Theorem ismea 42732

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 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥)))))

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.

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝑀 ∈ Meas → 𝑀 ∈ Meas)
2		fex 6988	. . . . 5 ⊢ ((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) → 𝑀 ∈ V)
3		id 22	. . . . . . . . . 10 ⊢ (𝑧 = 𝑀 → 𝑧 = 𝑀)
4		dmeq 5771	. . . . . . . . . 10 ⊢ (𝑧 = 𝑀 → dom 𝑧 = dom 𝑀)
5	3, 4	feq12d 6501	. . . . . . . . 9 ⊢ (𝑧 = 𝑀 → (𝑧:dom 𝑧⟶(0[,]+∞) ↔ 𝑀:dom 𝑀⟶(0[,]+∞)))
6	4	eleq1d 2897	. . . . . . . . 9 ⊢ (𝑧 = 𝑀 → (dom 𝑧 ∈ SAlg ↔ dom 𝑀 ∈ SAlg))
7	5, 6	anbi12d 632	. . . . . . . 8 ⊢ (𝑧 = 𝑀 → ((𝑧:dom 𝑧⟶(0[,]+∞) ∧ dom 𝑧 ∈ SAlg) ↔ (𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg)))
8		fveq1 6668	. . . . . . . . 9 ⊢ (𝑧 = 𝑀 → (𝑧‘∅) = (𝑀‘∅))
9	8	eqeq1d 2823	. . . . . . . 8 ⊢ (𝑧 = 𝑀 → ((𝑧‘∅) = 0 ↔ (𝑀‘∅) = 0))
10	7, 9	anbi12d 632	. . . . . . 7 ⊢ (𝑧 = 𝑀 → (((𝑧:dom 𝑧⟶(0[,]+∞) ∧ dom 𝑧 ∈ SAlg) ∧ (𝑧‘∅) = 0) ↔ ((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0)))
11	4	pweqd 4557	. . . . . . . 8 ⊢ (𝑧 = 𝑀 → 𝒫 dom 𝑧 = 𝒫 dom 𝑀)
12		fveq1 6668	. . . . . . . . . 10 ⊢ (𝑧 = 𝑀 → (𝑧‘∪ 𝑥) = (𝑀‘∪ 𝑥))
13		reseq1 5846	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑀 → (𝑧 ↾ 𝑥) = (𝑀 ↾ 𝑥))
14	13	fveq2d 6673	. . . . . . . . . 10 ⊢ (𝑧 = 𝑀 → (Σ^‘(𝑧 ↾ 𝑥)) = (Σ^‘(𝑀 ↾ 𝑥)))
15	12, 14	eqeq12d 2837	. . . . . . . . 9 ⊢ (𝑧 = 𝑀 → ((𝑧‘∪ 𝑥) = (Σ^‘(𝑧 ↾ 𝑥)) ↔ (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥))))
16	15	imbi2d 343	. . . . . . . 8 ⊢ (𝑧 = 𝑀 → (((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑧‘∪ 𝑥) = (Σ^‘(𝑧 ↾ 𝑥))) ↔ ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥)))))
17	11, 16	raleqbidv 3401	. . . . . . 7 ⊢ (𝑧 = 𝑀 → (∀𝑥 ∈ 𝒫 dom 𝑧((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑧‘∪ 𝑥) = (Σ^‘(𝑧 ↾ 𝑥))) ↔ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥)))))
18	10, 17	anbi12d 632	. . . . . 6 ⊢ (𝑧 = 𝑀 → ((((𝑧:dom 𝑧⟶(0[,]+∞) ∧ dom 𝑧 ∈ SAlg) ∧ (𝑧‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑧((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑧‘∪ 𝑥) = (Σ^‘(𝑧 ↾ 𝑥)))) ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥))))))
19		df-mea 42731	. . . . . 6 ⊢ Meas = {𝑧 ∣ (((𝑧:dom 𝑧⟶(0[,]+∞) ∧ dom 𝑧 ∈ SAlg) ∧ (𝑧‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑧((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑧‘∪ 𝑥) = (Σ^‘(𝑧 ↾ 𝑥))))}
20	18, 19	elab2g 3667	. . . . 5 ⊢ (𝑀 ∈ V → (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))))))
21	2, 20	syl 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 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥))))))
22	21	ad2antrr 724	. . 3 ⊢ ((((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥)))) → (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))))))
23	22	ibir 270	. 2 ⊢ ((((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥)))) → 𝑀 ∈ Meas)
24	18, 19	elab2g 3667	. 2 ⊢ (𝑀 ∈ Meas → (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))))))
25	1, 23, 24	pm5.21nii 382	1 ⊢ (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)))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3494  ∅c0 4290  𝒫 cpw 4538  ∪ cuni 4837  Disj wdisj 5030   class class class wbr 5065  dom cdm 5554   ↾ cres 5556  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155  ωcom 7579   ≼ cdom 8506  0cc0 10536  +∞cpnf 10671  [,]cicc 12740  SAlgcsalg 42592  Σ^{^}csumge0 42643  Meascmea 42730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-mea 42731

This theorem is referenced by:  dmmeasal  42733  meaf  42734  mea0  42735  meadjuni  42738  ismeannd  42748  psmeasure  42752