ftc2ditg - Metamath Proof Explorer

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Theorem ftc2ditg 26359

Description: Directed integral analogue of ftc2 26357. (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 13553	. . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋[,]𝑌) ⊆ ℝ)
4	1, 2, 3	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
5		ftc2ditg.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝑋[,]𝑌))
6	4, 5	sseldd 3932	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
7		ftc2ditg.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌))
8	4, 7	sseldd 3932	. 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 26358	. 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
13		fvexd 6898	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
14		cncff 25207	. . . . . . . 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 6951	. . . . . 6 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)))
17	16, 10	eqeltrrd 2862	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿¹)
18	1, 2, 7, 5, 13, 17	ditgswap 26172	. . . 4 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = -⨜[𝐵 → 𝐴]((ℝ D 𝐹)‘𝑡) d𝑡)
19	18	adantr 486	. . 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 26358	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐵 → 𝐴]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐴) − (𝐹‘𝐵)))
21	20	negeqd 11544	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → -⨜[𝐵 → 𝐴]((ℝ D 𝐹)‘𝑡) d𝑡 = -((𝐹‘𝐴) − (𝐹‘𝐵)))
22		cncff 25207	. . . . . . 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 7083	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℂ)
25	23, 7	ffvelcdmd 7083	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℂ)
26	24, 25	negsubdi2d 11678	. . . 4 ⊢ (𝜑 → -((𝐹‘𝐴) − (𝐹‘𝐵)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
27	26	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → -((𝐹‘𝐴) − (𝐹‘𝐵)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
28	19, 21, 27	3eqtrd 2800	. 2 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
29	6, 8, 12, 28	lecasei 11409	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 Vcvv 3451 ⊆ wss 3899 class class class wbr 5103 ↦ cmpt 5186 ⟶wf 6533 ‘cfv 6537 (class class class)co 7418 ℂcc 11191 ℝcr 11192 ≤ cle 11337 − cmin 11534 -cneg 11535 (,)cioo 13469 [,]cicc 13472 –cn→ccncf 25190 𝐿¹cibl 25931 ⨜cdit 26159 D cdv 26176

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-inf2 9635 ax-cc 10506 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271 ax-addf 11272

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-symdif 4199 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7691 df-ofr 7692 df-om 7876 df-1st 7999 df-2nd 8000 df-supp 8171 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-oadd 8473 df-omul 8474 df-er 8710 df-map 8842 df-pm 8843 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9347 df-fi 9396 df-sup 9427 df-inf 9428 df-oi 9497 df-dju 9975 df-card 10013 df-acn 10016 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-z 12687 df-dec 12808 df-uz 12959 df-q 13069 df-rp 13114 df-xneg 13234 df-xadd 13235 df-xmul 13236 df-ioo 13473 df-ioc 13474 df-ico 13475 df-icc 13476 df-fz 13633 df-fzo 13782 df-fl 13925 df-mod 14003 df-seq 14138 df-exp 14198 df-hash 14468 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-clim 15648 df-rlim 15649 df-sum 15847 df-struct 17318 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-ress 17402 df-plusg 17434 df-mulr 17435 df-starv 17436 df-sca 17437 df-vsca 17438 df-ip 17439 df-tset 17440 df-ple 17441 df-ds 17443 df-unif 17444 df-hom 17445 df-cco 17446 df-rest 17586 df-topn 17587 df-0g 17605 df-gsum 17606 df-topgen 17607 df-pt 17608 df-prds 17611 df-xrs 17667 df-qtop 17672 df-imas 17673 df-xps 17675 df-mre 17749 df-mrc 17750 df-acs 17752 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-submnd 18972 df-mulg 19271 df-cntz 19524 df-cmn 19989 df-psmet 21663 df-xmet 21664 df-met 21665 df-bl 21666 df-mopn 21667 df-fbas 21668 df-fg 21669 df-cnfld 21672 df-top 23205 df-topon 23222 df-topsp 23244 df-bases 23257 df-cld 23330 df-ntr 23331 df-cls 23332 df-nei 23409 df-lp 23447 df-perf 23448 df-cn 23538 df-cnp 23539 df-haus 23626 df-cmp 23698 df-tx 23874 df-hmeo 24067 df-fil 24158 df-fm 24250 df-flim 24251 df-flf 24252 df-xms 24632 df-ms 24633 df-tms 24634 df-cncf 25192 df-ovol 25778 df-vol 25779 df-mbf 25933 df-itg1 25934 df-itg2 25935 df-ibl 25936 df-itg 25937 df-0p 25984 df-ditg 26160 df-limc 26179 df-dv 26180

This theorem is used by: (None)

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