Theorem meaf 46764

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 46762	. . . . 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 768	. 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 6647	. 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 1542   ∈ wcel 2114  ∀wral 3052  ∅c0 4286  𝒫 cpw 4555  ∪ cuni 4864  Disj wdisj 5066   class class class wbr 5099  dom cdm 5625   ↾ cres 5627  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360  ωcom 7810   ≼ cdom 8885  0cc0 11030  +∞cpnf 11167  [,]cicc 13268  SAlgcsalg 46619  Σ^{^}csumge0 46673  Meascmea 46760

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 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pr 5378

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-mea 46761

This theorem is referenced by:  meacl  46769  meadjun  46773  meadjiunlem  46776  meadjiun  46777