Theorem measval 32797

Description: The value of the measures function applied on a sigma-algebra. (Contributed by Thierry Arnoux, 17-Oct-2016.)

Assertion

Ref	Expression
measval	⊢ (𝑆 ∈ ∪ ran sigAlgebra → (measures‘𝑆) = {𝑚 ∣ (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,𝑚,𝑦   𝑆,𝑚,𝑥

Allowed substitution hint:   𝑆(𝑦)

Proof of Theorem measval

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1136	. . . 4 ⊢ ((𝑚:𝑆⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦))) → 𝑚:𝑆⟶(0[,]+∞))
2	1	ss2abi 4023	. . 3 ⊢ {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[,]+∞)}
3		ovex 7390	. . . 4 ⊢ (0[,]+∞) ∈ V
4		mapex 8771	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ (0[,]+∞) ∈ V) → {𝑚 ∣ 𝑚:𝑆⟶(0[,]+∞)} ∈ V)
5	3, 4	mpan2 689	. . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → {𝑚 ∣ 𝑚:𝑆⟶(0[,]+∞)} ∈ V)
6		ssexg 5280	. . 3 ⊢ (({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[,]+∞)} ∧ {𝑚 ∣ 𝑚:𝑆⟶(0[,]+∞)} ∈ V) → {𝑚 ∣ (𝑚:𝑆⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)))} ∈ V)
7	2, 5, 6	sylancr 587	. 2 ⊢ (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)))} ∈ V)
8		feq2 6650	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑚:𝑠⟶(0[,]+∞) ↔ 𝑚:𝑆⟶(0[,]+∞)))
9		pweq 4574	. . . . . 6 ⊢ (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
10	9	raleqdv 3313	. . . . 5 ⊢ (𝑠 = 𝑆 → (∀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))))
11	8, 10	3anbi13d 1438	. . . 4 ⊢ (𝑠 = 𝑆 → ((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)))))
12	11	abbidv 2805	. . 3 ⊢ (𝑠 = 𝑆 → {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[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)))})
13		df-meas 32795	. . 3 ⊢ 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)))})
14	12, 13	fvmptg 6946	. 2 ⊢ ((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)))} ∈ V) → (measures‘𝑆) = {𝑚 ∣ (𝑚:𝑆⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)))})
15	7, 14	mpdan 685	1 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (measures‘𝑆) = {𝑚 ∣ (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   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2713  ∀wral 3064  Vcvv 3445   ⊆ wss 3910  ∅c0 4282  𝒫 cpw 4560  ∪ cuni 4865  Disj wdisj 5070   class class class wbr 5105  ran crn 5634  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  ωcom 7802   ≼ cdom 8881  0cc0 11051  +∞cpnf 11186  [,]cicc 13267  Σ^*cesum 32626  sigAlgebracsiga 32707  measurescmeas 32794

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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-meas 32795

This theorem is referenced by: (None)