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Theorem itgmulc2lem1 25809

Description: Lemma for itgmulc2 25811: positive real case. (Contributed by Mario Carneiro, 25-Aug-2014.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
itgmulc2lem1	⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐶 𝜑,𝑥 𝑥,𝑉 Allowed substitution hint: 𝐵(𝑥)

**Proof of Theorem itgmulc2lem1**
Step	Hyp	Ref	Expression
1		itgmulc2.5	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
2		itgmulc2.7	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵)
3		elrege0 13398	. . . . . . . 8 ⊢ (𝐵 ∈ (0[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))
4	1, 2, 3	sylanbrc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))
5		0e0icopnf 13402	. . . . . . . 8 ⊢ 0 ∈ (0[,)+∞)
6	5	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞))
7	4, 6	ifclda 4503	. . . . . 6 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ (0[,)+∞))
8	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, 𝐵, 0) ∈ (0[,)+∞))
9	8	fmpttd 7061	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)):ℝ⟶(0[,)+∞))
10		itgmulc2.3	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)
11	1, 2	iblpos 25770	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) ∈ ℝ)))
12	10, 11	mpbid 232	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) ∈ ℝ))
13	12	simprd 495	. . . 4 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) ∈ ℝ)
14		itgmulc2.4	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ)
15		itgmulc2.6	. . . . 5 ⊢ (𝜑 → 0 ≤ 𝐶)
16		elrege0 13398	. . . . 5 ⊢ (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
17	14, 15, 16	sylanbrc 584	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
18	9, 13, 17	itg2mulc 25724	. . 3 ⊢ (𝜑 → (∫₂‘((ℝ × {𝐶}) ∘_f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)))) = (𝐶 · (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)))))
19		reex 11120	. . . . . . 7 ⊢ ℝ ∈ V
20	19	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ V)
21	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐶 ∈ ℝ)
22		fconstmpt 5686	. . . . . . 7 ⊢ (ℝ × {𝐶}) = (𝑥 ∈ ℝ ↦ 𝐶)
23	22	a1i 11	. . . . . 6 ⊢ (𝜑 → (ℝ × {𝐶}) = (𝑥 ∈ ℝ ↦ 𝐶))
24		eqidd 2738	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)))
25	20, 21, 8, 23, 24	offval2 7644	. . . . 5 ⊢ (𝜑 → ((ℝ × {𝐶}) ∘_f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) = (𝑥 ∈ ℝ ↦ (𝐶 · if(𝑥 ∈ 𝐴, 𝐵, 0))))
26		ovif2 7459	. . . . . . 7 ⊢ (𝐶 · if(𝑥 ∈ 𝐴, 𝐵, 0)) = if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), (𝐶 · 0))
27		itgmulc2.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℂ)
28	27	mul01d 11336	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 · 0) = 0)
29	28	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐶 · 0) = 0)
30	29	ifeq2d 4488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), (𝐶 · 0)) = if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0))
31	26, 30	eqtrid 2784	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐶 · if(𝑥 ∈ 𝐴, 𝐵, 0)) = if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0))
32	31	mpteq2dva 5179	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ (𝐶 · if(𝑥 ∈ 𝐴, 𝐵, 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0)))
33	25, 32	eqtrd 2772	. . . 4 ⊢ (𝜑 → ((ℝ × {𝐶}) ∘_f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0)))
34	33	fveq2d 6838	. . 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 2774	. 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 25773	. . 3 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))))
37	36	oveq2d 7376	. 2 ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = (𝐶 · (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)))))
38	14	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)
39	38, 1	remulcld 11166	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℝ)
40		itgmulc2.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
41	27, 40, 10	iblmulc2 25808	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ 𝐿¹)
42	15	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐶)
43	38, 1, 42, 2	mulge0d 11718	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (𝐶 · 𝐵))
44	39, 41, 43	itgposval 25773	. 2 ⊢ (𝜑 → ∫𝐴(𝐶 · 𝐵) d𝑥 = (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (𝐶 · 𝐵), 0))))
45	35, 37, 44	3eqtr4d 2782	1 ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3430 ifcif 4467 {csn 4568 class class class wbr 5086 ↦ cmpt 5167 × cxp 5622 ‘cfv 6492 (class class class)co 7360 ∘_f cof 7622 ℂcc 11027 ℝcr 11028 0cc0 11029 · cmul 11034 +∞cpnf 11167 ≤ cle 11171 [,)cico 13291 MblFncmbf 25591 ∫₂citg2 25593 𝐿¹cibl 25594 ∫citg 25595

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-inf2 9553 ax-cc 10348 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-ofr 7625 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-oadd 8402 df-omul 8403 df-er 8636 df-map 8768 df-pm 8769 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-fi 9317 df-sup 9348 df-inf 9349 df-oi 9418 df-dju 9816 df-card 9854 df-acn 9857 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-rlim 15442 df-sum 15640 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-cn 23202 df-cnp 23203 df-cmp 23362 df-tx 23537 df-hmeo 23730 df-xms 24295 df-ms 24296 df-tms 24297 df-cncf 24855 df-ovol 25441 df-vol 25442 df-mbf 25596 df-itg1 25597 df-itg2 25598 df-ibl 25599 df-itg 25600 df-0p 25647

This theorem is referenced by: itgmulc2lem2 25810

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