Theorem ismeas 34598

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 3475	. . 3 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀 ∈ V)
2	1	a1i 11	. 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) → 𝑀 ∈ V))
3		simp1 1153	. . 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 7445	. . . 4 ⊢ (0[,]+∞) ∈ V
5		fex2 7931	. . . . . 6 ⊢ ((𝑀:𝑆⟶(0[,]+∞) ∧ 𝑆 ∈ ∪ ran sigAlgebra ∧ (0[,]+∞) ∈ V) → 𝑀 ∈ V)
6	5	3expb 1137	. . . . 5 ⊢ ((𝑀:𝑆⟶(0[,]+∞) ∧ (𝑆 ∈ ∪ ran sigAlgebra ∧ (0[,]+∞) ∈ V)) → 𝑀 ∈ V)
7	6	expcom 418	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ (0[,]+∞) ∈ V) → (𝑀:𝑆⟶(0[,]+∞) → 𝑀 ∈ V))
8	4, 7	mpan2 703	. . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑀:𝑆⟶(0[,]+∞) → 𝑀 ∈ V))
9	3, 8	syl5 35	. 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 34595	. . . 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 3458	. . . . . 6 ⊢ 𝑠 ∈ V
12		mapex 7935	. . . . . 6 ⊢ ((𝑠 ∈ V ∧ (0[,]+∞) ∈ V) → {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)} ∈ V)
13	11, 4, 12	mp2an 704	. . . . 5 ⊢ {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)} ∈ V
14		simp1 1153	. . . . . 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 4019	. . . . 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 5292	. . . 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 489	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
18		simpl 487	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝑠 = 𝑆)
19	17, 18	feq12d 6693	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (𝑚:𝑠⟶(0[,]+∞) ↔ 𝑀:𝑆⟶(0[,]+∞)))
20		fveq1 6880	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑚‘∅) = (𝑀‘∅))
21	20	eqeq1d 2764	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((𝑚‘∅) = 0 ↔ (𝑀‘∅) = 0))
22	21	adantl 486	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → ((𝑚‘∅) = 0 ↔ (𝑀‘∅) = 0))
23	18	pweqd 4578	. . . . . 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 34438	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → Σ*𝑦 ∈ 𝑥(𝑚‘𝑦) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))
27	24, 26	eqeq12d 2778	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → ((𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦) ↔ (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)))
28	27	imbi2d 343	. . . . . . 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 486	. . . . . 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 3337	. . . . 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 1463	. . . 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 33011	. . 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 417	. 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 382	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)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  {cab 2740  ∀wral 3078  Vcvv 3454  ∅c0 4285  𝒫 cpw 4561  ∪ cuni 4871  Disj wdisj 5075   class class class wbr 5108  ran crn 5661  ⟶wf 6532  ‘cfv 6536  (class class class)co 7412  ωcom 7860   ≼ cdom 8939  0cc0 11106  +∞cpnf 11246  [,]cicc 13381  Σ^*cesum 34426  sigAlgebracsiga 34507  measurescmeas 34594

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7415  df-esum 34427  df-meas 34595

This theorem is used by:  measbasedom  34601  measfrge0  34602  measvnul  34605  measvun  34608  measinb  34620  measres  34621  measdivcst  34623  measdivcstALTV  34624  cntmeas  34625  volmeas  34630  ddemeas  34635  omsmeas  34722  dstrvprob  34871