Theorem meaf 47032

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 47030	. . . . 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 220	. . . 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 498	. . 3 ⊢ (𝜑 → ((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0))
5	4	simplld 777	. 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 6677	. 2 ⊢ (𝜑 → (𝑀:𝑆⟶(0[,]+∞) ↔ 𝑀:dom 𝑀⟶(0[,]+∞)))
9	5, 8	mpbird 259	1 ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ∅c0 4287  𝒫 cpw 4557  ∪ cuni 4867  Disj wdisj 5069   class class class wbr 5102  dom cdm 5649   ↾ cres 5651  ⟶wf 6519  ‘cfv 6523  (class class class)co 7398  ωcom 7848   ≼ cdom 8927  0cc0 11075  +∞cpnf 11215  [,]cicc 13354  SAlgcsalg 46887  Σ^{^}csumge0 46941  Meascmea 47028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-mea 47029

This theorem is referenced by:  meacl  47037  meadjun  47041  meadjiunlem  47044  meadjiun  47045