Theorem ismeas 33228

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 3493	. . 3 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀 ∈ V)
2	1	a1i 11	. 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) → 𝑀 ∈ V))
3		simp1 1137	. . 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 7442	. . . 4 ⊢ (0[,]+∞) ∈ V
5		fex2 7924	. . . . . 6 ⊢ ((𝑀:𝑆⟶(0[,]+∞) ∧ 𝑆 ∈ ∪ ran sigAlgebra ∧ (0[,]+∞) ∈ V) → 𝑀 ∈ V)
6	5	3expb 1121	. . . . 5 ⊢ ((𝑀:𝑆⟶(0[,]+∞) ∧ (𝑆 ∈ ∪ ran sigAlgebra ∧ (0[,]+∞) ∈ V)) → 𝑀 ∈ V)
7	6	expcom 415	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ (0[,]+∞) ∈ V) → (𝑀:𝑆⟶(0[,]+∞) → 𝑀 ∈ V))
8	4, 7	mpan2 690	. . 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 33225	. . . 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 3479	. . . . . 6 ⊢ 𝑠 ∈ V
12		mapex 8826	. . . . . 6 ⊢ ((𝑠 ∈ V ∧ (0[,]+∞) ∈ V) → {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)} ∈ V)
13	11, 4, 12	mp2an 691	. . . . 5 ⊢ {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)} ∈ V
14		simp1 1137	. . . . . 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 4064	. . . . 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 5323	. . . 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 486	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
18		simpl 484	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝑠 = 𝑆)
19	17, 18	feq12d 6706	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (𝑚:𝑠⟶(0[,]+∞) ↔ 𝑀:𝑆⟶(0[,]+∞)))
20		fveq1 6891	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑚‘∅) = (𝑀‘∅))
21	20	eqeq1d 2735	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((𝑚‘∅) = 0 ↔ (𝑀‘∅) = 0))
22	21	adantl 483	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → ((𝑚‘∅) = 0 ↔ (𝑀‘∅) = 0))
23	18	pweqd 4620	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝒫 𝑠 = 𝒫 𝑆)
24		fveq1 6891	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (𝑚‘∪ 𝑥) = (𝑀‘∪ 𝑥))
25		fveq1 6891	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (𝑚‘𝑦) = (𝑀‘𝑦))
26	25	esumeq2sdv 33068	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → Σ*𝑦 ∈ 𝑥(𝑚‘𝑦) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))
27	24, 26	eqeq12d 2749	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → ((𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦) ↔ (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)))
28	27	imbi2d 341	. . . . . . 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 483	. . . . . 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 3343	. . . . 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 1437	. . . 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 31911	. . 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 414	. 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 381	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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  ∀wral 3062  Vcvv 3475  ∅c0 4323  𝒫 cpw 4603  ∪ cuni 4909  Disj wdisj 5114   class class class wbr 5149  ran crn 5678  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409  ωcom 7855   ≼ cdom 8937  0cc0 11110  +∞cpnf 11245  [,]cicc 13327  Σ^*cesum 33056  sigAlgebracsiga 33137  measurescmeas 33224

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-esum 33057  df-meas 33225

This theorem is referenced by:  measbasedom  33231  measfrge0  33232  measvnul  33235  measvun  33238  measinb  33250  measres  33251  measdivcst  33253  measdivcstALTV  33254  cntmeas  33255  volmeas  33260  ddemeas  33265  omsmeas  33353  dstrvprob  33501