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Theorem ovolshft 25499

Description: The Lebesgue outer measure function is shift-invariant. (Contributed by Mario Carneiro, 22-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.)

**Hypotheses**
Ref	Expression
ovolshft.1	⊢ (𝜑 → 𝐴 ⊆ ℝ)
ovolshft.2	⊢ (𝜑 → 𝐶 ∈ ℝ)
ovolshft.3	⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴})

**Assertion**
Ref	Expression
ovolshft	⊢ (𝜑 → (vol‘𝐴) = (vol‘𝐵))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐶 Allowed substitution hints: 𝜑(𝑥) 𝐵(𝑥)

Proof of Theorem ovolshft Dummy variables 𝑓 𝑔 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovolshft.1	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
2		ovolshft.2	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ)
3		ovolshft.3	. . . . 5 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴})
4		eqid 2741	. . . . 5 ⊢ {𝑧 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))} = {𝑧 ∈ ℝ^ ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^*, < ))}
5	1, 2, 3, 4	ovolshftlem2 25498	. . . 4 ⊢ (𝜑 → {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))} ⊆ {𝑧 ∈ ℝ^ ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^*, < ))})
6		ssrab2 4013	. . . . . 6 ⊢ {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴} ⊆ ℝ
7	3, 6	eqsstrdi 3960	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
8	2	renegcld 11573	. . . . 5 ⊢ (𝜑 → -𝐶 ∈ ℝ)
9	1, 2, 3	shft2rab 25496	. . . . 5 ⊢ (𝜑 → 𝐴 = {𝑤 ∈ ℝ ∣ (𝑤 − -𝐶) ∈ 𝐵})
10		eqid 2741	. . . . 5 ⊢ {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))} = {𝑦 ∈ ℝ^ ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))}
11	7, 8, 9, 10	ovolshftlem2 25498	. . . 4 ⊢ (𝜑 → {𝑧 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))} ⊆ {𝑦 ∈ ℝ^ ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))})
12	5, 11	eqssd 3933	. . 3 ⊢ (𝜑 → {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))} = {𝑧 ∈ ℝ^ ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^*, < ))})
13	12	infeq1d 9385	. 2 ⊢ (𝜑 → inf({𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))}, ℝ^, < ) = inf({𝑧 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))}, ℝ^, < ))
14	10	ovolval 25461	. . 3 ⊢ (𝐴 ⊆ ℝ → (vol‘𝐴) = inf({𝑦 ∈ ℝ^ ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))}, ℝ^, < ))
15	1, 14	syl 17	. 2 ⊢ (𝜑 → (vol‘𝐴) = inf({𝑦 ∈ ℝ^ ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))}, ℝ^, < ))
16	4	ovolval 25461	. . 3 ⊢ (𝐵 ⊆ ℝ → (vol‘𝐵) = inf({𝑧 ∈ ℝ^ ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))}, ℝ^, < ))
17	7, 16	syl 17	. 2 ⊢ (𝜑 → (vol‘𝐵) = inf({𝑧 ∈ ℝ^ ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))}, ℝ^, < ))
18	13, 15, 17	3eqtr4d 2786	1 ⊢ (𝜑 → (vol‘𝐴) = (vol‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ∃wrex 3065 {crab 3393 ∩ cin 3883 ⊆ wss 3884 ∪ cuni 4840 × cxp 5618 ran crn 5621 ∘ ccom 5624 ‘cfv 6488 (class class class)co 7359 ↑_m cmap 8767 supcsup 9347 infcinf 9348 ℝcr 11033 1c1 11035 + caddc 11037 ℝ^*cxr 11174 < clt 11175 ≤ cle 11176 − cmin 11373 -cneg 11374 ℕcn 12169 (,)cioo 13293 seqcseq 13958 abscabs 15191 vol*covol 25450

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7681 ax-cnex 11090 ax-resscn 11091 ax-1cn 11092 ax-icn 11093 ax-addcl 11094 ax-addrcl 11095 ax-mulcl 11096 ax-mulrcl 11097 ax-mulcom 11098 ax-addass 11099 ax-mulass 11100 ax-distr 11101 ax-i2m1 11102 ax-1ne0 11103 ax-1rid 11104 ax-rnegex 11105 ax-rrecex 11106 ax-cnre 11107 ax-pre-lttri 11108 ax-pre-lttrn 11109 ax-pre-ltadd 11110 ax-pre-mulgt0 11111 ax-pre-sup 11112

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3725 df-csb 3833 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-pss 3904 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-riota 7316 df-ov 7362 df-oprab 7363 df-mpo 7364 df-om 7810 df-1st 7933 df-2nd 7934 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8343 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-sup 9349 df-inf 9350 df-pnf 11177 df-mnf 11178 df-xr 11179 df-ltxr 11180 df-le 11181 df-sub 11375 df-neg 11376 df-div 11804 df-nn 12170 df-2 12239 df-3 12240 df-n0 12433 df-z 12520 df-uz 12784 df-rp 12938 df-ioo 13297 df-ico 13299 df-fz 13457 df-seq 13959 df-exp 14019 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-ovol 25452

This theorem is referenced by: shftmbl 25526 vitalilem4 25599

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