ftc2ditg - Metamath Proof Explorer

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Theorem ftc2ditg 26205

Description: Directed integral analogue of ftc2 26203. (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 13451	. . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋[,]𝑌) ⊆ ℝ)
4	1, 2, 3	syl2anc 595	. . 3 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
5		ftc2ditg.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝑋[,]𝑌))
6	4, 5	sseldd 3938	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
7		ftc2ditg.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌))
8	4, 7	sseldd 3938	. 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 26204	. 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
13		fvexd 6896	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
14		cncff 25052	. . . . . . . 8 ⊢ ((ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ) → (ℝ D 𝐹):(𝑋(,)𝑌)⟶ℂ)
15	9, 14	syl 18	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹):(𝑋(,)𝑌)⟶ℂ)
16	15	feqmptd 6949	. . . . . 6 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)))
17	16, 10	eqeltrrd 2864	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿¹)
18	1, 2, 7, 5, 13, 17	ditgswap 26018	. . . 4 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = -⨜[𝐵 → 𝐴]((ℝ D 𝐹)‘𝑡) d𝑡)
19	18	adantr 485	. . 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 26204	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐵 → 𝐴]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐴) − (𝐹‘𝐵)))
21	20	negeqd 11446	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → -⨜[𝐵 → 𝐴]((ℝ D 𝐹)‘𝑡) d𝑡 = -((𝐹‘𝐴) − (𝐹‘𝐵)))
22		cncff 25052	. . . . . . 7 ⊢ (𝐹 ∈ ((𝑋[,]𝑌)–cn→ℂ) → 𝐹:(𝑋[,]𝑌)⟶ℂ)
23	11, 22	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹:(𝑋[,]𝑌)⟶ℂ)
24	23, 5	ffvelcdmd 7080	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℂ)
25	23, 7	ffvelcdmd 7080	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℂ)
26	24, 25	negsubdi2d 11580	. . . 4 ⊢ (𝜑 → -((𝐹‘𝐴) − (𝐹‘𝐵)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
27	26	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → -((𝐹‘𝐴) − (𝐹‘𝐵)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
28	19, 21, 27	3eqtrd 2802	. 2 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
29	6, 8, 12, 28	lecasei 11311	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 Vcvv 3455 ⊆ wss 3905 class class class wbr 5109 ↦ cmpt 5192 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ℂcc 11093 ℝcr 11094 ≤ cle 11239 − cmin 11436 -cneg 11437 (,)cioo 13367 [,]cicc 13370 –cn→ccncf 25035 𝐿¹cibl 25776 ⨜cdit 26005 D cdv 26022

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cc 10414 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-symdif 4206 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-disj 5077 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-ofr 7675 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-oadd 8453 df-omul 8454 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-dju 9883 df-card 9921 df-acn 9924 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13531 df-fzo 13679 df-fl 13821 df-mod 13899 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-clim 15535 df-rlim 15536 df-sum 15734 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-pt 17492 df-prds 17495 df-xrs 17551 df-qtop 17556 df-imas 17557 df-xps 17559 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-mulg 19129 df-cntz 19382 df-cmn 19847 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-fbas 21519 df-fg 21520 df-cnfld 21523 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-cld 23176 df-ntr 23177 df-cls 23178 df-nei 23255 df-lp 23293 df-perf 23294 df-cn 23384 df-cnp 23385 df-haus 23472 df-cmp 23544 df-tx 23719 df-hmeo 23912 df-fil 24003 df-fm 24095 df-flim 24096 df-flf 24097 df-xms 24477 df-ms 24478 df-tms 24479 df-cncf 25037 df-ovol 25623 df-vol 25624 df-mbf 25778 df-itg1 25779 df-itg2 25780 df-ibl 25781 df-itg 25782 df-0p 25829 df-ditg 26006 df-limc 26025 df-dv 26026

This theorem is referenced by: (None)

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