ditgswap - Metamath Proof Explorer

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Theorem ditgswap 26180

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 13543	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐴 ∈ (𝑋[,]𝑌) ↔ (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌)))
5	2, 3, 4	syl2anc 596	. . . 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 1160	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
8		ditgcl.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌))
9		elicc2 13543	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐵 ∈ (𝑋[,]𝑌) ↔ (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌)))
10	2, 3, 9	syl2anc 596	. . . 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 1160	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ)
13		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
14	7	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ)
15	12	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ)
16	13, 14, 15	ditgneg 26178	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥)
17	13	ditgpos 26177	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥)
18	17	negeqd 11552	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → -⨜[𝐴 → 𝐵]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥)
19	16, 18	eqtr4d 2798	. 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -⨜[𝐴 → 𝐵]𝐶 d𝑥)
20	2	rexrd 11360	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
21	11	simp2d 1161	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≤ 𝐵)
22		iooss1 13512	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑋 ≤ 𝐵) → (𝐵(,)𝐴) ⊆ (𝑋(,)𝐴))
23	20, 21, 22	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝐵(,)𝐴) ⊆ (𝑋(,)𝐴))
24	3	rexrd 11360	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ ℝ^*)
25	6	simp3d 1162	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≤ 𝑌)
26		iooss2 13513	. . . . . . . . . 10 ⊢ ((𝑌 ∈ ℝ^* ∧ 𝐴 ≤ 𝑌) → (𝑋(,)𝐴) ⊆ (𝑋(,)𝑌))
27	24, 25, 26	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝑋(,)𝐴) ⊆ (𝑋(,)𝑌))
28	23, 27	sstrd 3918	. . . . . . . 8 ⊢ (𝜑 → (𝐵(,)𝐴) ⊆ (𝑋(,)𝑌))
29	28	sselda 3908	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐴)) → 𝑥 ∈ (𝑋(,)𝑌))
30		ditgcl.i	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ 𝐿¹)
31		iblmbf 26089	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ 𝐿¹ → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ MblFn)
32	30, 31	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ MblFn)
33		ditgcl.c	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐶 ∈ 𝑉)
34	32, 33	mbfmptcl 25958	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐶 ∈ ℂ)
35	29, 34	syldan 603	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐴)) → 𝐶 ∈ ℂ)
36		ioombl 25887	. . . . . . . 8 ⊢ (𝐵(,)𝐴) ∈ dom vol
37	36	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐵(,)𝐴) ∈ dom vol)
38	28, 37, 33, 30	iblss 26126	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐵(,)𝐴) ↦ 𝐶) ∈ 𝐿¹)
39	35, 38	itgcl 26105	. . . . 5 ⊢ (𝜑 → ∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ)
40	39	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ)
41	40	negnegd 11661	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → - -∫(𝐵(,)𝐴)𝐶 d𝑥 = ∫(𝐵(,)𝐴)𝐶 d𝑥)
42		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ≤ 𝐴)
43	12	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ∈ ℝ)
44	7	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐴 ∈ ℝ)
45	42, 43, 44	ditgneg 26178	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = -∫(𝐵(,)𝐴)𝐶 d𝑥)
46	45	negeqd 11552	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → -⨜[𝐴 → 𝐵]𝐶 d𝑥 = - -∫(𝐵(,)𝐴)𝐶 d𝑥)
47	42	ditgpos 26177	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = ∫(𝐵(,)𝐴)𝐶 d𝑥)
48	41, 46, 47	3eqtr4rd 2806	. 2 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -⨜[𝐴 → 𝐵]𝐶 d𝑥)
49	7, 12, 19, 48	lecasei 11417	1 ⊢ (𝜑 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -⨜[𝐴 → 𝐵]𝐶 d𝑥)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ⊆ wss 3895 class class class wbr 5102 ↦ cmpt 5185 dom cdm 5650 (class class class)co 7420 ℂcc 11199 ℝcr 11200 ℝ^*cxr 11343 ≤ cle 11345 -cneg 11543 (,)cioo 13477 [,]cicc 13480 volcvol 25785 MblFncmbf 25936 𝐿¹cibl 25939 ∫citg 25940 ⨜cdit 26167

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9643 ax-cnex 11257 ax-resscn 11258 ax-1cn 11259 ax-icn 11260 ax-addcl 11261 ax-addrcl 11262 ax-mulcl 11263 ax-mulrcl 11264 ax-mulcom 11265 ax-addass 11266 ax-mulass 11267 ax-distr 11268 ax-i2m1 11269 ax-1ne0 11270 ax-1rid 11271 ax-rnegex 11272 ax-rrecex 11273 ax-cnre 11274 ax-pre-lttri 11275 ax-pre-lttrn 11276 ax-pre-ltadd 11277 ax-pre-mulgt0 11278 ax-pre-sup 11279 ax-addf 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-disj 5070 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-ofr 7694 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8300 df-wrecs 8331 df-recs 8380 df-rdg 8419 df-1o 8477 df-2o 8478 df-er 8718 df-map 8850 df-pm 8851 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9435 df-inf 9436 df-oi 9505 df-dju 9983 df-card 10021 df-pnf 11346 df-mnf 11347 df-xr 11348 df-ltxr 11349 df-le 11350 df-sub 11544 df-neg 11545 df-div 11975 df-nn 12337 df-2 12406 df-3 12407 df-4 12408 df-n0 12608 df-z 12695 df-uz 12967 df-q 13077 df-rp 13122 df-xadd 13243 df-ioo 13481 df-ico 13483 df-icc 13484 df-fz 13641 df-fzo 13790 df-fl 13933 df-mod 14011 df-seq 14146 df-exp 14206 df-hash 14476 df-cj 15267 df-re 15268 df-im 15269 df-sqrt 15403 df-abs 15404 df-clim 15656 df-rlim 15657 df-sum 15855 df-xmet 21672 df-met 21673 df-ovol 25786 df-vol 25787 df-mbf 25941 df-itg1 25942 df-itg2 25943 df-ibl 25944 df-itg 25945 df-0p 25992 df-ditg 26168

This theorem is used by: ditgsplit 26182 ftc2ditg 26367

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