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Theorem ovolshft 24675

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 2738	. . . . 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 24674	. . . 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 3975	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
8	2	renegcld 11402	. . . . 5 ⊢ (𝜑 → -𝐶 ∈ ℝ)
9	1, 2, 3	shft2rab 24672	. . . . 5 ⊢ (𝜑 → 𝐴 = {𝑤 ∈ ℝ ∣ (𝑤 − -𝐶) ∈ 𝐵})
10		eqid 2738	. . . . 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 24674	. . . 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 3938	. . 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 9236	. 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 24637	. . 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 24637	. . 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 2788	1 ⊢ (𝜑 → (vol‘𝐴) = (vol‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 {crab 3068 ∩ cin 3886 ⊆ wss 3887 ∪ cuni 4839 × cxp 5587 ran crn 5590 ∘ ccom 5593 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 supcsup 9199 infcinf 9200 ℝcr 10870 1c1 10872 + caddc 10874 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 − cmin 11205 -cneg 11206 ℕcn 11973 (,)cioo 13079 seqcseq 13721 abscabs 14945 vol*covol 24626

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-ioo 13083 df-ico 13085 df-fz 13240 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-ovol 24628

This theorem is referenced by: shftmbl 24702 vitalilem4 24775

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