Theorem ismeas 33686

Description: The property of being a measure. (Contributed by Thierry Arnoux, 10-Sep-2016.) (Revised by Thierry Arnoux, 19-Oct-2016.)

Assertion

Ref	Expression
ismeas	⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)))))

Distinct variable groups:   𝑥,𝑦,𝑀   𝑥,𝑆

Allowed substitution hint:   𝑆(𝑦)

Proof of Theorem ismeas

Dummy variables 𝑚 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3485	. . 3 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀 ∈ V)
2	1	a1i 11	. 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) → 𝑀 ∈ V))
3		simp1 1133	. . 3 ⊢ ((𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))) → 𝑀:𝑆⟶(0[,]+∞))
4		ovex 7434	. . . 4 ⊢ (0[,]+∞) ∈ V
5		fex2 7917	. . . . . 6 ⊢ ((𝑀:𝑆⟶(0[,]+∞) ∧ 𝑆 ∈ ∪ ran sigAlgebra ∧ (0[,]+∞) ∈ V) → 𝑀 ∈ V)
6	5	3expb 1117	. . . . 5 ⊢ ((𝑀:𝑆⟶(0[,]+∞) ∧ (𝑆 ∈ ∪ ran sigAlgebra ∧ (0[,]+∞) ∈ V)) → 𝑀 ∈ V)
7	6	expcom 413	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ (0[,]+∞) ∈ V) → (𝑀:𝑆⟶(0[,]+∞) → 𝑀 ∈ V))
8	4, 7	mpan2 688	. . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑀:𝑆⟶(0[,]+∞) → 𝑀 ∈ V))
9	3, 8	syl5 34	. 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ((𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))) → 𝑀 ∈ V))
10		df-meas 33683	. . . 4 ⊢ measures = (𝑠 ∈ ∪ ran sigAlgebra ↦ {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)))})
11		vex 3470	. . . . . 6 ⊢ 𝑠 ∈ V
12		mapex 8822	. . . . . 6 ⊢ ((𝑠 ∈ V ∧ (0[,]+∞) ∈ V) → {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)} ∈ V)
13	11, 4, 12	mp2an 689	. . . . 5 ⊢ {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)} ∈ V
14		simp1 1133	. . . . . 6 ⊢ ((𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦))) → 𝑚:𝑠⟶(0[,]+∞))
15	14	ss2abi 4055	. . . . 5 ⊢ {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)))} ⊆ {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)}
16	13, 15	ssexi 5312	. . . 4 ⊢ {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)))} ∈ V
17		simpr 484	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
18		simpl 482	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝑠 = 𝑆)
19	17, 18	feq12d 6695	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (𝑚:𝑠⟶(0[,]+∞) ↔ 𝑀:𝑆⟶(0[,]+∞)))
20		fveq1 6880	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑚‘∅) = (𝑀‘∅))
21	20	eqeq1d 2726	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((𝑚‘∅) = 0 ↔ (𝑀‘∅) = 0))
22	21	adantl 481	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → ((𝑚‘∅) = 0 ↔ (𝑀‘∅) = 0))
23	18	pweqd 4611	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝒫 𝑠 = 𝒫 𝑆)
24		fveq1 6880	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (𝑚‘∪ 𝑥) = (𝑀‘∪ 𝑥))
25		fveq1 6880	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (𝑚‘𝑦) = (𝑀‘𝑦))
26	25	esumeq2sdv 33526	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → Σ*𝑦 ∈ 𝑥(𝑚‘𝑦) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))
27	24, 26	eqeq12d 2740	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → ((𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦) ↔ (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)))
28	27	imbi2d 340	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)) ↔ ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))))
29	28	adantl 481	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)) ↔ ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))))
30	23, 29	raleqbidv 3334	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)) ↔ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))))
31	19, 22, 30	3anbi123d 1432	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → ((𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦))) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)))))
32	10, 16, 31	abfmpel 32349	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑀 ∈ V) → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)))))
33	32	ex 412	. 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑀 ∈ V → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))))))
34	2, 9, 33	pm5.21ndd 379	1 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2701  ∀wral 3053  Vcvv 3466  ∅c0 4314  𝒫 cpw 4594  ∪ cuni 4899  Disj wdisj 5103   class class class wbr 5138  ran crn 5667  ⟶wf 6529  ‘cfv 6533  (class class class)co 7401  ωcom 7848   ≼ cdom 8933  0cc0 11106  +∞cpnf 11242  [,]cicc 13324  Σ^*cesum 33514  sigAlgebracsiga 33595  measurescmeas 33682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-esum 33515  df-meas 33683

This theorem is referenced by:  measbasedom  33689  measfrge0  33690  measvnul  33693  measvun  33696  measinb  33708  measres  33709  measdivcst  33711  measdivcstALTV  33712  cntmeas  33713  volmeas  33718  ddemeas  33723  omsmeas  33811  dstrvprob  33959