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Theorem ismea 42913
Description: Express the predicate "𝑀 is a measure." Definition 112A of [Fremlin1] p. 14. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
ismea (𝑀 ∈ Meas ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = (Σ^‘(𝑀𝑥)))))
Distinct variable groups:   𝑥,𝑀   𝑥,𝑦
Allowed substitution hint:   𝑀(𝑦)

Proof of Theorem ismea
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 id 22 . 2 (𝑀 ∈ Meas → 𝑀 ∈ Meas)
2 fex 6962 . . . . 5 ((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) → 𝑀 ∈ V)
3 id 22 . . . . . . . . . 10 (𝑧 = 𝑀𝑧 = 𝑀)
4 dmeq 5745 . . . . . . . . . 10 (𝑧 = 𝑀 → dom 𝑧 = dom 𝑀)
53, 4feq12d 6475 . . . . . . . . 9 (𝑧 = 𝑀 → (𝑧:dom 𝑧⟶(0[,]+∞) ↔ 𝑀:dom 𝑀⟶(0[,]+∞)))
64eleq1d 2896 . . . . . . . . 9 (𝑧 = 𝑀 → (dom 𝑧 ∈ SAlg ↔ dom 𝑀 ∈ SAlg))
75, 6anbi12d 633 . . . . . . . 8 (𝑧 = 𝑀 → ((𝑧:dom 𝑧⟶(0[,]+∞) ∧ dom 𝑧 ∈ SAlg) ↔ (𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg)))
8 fveq1 6642 . . . . . . . . 9 (𝑧 = 𝑀 → (𝑧‘∅) = (𝑀‘∅))
98eqeq1d 2823 . . . . . . . 8 (𝑧 = 𝑀 → ((𝑧‘∅) = 0 ↔ (𝑀‘∅) = 0))
107, 9anbi12d 633 . . . . . . 7 (𝑧 = 𝑀 → (((𝑧:dom 𝑧⟶(0[,]+∞) ∧ dom 𝑧 ∈ SAlg) ∧ (𝑧‘∅) = 0) ↔ ((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0)))
114pweqd 4531 . . . . . . . 8 (𝑧 = 𝑀 → 𝒫 dom 𝑧 = 𝒫 dom 𝑀)
12 fveq1 6642 . . . . . . . . . 10 (𝑧 = 𝑀 → (𝑧 𝑥) = (𝑀 𝑥))
13 reseq1 5820 . . . . . . . . . . 11 (𝑧 = 𝑀 → (𝑧𝑥) = (𝑀𝑥))
1413fveq2d 6647 . . . . . . . . . 10 (𝑧 = 𝑀 → (Σ^‘(𝑧𝑥)) = (Σ^‘(𝑀𝑥)))
1512, 14eqeq12d 2837 . . . . . . . . 9 (𝑧 = 𝑀 → ((𝑧 𝑥) = (Σ^‘(𝑧𝑥)) ↔ (𝑀 𝑥) = (Σ^‘(𝑀𝑥))))
1615imbi2d 344 . . . . . . . 8 (𝑧 = 𝑀 → (((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑧 𝑥) = (Σ^‘(𝑧𝑥))) ↔ ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = (Σ^‘(𝑀𝑥)))))
1711, 16raleqbidv 3386 . . . . . . 7 (𝑧 = 𝑀 → (∀𝑥 ∈ 𝒫 dom 𝑧((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑧 𝑥) = (Σ^‘(𝑧𝑥))) ↔ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = (Σ^‘(𝑀𝑥)))))
1810, 17anbi12d 633 . . . . . 6 (𝑧 = 𝑀 → ((((𝑧:dom 𝑧⟶(0[,]+∞) ∧ dom 𝑧 ∈ SAlg) ∧ (𝑧‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑧((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑧 𝑥) = (Σ^‘(𝑧𝑥)))) ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = (Σ^‘(𝑀𝑥))))))
19 df-mea 42912 . . . . . 6 Meas = {𝑧 ∣ (((𝑧:dom 𝑧⟶(0[,]+∞) ∧ dom 𝑧 ∈ SAlg) ∧ (𝑧‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑧((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑧 𝑥) = (Σ^‘(𝑧𝑥))))}
2018, 19elab2g 3645 . . . . 5 (𝑀 ∈ V → (𝑀 ∈ Meas ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = (Σ^‘(𝑀𝑥))))))
212, 20syl 17 . . . 4 ((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) → (𝑀 ∈ Meas ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = (Σ^‘(𝑀𝑥))))))
2221ad2antrr 725 . . 3 ((((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = (Σ^‘(𝑀𝑥)))) → (𝑀 ∈ Meas ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = (Σ^‘(𝑀𝑥))))))
2322ibir 271 . 2 ((((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = (Σ^‘(𝑀𝑥)))) → 𝑀 ∈ Meas)
2418, 19elab2g 3645 . 2 (𝑀 ∈ Meas → (𝑀 ∈ Meas ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = (Σ^‘(𝑀𝑥))))))
251, 23, 24pm5.21nii 383 1 (𝑀 ∈ Meas ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = (Σ^‘(𝑀𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3126  Vcvv 3471  c0 4266  𝒫 cpw 4512   cuni 4811  Disj wdisj 5004   class class class wbr 5039  dom cdm 5528  cres 5530  wf 6324  cfv 6328  (class class class)co 7130  ωcom 7555  cdom 8482  0cc0 10514  +∞cpnf 10649  [,]cicc 12719  SAlgcsalg 42773  Σ^csumge0 42824  Meascmea 42911
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-mea 42912
This theorem is referenced by:  dmmeasal  42914  meaf  42915  mea0  42916  meadjuni  42919  ismeannd  42929  psmeasure  42933
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