Theorem omsfval 34299

Description: Value of the outer measure evaluated for a given set 𝐴. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)

Assertion

Ref	Expression
omsfval	⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) → ((toOMeas‘𝑅)‘𝐴) = inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < ))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴,𝑦,𝑧   𝑥,𝑄,𝑦,𝑧   𝑥,𝑉,𝑦,𝑧

Proof of Theorem omsfval

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1137	. . . 4 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) → 𝑅:𝑄⟶(0[,]+∞))
2		simp1 1136	. . . 4 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) → 𝑄 ∈ 𝑉)
3	1, 2	fexd 7156	. . 3 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) → 𝑅 ∈ V)
4		omsval 34298	. . 3 ⊢ (𝑅 ∈ V → (toOMeas‘𝑅) = (𝑎 ∈ 𝒫 ∪ dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < )))
5	3, 4	syl 17	. 2 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) → (toOMeas‘𝑅) = (𝑎 ∈ 𝒫 ∪ dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < )))
6		simpr 484	. . . . . . . 8 ⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
7	6	sseq1d 3961	. . . . . . 7 ⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) ∧ 𝑎 = 𝐴) → (𝑎 ⊆ ∪ 𝑧 ↔ 𝐴 ⊆ ∪ 𝑧))
8	7	anbi1d 631	. . . . . 6 ⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) ∧ 𝑎 = 𝐴) → ((𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω) ↔ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)))
9	8	rabbidv 3402	. . . . 5 ⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) ∧ 𝑎 = 𝐴) → {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} = {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)})
10	9	mpteq1d 5176	. . . 4 ⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) ∧ 𝑎 = 𝐴) → (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)) = (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)))
11	10	rneqd 5873	. . 3 ⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) ∧ 𝑎 = 𝐴) → ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)) = ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)))
12	11	infeq1d 9357	. 2 ⊢ (((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) ∧ 𝑎 = 𝐴) → inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < ) = inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < ))
13		simp3 1138	. . . 4 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) → 𝐴 ⊆ ∪ 𝑄)
14		fdm 6655	. . . . . 6 ⊢ (𝑅:𝑄⟶(0[,]+∞) → dom 𝑅 = 𝑄)
15	14	3ad2ant2 1134	. . . . 5 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) → dom 𝑅 = 𝑄)
16	15	unieqd 4867	. . . 4 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) → ∪ dom 𝑅 = ∪ 𝑄)
17	13, 16	sseqtrrd 3967	. . 3 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) → 𝐴 ⊆ ∪ dom 𝑅)
18	2	uniexd 7670	. . . . 5 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) → ∪ 𝑄 ∈ V)
19		ssexg 5256	. . . . 5 ⊢ ((𝐴 ⊆ ∪ 𝑄 ∧ ∪ 𝑄 ∈ V) → 𝐴 ∈ V)
20	13, 18, 19	syl2anc 584	. . . 4 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) → 𝐴 ∈ V)
21		elpwg 4548	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 ∪ dom 𝑅 ↔ 𝐴 ⊆ ∪ dom 𝑅))
22	20, 21	syl 17	. . 3 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) → (𝐴 ∈ 𝒫 ∪ dom 𝑅 ↔ 𝐴 ⊆ ∪ dom 𝑅))
23	17, 22	mpbird 257	. 2 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) → 𝐴 ∈ 𝒫 ∪ dom 𝑅)
24		xrltso 13035	. . . 4 ⊢ < Or ℝ*
25		iccssxr 13325	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ*
26		soss 5539	. . . . 5 ⊢ ((0[,]+∞) ⊆ ℝ* → ( < Or ℝ* → < Or (0[,]+∞)))
27	25, 26	ax-mp 5	. . . 4 ⊢ ( < Or ℝ* → < Or (0[,]+∞))
28	24, 27	mp1i 13	. . 3 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) → < Or (0[,]+∞))
29	28	infexd 9363	. 2 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) → inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < ) ∈ V)
30	5, 12, 23, 29	fvmptd 6931	1 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) → ((toOMeas‘𝑅)‘𝐴) = inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  {crab 3395  Vcvv 3436   ⊆ wss 3897  𝒫 cpw 4545  ∪ cuni 4854   class class class wbr 5086   ↦ cmpt 5167   Or wor 5518  dom cdm 5611  ran crn 5612  ⟶wf 6472  ‘cfv 6476  (class class class)co 7341  ωcom 7791   ≼ cdom 8862  infcinf 9320  0cc0 11001  +∞cpnf 11138  ℝ^*cxr 11140   < clt 11141  [,]cicc 13243  Σ^*cesum 34032  toOMeascoms 34296

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 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-pre-lttri 11075  ax-pre-lttrn 11076

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-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-po 5519  df-so 5520  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-sup 9321  df-inf 9322  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-icc 13247  df-esum 34033  df-oms 34297

This theorem is referenced by:  omsf  34301  oms0  34302  omsmon  34303  omssubaddlem  34304  omssubadd  34305