Theorem meaf 46639

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 46637	. . . . 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 6644	. 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 1541   ∈ wcel 2113  ∀wral 3049  ∅c0 4283  𝒫 cpw 4552  ∪ cuni 4861  Disj wdisj 5063   class class class wbr 5096  dom cdm 5622   ↾ cres 5624  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356  ωcom 7806   ≼ cdom 8879  0cc0 11024  +∞cpnf 11161  [,]cicc 13262  SAlgcsalg 46494  Σ^{^}csumge0 46548  Meascmea 46635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-mea 46636

This theorem is referenced by:  meacl  46644  meadjun  46648  meadjiunlem  46651  meadjiun  46652