Theorem measval 34342

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 1137	. . . 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 4006	. . 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 7400	. . . 4 ⊢ (0[,]+∞) ∈ V
4		mapex 7892	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ (0[,]+∞) ∈ V) → {𝑚 ∣ 𝑚:𝑆⟶(0[,]+∞)} ∈ V)
5	3, 4	mpan2 692	. . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → {𝑚 ∣ 𝑚:𝑆⟶(0[,]+∞)} ∈ V)
6		ssexg 5264	. . 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 588	. 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 6647	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑚:𝑠⟶(0[,]+∞) ↔ 𝑚:𝑆⟶(0[,]+∞)))
9		pweq 4555	. . . . . 6 ⊢ (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
10	9	raleqdv 3295	. . . . 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 1441	. . . 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 2802	. . 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 34340	. . 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 6945	. 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 688	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 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2714  ∀wral 3051  Vcvv 3429   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541  ∪ cuni 4850  Disj wdisj 5052   class class class wbr 5085  ran crn 5632  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  ωcom 7817   ≼ cdom 8891  0cc0 11038  +∞cpnf 11176  [,]cicc 13301  Σ^*cesum 34171  sigAlgebracsiga 34252  measurescmeas 34339

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-ov 7370  df-meas 34340

This theorem is referenced by: (None)