Theorem ovnval2 46731

Description: Value of the Lebesgue outer measure of a subset 𝐴 of the space of multidimensional real numbers. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypotheses

Ref	Expression
ovnval2.1	⊢ (𝜑 → 𝑋 ∈ Fin)
ovnval2.2	⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋))
ovnval2.3	⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}

Assertion

Ref	Expression
ovnval2	⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) = if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )))

Distinct variable groups:   𝐴,𝑖,𝑧   𝑖,𝑋,𝑗,𝑘,𝑧

Allowed substitution hints:   𝜑(𝑧,𝑖,𝑗,𝑘)   𝐴(𝑗,𝑘)   𝑀(𝑧,𝑖,𝑗,𝑘)

Proof of Theorem ovnval2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovnval2.1	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
2	1	ovnval 46727	. 2 ⊢ (𝜑 → (voln*‘𝑋) = (𝑦 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < ))))
3		biidd 262	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑋 = ∅ ↔ 𝑋 = ∅))
4		sseq1 3957	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ↔ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)))
5	4	anbi1d 631	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ((𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) ↔ (𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
6	5	rexbidv 3158	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) ↔ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
7	6	rabbidv 3404	. . . . . 6 ⊢ (𝑦 = 𝐴 → {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))})
8		ovnval2.3	. . . . . 6 ⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}
9	7, 8	eqtr4di 2787	. . . . 5 ⊢ (𝑦 = 𝐴 → {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} = 𝑀)
10	9	infeq1d 9379	. . . 4 ⊢ (𝑦 = 𝐴 → inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < ) = inf(𝑀, ℝ*, < ))
11	3, 10	ifbieq2d 4504	. . 3 ⊢ (𝑦 = 𝐴 → if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < )) = if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )))
12	11	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < )) = if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )))
13		ovnval2.2	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋))
14		ovexd 7391	. . . . 5 ⊢ (𝜑 → (ℝ ↑m 𝑋) ∈ V)
15	14, 13	ssexd 5267	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
16		elpwg 4555	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (ℝ ↑m 𝑋) ↔ 𝐴 ⊆ (ℝ ↑m 𝑋)))
17	15, 16	syl 17	. . 3 ⊢ (𝜑 → (𝐴 ∈ 𝒫 (ℝ ↑m 𝑋) ↔ 𝐴 ⊆ (ℝ ↑m 𝑋)))
18	13, 17	mpbird 257	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
19		c0ex 11124	. . . 4 ⊢ 0 ∈ V
20	19	a1i 11	. . 3 ⊢ (𝜑 → 0 ∈ V)
21	8	infeq1i 9380	. . . 4 ⊢ inf(𝑀, ℝ*, < ) = inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < )
22		xrltso 13053	. . . . . 6 ⊢ < Or ℝ*
23	22	infex 9396	. . . . 5 ⊢ inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < ) ∈ V
24	23	a1i 11	. . . 4 ⊢ (𝜑 → inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < ) ∈ V)
25	21, 24	eqeltrid 2838	. . 3 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) ∈ V)
26	20, 25	ifcld 4524	. 2 ⊢ (𝜑 → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) ∈ V)
27	2, 12, 18, 26	fvmptd 6946	1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) = if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3058  {crab 3397  Vcvv 3438   ⊆ wss 3899  ∅c0 4283  ifcif 4477  𝒫 cpw 4552  ∪ ciun 4944   ↦ cmpt 5177   × cxp 5620   ∘ ccom 5626  ‘cfv 6490  (class class class)co 7356   ↑_m cmap 8761  Xcixp 8833  Fincfn 8881  infcinf 9342  ℝcr 11023  0cc0 11024  ℝ^*cxr 11163   < clt 11164  ℕcn 12143  [,)cico 13261  ∏cprod 15824  volcvol 25418  Σ^{^}csumge0 46548  voln*covoln 46722

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-mulcl 11086  ax-i2m1 11092  ax-pre-lttri 11098  ax-pre-lttrn 11099

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-po 5530  df-so 5531  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-ixp 8834  df-en 8882  df-dom 8883  df-sdom 8884  df-sup 9343  df-inf 9344  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-seq 13923  df-prod 15825  df-ovoln 46723

This theorem is referenced by:  ovn0val  46736  ovnn0val  46737  ovnval2b  46738  ovn0  46752  ovnhoilem1  46787  ovnovollem3  46844