Theorem ismeas 34825

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 3472	. . 3 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀 ∈ V)
2	1	a1i 11	. 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) → 𝑀 ∈ V))
3		simp1 1154	. . 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 7451	. . . 4 ⊢ (0[,]+∞) ∈ V
5		fex2 7946	. . . . . 6 ⊢ ((𝑀:𝑆⟶(0[,]+∞) ∧ 𝑆 ∈ ∪ ran sigAlgebra ∧ (0[,]+∞) ∈ V) → 𝑀 ∈ V)
6	5	3expb 1138	. . . . 5 ⊢ ((𝑀:𝑆⟶(0[,]+∞) ∧ (𝑆 ∈ ∪ ran sigAlgebra ∧ (0[,]+∞) ∈ V)) → 𝑀 ∈ V)
7	6	expcom 419	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ (0[,]+∞) ∈ V) → (𝑀:𝑆⟶(0[,]+∞) → 𝑀 ∈ V))
8	4, 7	mpan2 704	. . 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 34822	. . . 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 3455	. . . . . 6 ⊢ 𝑠 ∈ V
12		mapex 7950	. . . . . 6 ⊢ ((𝑠 ∈ V ∧ (0[,]+∞) ∈ V) → {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)} ∈ V)
13	11, 4, 12	mp2an 705	. . . . 5 ⊢ {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)} ∈ V
14		simp1 1154	. . . . . 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 4014	. . . . 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 490	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
18		simpl 488	. . . . . 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 6882	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑚‘∅) = (𝑀‘∅))
21	20	eqeq1d 2763	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((𝑚‘∅) = 0 ↔ (𝑀‘∅) = 0))
22	21	adantl 487	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → ((𝑚‘∅) = 0 ↔ (𝑀‘∅) = 0))
23	18	pweqd 4574	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝒫 𝑠 = 𝒫 𝑆)
24		fveq1 6882	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (𝑚‘∪ 𝑥) = (𝑀‘∪ 𝑥))
25		fveq1 6882	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (𝑚‘𝑦) = (𝑀‘𝑦))
26	25	esumeq2sdv 34664	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → Σ*𝑦 ∈ 𝑥(𝑚‘𝑦) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))
27	24, 26	eqeq12d 2777	. . . . . . . 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 487	. . . . . 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 3335	. . . . 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 1464	. . . 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 33242	. . 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 418	. 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 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077  Vcvv 3451  ∅c0 4279  𝒫 cpw 4557  ∪ cuni 4867  Disj wdisj 5070   class class class wbr 5103  ran crn 5652  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418  ωcom 7875   ≼ cdom 8964  0cc0 11193  +∞cpnf 11333  [,]cicc 13472  Σ^*cesum 34652  sigAlgebracsiga 34733  measurescmeas 34821

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7421  df-esum 34653  df-meas 34822

This theorem is used by:  measbasedom  34828  measfrge0  34829  measvnul  34832  measvun  34835  measinb  34847  measres  34848  measdivcst  34850  measdivcstALTV  34851  cntmeas  34852  volmeas  34857  ddemeas  34862  omsmeas  34948  dstrvprob  35097