Theorem meaf 46444

Description: A measure is a function that maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
meaf.m	⊢ (𝜑 → 𝑀 ∈ Meas)
meaf.s	⊢ 𝑆 = dom 𝑀

Assertion

Ref	Expression
meaf	⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞))

Proof of Theorem meaf

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		meaf.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ Meas)
2		ismea 46442	. . . . 5 ⊢ (𝑀 ∈ Meas ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥)))))
3	1, 2	sylib 218	. . . 4 ⊢ (𝜑 → (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥)))))
4	3	simpld 494	. . 3 ⊢ (𝜑 → ((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0))
5	4	simplld 767	. 2 ⊢ (𝜑 → 𝑀:dom 𝑀⟶(0[,]+∞))
6		meaf.s	. . . 4 ⊢ 𝑆 = dom 𝑀
7	6	a1i 11	. . 3 ⊢ (𝜑 → 𝑆 = dom 𝑀)
8	7	feq2d 6636	. 2 ⊢ (𝜑 → (𝑀:𝑆⟶(0[,]+∞) ↔ 𝑀:dom 𝑀⟶(0[,]+∞)))
9	5, 8	mpbird 257	1 ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∅c0 4284  𝒫 cpw 4551  ∪ cuni 4858  Disj wdisj 5059   class class class wbr 5092  dom cdm 5619   ↾ cres 5621  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  ωcom 7799   ≼ cdom 8870  0cc0 11009  +∞cpnf 11146  [,]cicc 13251  SAlgcsalg 46299  Σ^{^}csumge0 46353  Meascmea 46440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-mea 46441

This theorem is referenced by:  meacl  46449  meadjun  46453  meadjiunlem  46456  meadjiun  46457