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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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