ovolval4 - Mathbox for Glauco Siliprandi

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Theorem ovolval4 47225

Description: The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 47221, but here 𝑓 may represent unordered interval bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
ovolval4.a	⊢ (𝜑 → 𝐴 ⊆ ℝ)
ovolval4.m	⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))}

**Assertion**
Ref	Expression
ovolval4	⊢ (𝜑 → (vol‘𝐴) = inf(𝑀, ℝ^, < ))

Distinct variable groups: 𝐴,𝑓,𝑦 𝜑,𝑦 Allowed substitution hints: 𝜑(𝑓) 𝑀(𝑦,𝑓)

Proof of Theorem ovolval4 Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovolval4.a	. 2 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
2		ovolval4.m	. 2 ⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))}
3		2fveq3 6872	. . . 4 ⊢ (𝑘 = 𝑛 → (1^st ‘(𝑓‘𝑘)) = (1^st ‘(𝑓‘𝑛)))
4		2fveq3 6872	. . . . . 6 ⊢ (𝑘 = 𝑛 → (2^nd ‘(𝑓‘𝑘)) = (2^nd ‘(𝑓‘𝑛)))
5	3, 4	breq12d 5113	. . . . 5 ⊢ (𝑘 = 𝑛 → ((1^st ‘(𝑓‘𝑘)) ≤ (2^nd ‘(𝑓‘𝑘)) ↔ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛))))
6	5, 4, 3	ifbieq12d 4509	. . . 4 ⊢ (𝑘 = 𝑛 → if((1^st ‘(𝑓‘𝑘)) ≤ (2^nd ‘(𝑓‘𝑘)), (2^nd ‘(𝑓‘𝑘)), (1^st ‘(𝑓‘𝑘))) = if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛))))
7	3, 6	opeq12d 4839	. . 3 ⊢ (𝑘 = 𝑛 → ⟨(1^st ‘(𝑓‘𝑘)), if((1^st ‘(𝑓‘𝑘)) ≤ (2^nd ‘(𝑓‘𝑘)), (2^nd ‘(𝑓‘𝑘)), (1^st ‘(𝑓‘𝑘)))⟩ = ⟨(1^st ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛)))⟩)
8	7	cbvmptv 5204	. 2 ⊢ (𝑘 ∈ ℕ ↦ ⟨(1^st ‘(𝑓‘𝑘)), if((1^st ‘(𝑓‘𝑘)) ≤ (2^nd ‘(𝑓‘𝑘)), (2^nd ‘(𝑓‘𝑘)), (1^st ‘(𝑓‘𝑘)))⟩) = (𝑛 ∈ ℕ ↦ ⟨(1^st ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛)))⟩)
9	1, 2, 8	ovolval4lem2 47224	1 ⊢ (𝜑 → (vol‘𝐴) = inf(𝑀, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1560 ∃wrex 3086 {crab 3414 ⊆ wss 3904 ifcif 4480 ⟨cop 4588 ∪ cuni 4865 class class class wbr 5100 ↦ cmpt 5181 × cxp 5645 ran crn 5648 ∘ ccom 5651 ‘cfv 6521 (class class class)co 7396 1^st c1st 7968 2^nd c2nd 7969 ↑_m cmap 8808 infcinf 9387 ℝcr 11072 ℝ^*cxr 11215 < clt 11216 ≤ cle 11217 ℕcn 12210 (,)cioo 13349 vol*covol 25524 volcvol 25525 Σ^{^}csumge0 46936

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-inf2 9596 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-pm 8811 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fi 9357 df-sup 9388 df-inf 9389 df-oi 9458 df-dju 9859 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-n0 12482 df-z 12569 df-uz 12840 df-q 12950 df-rp 12994 df-xneg 13114 df-xadd 13115 df-xmul 13116 df-ioo 13353 df-ico 13355 df-icc 13356 df-fz 13513 df-fzo 13660 df-fl 13802 df-seq 14015 df-exp 14075 df-hash 14344 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-clim 15515 df-rlim 15516 df-sum 15714 df-rest 17451 df-topgen 17472 df-psmet 21416 df-xmet 21417 df-met 21418 df-bl 21419 df-mopn 21420 df-top 22954 df-topon 22971 df-bases 23006 df-cmp 23447 df-ovol 25526 df-vol 25527 df-sumge0 46937

This theorem is referenced by: ovolval5 47229

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