ftc2ditg - Metamath Proof Explorer

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Theorem ftc2ditg 24028

Description: Directed integral analogue of ftc2 24026. (Contributed by Mario Carneiro, 3-Sep-2014.)

**Hypotheses**
Ref	Expression
ftc2ditg.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
ftc2ditg.y	⊢ (𝜑 → 𝑌 ∈ ℝ)
ftc2ditg.a	⊢ (𝜑 → 𝐴 ∈ (𝑋[,]𝑌))
ftc2ditg.b	⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌))
ftc2ditg.c	⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ))
ftc2ditg.i	⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿¹)
ftc2ditg.f	⊢ (𝜑 → 𝐹 ∈ ((𝑋[,]𝑌)–cn→ℂ))

**Assertion**
Ref	Expression
ftc2ditg	⊢ (𝜑 → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))

Distinct variable groups: 𝑡,𝐴 𝑡,𝐵 𝑡,𝐹 𝜑,𝑡 𝑡,𝑋 𝑡,𝑌

**Proof of Theorem ftc2ditg**
Step	Hyp	Ref	Expression
1		ftc2ditg.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ)
2		ftc2ditg.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ ℝ)
3		iccssre 12459	. . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋[,]𝑌) ⊆ ℝ)
4	1, 2, 3	syl2anc 573	. . 3 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
5		ftc2ditg.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝑋[,]𝑌))
6	4, 5	sseldd 3753	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
7		ftc2ditg.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌))
8	4, 7	sseldd 3753	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9		ftc2ditg.c	. . 3 ⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ))
10		ftc2ditg.i	. . 3 ⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿¹)
11		ftc2ditg.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝑋[,]𝑌)–cn→ℂ))
12	1, 2, 5, 7, 9, 10, 11	ftc2ditglem 24027	. 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
13		fvexd 6346	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
14		cncff 22915	. . . . . . . 8 ⊢ ((ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ) → (ℝ D 𝐹):(𝑋(,)𝑌)⟶ℂ)
15	9, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹):(𝑋(,)𝑌)⟶ℂ)
16	15	feqmptd 6393	. . . . . 6 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)))
17	16, 10	eqeltrrd 2851	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿¹)
18	1, 2, 7, 5, 13, 17	ditgswap 23842	. . . 4 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = -⨜[𝐵 → 𝐴]((ℝ D 𝐹)‘𝑡) d𝑡)
19	18	adantr 466	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = -⨜[𝐵 → 𝐴]((ℝ D 𝐹)‘𝑡) d𝑡)
20	1, 2, 7, 5, 9, 10, 11	ftc2ditglem 24027	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐵 → 𝐴]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐴) − (𝐹‘𝐵)))
21	20	negeqd 10480	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → -⨜[𝐵 → 𝐴]((ℝ D 𝐹)‘𝑡) d𝑡 = -((𝐹‘𝐴) − (𝐹‘𝐵)))
22		cncff 22915	. . . . . . 7 ⊢ (𝐹 ∈ ((𝑋[,]𝑌)–cn→ℂ) → 𝐹:(𝑋[,]𝑌)⟶ℂ)
23	11, 22	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:(𝑋[,]𝑌)⟶ℂ)
24	23, 5	ffvelrnd 6505	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℂ)
25	23, 7	ffvelrnd 6505	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℂ)
26	24, 25	negsubdi2d 10613	. . . 4 ⊢ (𝜑 → -((𝐹‘𝐴) − (𝐹‘𝐵)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
27	26	adantr 466	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → -((𝐹‘𝐴) − (𝐹‘𝐵)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
28	19, 21, 27	3eqtrd 2809	. 2 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
29	6, 8, 12, 28	lecasei 10348	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 Vcvv 3351 ⊆ wss 3723 class class class wbr 4787 ↦ cmpt 4864 ⟶wf 6026 ‘cfv 6030 (class class class)co 6795 ℂcc 10139 ℝcr 10140 ≤ cle 10280 − cmin 10471 -cneg 10472 (,)cioo 12379 [,]cicc 12382 –cn→ccncf 22898 𝐿¹cibl 23604 ⨜cdit 23829 D cdv 23846

