanbrc 698 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) |
19 | 10, 14, 18 | itg2mulc 23514 | . . 3 ⊢ (𝜑 → (∫2‘((ℝ × {𝐶}) ∘𝑓 · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)))) = (𝐶 · (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))))) |
20 | reex 10027 | . . . . . . 7 ⊢ ℝ ∈ V | |
21 | 20 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ∈ V) |
22 | 15 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐶 ∈ ℝ) |
23 | fconstmpt 5163 | . . . . . . 7 ⊢ (ℝ × {𝐶}) = (𝑥 ∈ ℝ ↦ 𝐶) | |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (ℝ × {𝐶}) = (𝑥 ∈ ℝ ↦ 𝐶)) |
25 | eqidd 2623 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) | |
26 | 21, 22, 8, 24, 25 | offval2 6914 | . . . . 5 ⊢ (𝜑 → ((ℝ × {𝐶}) ∘𝑓 · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) = (𝑥 ∈ ℝ ↦ (𝐶 · if(𝑥 ∈ 𝐴, 𝐵, 0)))) |
27 | ovif2 6738 | . . . . . . 7 ⊢ (𝐶 · if(𝑥 ∈ 𝐴, 𝐵, 0)) = if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), (𝐶 · 0)) | |
28 | itgmulc2.1 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
29 | 28 | mul01d 10235 | . . . . . . . . 9 ⊢ (𝜑 → (𝐶 · 0) = 0) |
30 | 29 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐶 · 0) = 0) |
31 | 30 | ifeq2d 4105 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), (𝐶 · 0)) = if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0)) |
32 | 27, 31 | syl5eq 2668 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐶 · if(𝑥 ∈ 𝐴, 𝐵, 0)) = if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0)) |
33 | 32 | mpteq2dva 4744 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ (𝐶 · if(𝑥 ∈ 𝐴, 𝐵, 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0))) |
34 | 26, 33 | eqtrd 2656 | . . . 4 ⊢ (𝜑 → ((ℝ × {𝐶}) ∘𝑓 · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0))) |
35 | 34 | fveq2d 6195 | . . 3 ⊢ (𝜑 → (∫2‘((ℝ × {𝐶}) ∘𝑓 · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)))) = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0)))) |
36 | 19, 35 | eqtr3d 2658 | . 2 ⊢ (𝜑 → (𝐶 · (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)))) = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0)))) |
37 | 1, 11, 2 | itgposval 23562 | . . 3 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)))) |
38 | 37 | oveq2d 6666 | . 2 ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = (𝐶 · (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))))) |
39 | 15 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ) |
40 | 39, 1 | remulcld 10070 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℝ) |
41 | itgmulc2.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
42 | 28, 41, 11 | iblmulc2 23597 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ 𝐿1) |
43 | 16 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐶) |
44 | 39, 1, 43, 2 | mulge0d 10604 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (𝐶 · 𝐵)) |
45 | 40, 42, 44 | itgposval 23562 | . 2 ⊢ (𝜑 → ∫𝐴(𝐶 · 𝐵) d𝑥 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0)))) |
46 | 36, 38, 45 | 3eqtr4d 2666 | 1 ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ifcif 4086 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 ℂcc 9934 ℝcr 9935 0cc0 9936 · cmul 9941 +∞cpnf 10071 ≤ cle 10075 [,)cico 12177 MblFncmbf 23383 ∫2citg2 23385 𝐿1cibl 23386 ∫citg 23387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cn 21031 df-cnp 21032 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-ovol 23233 df-vol 23234 df-mbf 23388 df-itg1 23389 df-itg2 23390 df-ibl 23391 df-itg 23392 df-0p 23437 |
This theorem is referenced by: itgmulc2lem2 23599 |
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