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Theorem ditgswap 24462

Description: Reverse a directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)

**Hypotheses**
Ref	Expression
ditgcl.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
ditgcl.y	⊢ (𝜑 → 𝑌 ∈ ℝ)
ditgcl.a	⊢ (𝜑 → 𝐴 ∈ (𝑋[,]𝑌))
ditgcl.b	⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌))
ditgcl.c	⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐶 ∈ 𝑉)
ditgcl.i	⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ 𝐿¹)

**Assertion**
Ref	Expression
ditgswap	⊢ (𝜑 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -⨜[𝐴 → 𝐵]𝐶 d𝑥)

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

**Proof of Theorem ditgswap**
Step	Hyp	Ref	Expression
1		ditgcl.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝑋[,]𝑌))
2		ditgcl.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ ℝ)
3		ditgcl.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℝ)
4		elicc2 12790	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐴 ∈ (𝑋[,]𝑌) ↔ (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌)))
5	2, 3, 4	syl2anc 587	. . . 4 ⊢ (𝜑 → (𝐴 ∈ (𝑋[,]𝑌) ↔ (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌)))
6	1, 5	mpbid 235	. . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌))
7	6	simp1d 1139	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
8		ditgcl.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌))
9		elicc2 12790	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐵 ∈ (𝑋[,]𝑌) ↔ (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌)))
10	2, 3, 9	syl2anc 587	. . . 4 ⊢ (𝜑 → (𝐵 ∈ (𝑋[,]𝑌) ↔ (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌)))
11	8, 10	mpbid 235	. . 3 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌))
12	11	simp1d 1139	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ)
13		simpr 488	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
14	7	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ)
15	12	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ)
16	13, 14, 15	ditgneg 24460	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥)
17	13	ditgpos 24459	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥)
18	17	negeqd 10869	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → -⨜[𝐴 → 𝐵]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥)
19	16, 18	eqtr4d 2836	. 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -⨜[𝐴 → 𝐵]𝐶 d𝑥)
20	2	rexrd 10680	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
21	11	simp2d 1140	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≤ 𝐵)
22		iooss1 12761	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑋 ≤ 𝐵) → (𝐵(,)𝐴) ⊆ (𝑋(,)𝐴))
23	20, 21, 22	syl2anc 587	. . . . . . . . 9 ⊢ (𝜑 → (𝐵(,)𝐴) ⊆ (𝑋(,)𝐴))
24	3	rexrd 10680	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ ℝ^*)
25	6	simp3d 1141	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≤ 𝑌)
26		iooss2 12762	. . . . . . . . . 10 ⊢ ((𝑌 ∈ ℝ^* ∧ 𝐴 ≤ 𝑌) → (𝑋(,)𝐴) ⊆ (𝑋(,)𝑌))
27	24, 25, 26	syl2anc 587	. . . . . . . . 9 ⊢ (𝜑 → (𝑋(,)𝐴) ⊆ (𝑋(,)𝑌))
28	23, 27	sstrd 3925	. . . . . . . 8 ⊢ (𝜑 → (𝐵(,)𝐴) ⊆ (𝑋(,)𝑌))
29	28	sselda 3915	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐴)) → 𝑥 ∈ (𝑋(,)𝑌))
30		ditgcl.i	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ 𝐿¹)
31		iblmbf 24371	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ 𝐿¹ → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ MblFn)
32	30, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ MblFn)
33		ditgcl.c	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐶 ∈ 𝑉)
34	32, 33	mbfmptcl 24240	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐶 ∈ ℂ)
35	29, 34	syldan 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐴)) → 𝐶 ∈ ℂ)
36		ioombl 24169	. . . . . . . 8 ⊢ (𝐵(,)𝐴) ∈ dom vol
37	36	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐵(,)𝐴) ∈ dom vol)
38	28, 37, 33, 30	iblss 24408	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐵(,)𝐴) ↦ 𝐶) ∈ 𝐿¹)
39	35, 38	itgcl 24387	. . . . 5 ⊢ (𝜑 → ∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ)
40	39	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ)
41	40	negnegd 10977	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → --∫(𝐵(,)𝐴)𝐶 d𝑥 = ∫(𝐵(,)𝐴)𝐶 d𝑥)
42		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ≤ 𝐴)
43	12	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ∈ ℝ)
44	7	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐴 ∈ ℝ)
45	42, 43, 44	ditgneg 24460	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = -∫(𝐵(,)𝐴)𝐶 d𝑥)
46	45	negeqd 10869	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → -⨜[𝐴 → 𝐵]𝐶 d𝑥 = --∫(𝐵(,)𝐴)𝐶 d𝑥)
47	42	ditgpos 24459	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = ∫(𝐵(,)𝐴)𝐶 d𝑥)
48	41, 46, 47	3eqtr4rd 2844	. 2 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -⨜[𝐴 → 𝐵]𝐶 d𝑥)
49	7, 12, 19, 48	lecasei 10735	1 ⊢ (𝜑 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -⨜[𝐴 → 𝐵]𝐶 d𝑥)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 (class class class)co 7135 ℂcc 10524 ℝcr 10525 ℝ^*cxr 10663 ≤ cle 10665 -cneg 10860 (,)cioo 12726 [,]cicc 12729 volcvol 24067 MblFncmbf 24218 𝐿¹cibl 24221 ∫citg 24222 ⨜cdit 24449

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xadd 12496 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-xmet 20084 df-met 20085 df-ovol 24068 df-vol 24069 df-mbf 24223 df-itg1 24224 df-itg2 24225 df-ibl 24226 df-itg 24227 df-0p 24274 df-ditg 24450

This theorem is referenced by: ditgsplit 24464 ftc2ditg 24649

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