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7099 ax-inf2 8705 ax-cc 9462 ax-cnex 10197 ax-resscn 10198 ax-1cn 10199 ax-icn 10200 ax-addcl 10201 ax-addrcl 10202 ax-mulcl 10203 ax-mulrcl 10204 ax-mulcom 10205 ax-addass 10206 ax-mulass 10207 ax-distr 10208 ax-i2m1 10209 ax-1ne0 10210 ax-1rid 10211 ax-rnegex 10212 ax-rrecex 10213 ax-cnre 10214 ax-pre-lttri 10215 ax-pre-lttrn 10216 ax-pre-ltadd 10217 ax-pre-mulgt0 10218 ax-pre-sup 10219 ax-addf 10220 ax-mulf 10221

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-iin 4658 df-disj 4756 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6756 df-ov 6798 df-oprab 6799 df-mpt2 6800 df-of 7047 df-ofr 7048 df-om 7216 df-1st 7318 df-2nd 7319 df-supp 7450 df-wrecs 7562 df-recs 7624 df-rdg 7662 df-1o 7716 df-2o 7717 df-oadd 7720 df-omul 7721 df-er 7899 df-map 8014 df-pm 8015 df-ixp 8066 df-en 8113 df-dom 8114 df-sdom 8115 df-fin 8116 df-fsupp 8435 df-fi 8476 df-sup 8507 df-inf 8508 df-oi 8574 df-card 8968 df-acn 8971 df-cda 9195 df-pnf 10281 df-mnf 10282 df-xr 10283 df-ltxr 10284 df-le 10285 df-sub 10473 df-neg 10474 df-div 10890 df-nn 11226 df-2 11284 df-3 11285 df-4 11286 df-5 11287 df-6 11288 df-7 11289 df-8 11290 df-9 11291 df-n0 11499 df-z 11584 df-dec 11700 df-uz 11893 df-q 11996 df-rp 12035 df-xneg 12150 df-xadd 12151 df-xmul 12152 df-ioo 12383 df-ioc 12384 df-ico 12385 df-icc 12386 df-fz 12533 df-fzo 12673 df-fl 12800 df-mod 12876 df-seq 13008 df-exp 13067 df-hash 13321 df-cj 14046 df-re 14047 df-im 14048 df-sqrt 14182 df-abs 14183 df-clim 14426 df-rlim 14427 df-sum 14624 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-starv 16163 df-sca 16164 df-vsca 16165 df-ip 16166 df-tset 16167 df-ple 16168 df-ds 16171 df-unif 16172 df-hom 16173 df-cco 16174 df-rest 16290 df-topn 16291 df-0g 16309 df-gsum 16310 df-topgen 16311 df-pt 16312 df-prds 16315 df-xrs 16369 df-qtop 16374 df-imas 16375 df-xps 16377 df-mre 16453 df-mrc 16454 df-acs 16456 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-submnd 17543 df-mulg 17748 df-cntz 17956 df-cmn 18401 df-psmet 19952 df-xmet 19953 df-met 19954 df-bl 19955 df-mopn 19956 df-fbas 19957 df-fg 19958 df-cnfld 19961 df-top 20918 df-topon 20935 df-topsp 20957 df-bases 20970 df-cld 21043 df-ntr 21044 df-cls 21045 df-nei 21122 df-lp 21160 df-perf 21161 df-cn 21251 df-cnp 21252 df-haus 21339 df-cmp 21410 df-tx 21585 df-hmeo 21778 df-fil 21869 df-fm 21961 df-flim 21962 df-flf 21963 df-xms 22344 df-ms 22345 df-tms 22346 df-cncf 22900 df-ovol 23451 df-vol 23452 df-mbf 23606 df-itg1 23607 df-itg2 23608 df-ibl 23609 df-itg 23610 df-0p 23656 df-ditg 23830 df-limc 23849 df-dv 23850

This theorem is referenced by: (None)

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