Theorem ismeas 32887

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 3464	. . 3 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀 ∈ V)
2	1	a1i 11	. 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) → 𝑀 ∈ V))
3		simp1 1136	. . 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 7395	. . . 4 ⊢ (0[,]+∞) ∈ V
5		fex2 7875	. . . . . 6 ⊢ ((𝑀:𝑆⟶(0[,]+∞) ∧ 𝑆 ∈ ∪ ran sigAlgebra ∧ (0[,]+∞) ∈ V) → 𝑀 ∈ V)
6	5	3expb 1120	. . . . 5 ⊢ ((𝑀:𝑆⟶(0[,]+∞) ∧ (𝑆 ∈ ∪ ran sigAlgebra ∧ (0[,]+∞) ∈ V)) → 𝑀 ∈ V)
7	6	expcom 414	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ (0[,]+∞) ∈ V) → (𝑀:𝑆⟶(0[,]+∞) → 𝑀 ∈ V))
8	4, 7	mpan2 689	. . 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 32884	. . . 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 3450	. . . . . 6 ⊢ 𝑠 ∈ V
12		mapex 8778	. . . . . 6 ⊢ ((𝑠 ∈ V ∧ (0[,]+∞) ∈ V) → {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)} ∈ V)
13	11, 4, 12	mp2an 690	. . . . 5 ⊢ {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)} ∈ V
14		simp1 1136	. . . . . 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 4028	. . . . 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 5284	. . . 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 485	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
18		simpl 483	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝑠 = 𝑆)
19	17, 18	feq12d 6661	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (𝑚:𝑠⟶(0[,]+∞) ↔ 𝑀:𝑆⟶(0[,]+∞)))
20		fveq1 6846	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑚‘∅) = (𝑀‘∅))
21	20	eqeq1d 2733	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((𝑚‘∅) = 0 ↔ (𝑀‘∅) = 0))
22	21	adantl 482	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → ((𝑚‘∅) = 0 ↔ (𝑀‘∅) = 0))
23	18	pweqd 4582	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝒫 𝑠 = 𝒫 𝑆)
24		fveq1 6846	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (𝑚‘∪ 𝑥) = (𝑀‘∪ 𝑥))
25		fveq1 6846	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (𝑚‘𝑦) = (𝑀‘𝑦))
26	25	esumeq2sdv 32727	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → Σ*𝑦 ∈ 𝑥(𝑚‘𝑦) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))
27	24, 26	eqeq12d 2747	. . . . . . . 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 482	. . . . . 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 3317	. . . . 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 1436	. . . 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 31638	. . 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 413	. 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 380	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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2708  ∀wral 3060  Vcvv 3446  ∅c0 4287  𝒫 cpw 4565  ∪ cuni 4870  Disj wdisj 5075   class class class wbr 5110  ran crn 5639  ⟶wf 6497  ‘cfv 6501  (class class class)co 7362  ωcom 7807   ≼ cdom 8888  0cc0 11060  +∞cpnf 11195  [,]cicc 13277  Σ^*cesum 32715  sigAlgebracsiga 32796  measurescmeas 32883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-ov 7365  df-esum 32716  df-meas 32884

This theorem is referenced by:  measbasedom  32890  measfrge0  32891  measvnul  32894  measvun  32897  measinb  32909  measres  32910  measdivcst  32912  measdivcstALTV  32913  cntmeas  32914  volmeas  32919  ddemeas  32924  omsmeas  33012  dstrvprob  33160