ditgswap - Metamath Proof Explorer

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Theorem ditgswap 25793

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 13317	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐴 ∈ (𝑋[,]𝑌) ↔ (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌)))
5	2, 3, 4	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐴 ∈ (𝑋[,]𝑌) ↔ (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌)))
6	1, 5	mpbid 232	. . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌))
7	6	simp1d 1142	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
8		ditgcl.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌))
9		elicc2 13317	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐵 ∈ (𝑋[,]𝑌) ↔ (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌)))
10	2, 3, 9	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐵 ∈ (𝑋[,]𝑌) ↔ (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌)))
11	8, 10	mpbid 232	. . 3 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌))
12	11	simp1d 1142	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ)
13		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
14	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ)
15	12	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ)
16	13, 14, 15	ditgneg 25791	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥)
17	13	ditgpos 25790	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥)
18	17	negeqd 11360	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → -⨜[𝐴 → 𝐵]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥)
19	16, 18	eqtr4d 2769	. 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -⨜[𝐴 → 𝐵]𝐶 d𝑥)
20	2	rexrd 11168	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
21	11	simp2d 1143	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≤ 𝐵)
22		iooss1 13286	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑋 ≤ 𝐵) → (𝐵(,)𝐴) ⊆ (𝑋(,)𝐴))
23	20, 21, 22	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐵(,)𝐴) ⊆ (𝑋(,)𝐴))
24	3	rexrd 11168	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ ℝ^*)
25	6	simp3d 1144	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≤ 𝑌)
26		iooss2 13287	. . . . . . . . . 10 ⊢ ((𝑌 ∈ ℝ^* ∧ 𝐴 ≤ 𝑌) → (𝑋(,)𝐴) ⊆ (𝑋(,)𝑌))
27	24, 25, 26	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑋(,)𝐴) ⊆ (𝑋(,)𝑌))
28	23, 27	sstrd 3940	. . . . . . . 8 ⊢ (𝜑 → (𝐵(,)𝐴) ⊆ (𝑋(,)𝑌))
29	28	sselda 3929	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐴)) → 𝑥 ∈ (𝑋(,)𝑌))
30		ditgcl.i	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ 𝐿¹)
31		iblmbf 25701	. . . . . . . . 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 25570	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐶 ∈ ℂ)
35	29, 34	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐴)) → 𝐶 ∈ ℂ)
36		ioombl 25499	. . . . . . . 8 ⊢ (𝐵(,)𝐴) ∈ dom vol
37	36	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐵(,)𝐴) ∈ dom vol)
38	28, 37, 33, 30	iblss 25739	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐵(,)𝐴) ↦ 𝐶) ∈ 𝐿¹)
39	35, 38	itgcl 25718	. . . . 5 ⊢ (𝜑 → ∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ)
40	39	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ ℂ)
41	40	negnegd 11469	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → --∫(𝐵(,)𝐴)𝐶 d𝑥 = ∫(𝐵(,)𝐴)𝐶 d𝑥)
42		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ≤ 𝐴)
43	12	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐵 ∈ ℝ)
44	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝐴 ∈ ℝ)
45	42, 43, 44	ditgneg 25791	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = -∫(𝐵(,)𝐴)𝐶 d𝑥)
46	45	negeqd 11360	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → -⨜[𝐴 → 𝐵]𝐶 d𝑥 = --∫(𝐵(,)𝐴)𝐶 d𝑥)
47	42	ditgpos 25790	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = ∫(𝐵(,)𝐴)𝐶 d𝑥)
48	41, 46, 47	3eqtr4rd 2777	. 2 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -⨜[𝐴 → 𝐵]𝐶 d𝑥)
49	7, 12, 19, 48	lecasei 11225	1 ⊢ (𝜑 → ⨜[𝐵 → 𝐴]𝐶 d𝑥 = -⨜[𝐴 → 𝐵]𝐶 d𝑥)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ⊆ wss 3897 class class class wbr 5093 ↦ cmpt 5174 dom cdm 5619 (class class class)co 7352 ℂcc 11010 ℝcr 11011 ℝ^*cxr 11151 ≤ cle 11153 -cneg 11351 (,)cioo 13251 [,]cicc 13254 volcvol 25397 MblFncmbf 25548 𝐿¹cibl 25551 ∫citg 25552 ⨜cdit 25780

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-inf2 9537 ax-cnex 11068 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088 ax-pre-mulgt0 11089 ax-pre-sup 11090 ax-addf 11091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-disj 5061 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-ofr 7617 df-om 7803 df-1st 7927 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-pm 8759 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-sup 9332 df-inf 9333 df-oi 9402 df-dju 9800 df-card 9838 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158 df-sub 11352 df-neg 11353 df-div 11781 df-nn 12132 df-2 12194 df-3 12195 df-4 12196 df-n0 12388 df-z 12475 df-uz 12739 df-q 12853 df-rp 12897 df-xadd 13018 df-ioo 13255 df-ico 13257 df-icc 13258 df-fz 13414 df-fzo 13561 df-fl 13702 df-mod 13780 df-seq 13915 df-exp 13975 df-hash 14244 df-cj 15012 df-re 15013 df-im 15014 df-sqrt 15148 df-abs 15149 df-clim 15401 df-rlim 15402 df-sum 15600 df-xmet 21290 df-met 21291 df-ovol 25398 df-vol 25399 df-mbf 25553 df-itg1 25554 df-itg2 25555 df-ibl 25556 df-itg 25557 df-0p 25604 df-ditg 25781

This theorem is referenced by: ditgsplit 25795 ftc2ditg 25986

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