Theorem ismea 47430

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 23	. 2 ⊢ (𝑀 ∈ Meas → 𝑀 ∈ Meas)
2		fex 7230	. . . . 5 ⊢ ((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) → 𝑀 ∈ V)
3		id 23	. . . . . . . . . 10 ⊢ (𝑧 = 𝑀 → 𝑧 = 𝑀)
4		dmeq 5885	. . . . . . . . . 10 ⊢ (𝑧 = 𝑀 → dom 𝑧 = dom 𝑀)
5	3, 4	feq12d 6695	. . . . . . . . 9 ⊢ (𝑧 = 𝑀 → (𝑧:dom 𝑧⟶(0[,]+∞) ↔ 𝑀:dom 𝑀⟶(0[,]+∞)))
6	4	eleq1d 2846	. . . . . . . . 9 ⊢ (𝑧 = 𝑀 → (dom 𝑧 ∈ SAlg ↔ dom 𝑀 ∈ SAlg))
7	5, 6	anbi12d 644	. . . . . . . 8 ⊢ (𝑧 = 𝑀 → ((𝑧:dom 𝑧⟶(0[,]+∞) ∧ dom 𝑧 ∈ SAlg) ↔ (𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg)))
8		fveq1 6882	. . . . . . . . 9 ⊢ (𝑧 = 𝑀 → (𝑧‘∅) = (𝑀‘∅))
9	8	eqeq1d 2763	. . . . . . . 8 ⊢ (𝑧 = 𝑀 → ((𝑧‘∅) = 0 ↔ (𝑀‘∅) = 0))
10	7, 9	anbi12d 644	. . . . . . 7 ⊢ (𝑧 = 𝑀 → (((𝑧:dom 𝑧⟶(0[,]+∞) ∧ dom 𝑧 ∈ SAlg) ∧ (𝑧‘∅) = 0) ↔ ((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0)))
11	4	pweqd 4574	. . . . . . . 8 ⊢ (𝑧 = 𝑀 → 𝒫 dom 𝑧 = 𝒫 dom 𝑀)
12		fveq1 6882	. . . . . . . . . 10 ⊢ (𝑧 = 𝑀 → (𝑧‘∪ 𝑥) = (𝑀‘∪ 𝑥))
13		reseq1 5964	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑀 → (𝑧 ↾ 𝑥) = (𝑀 ↾ 𝑥))
14	13	fveq2d 6887	. . . . . . . . . 10 ⊢ (𝑧 = 𝑀 → (Σ^‘(𝑧 ↾ 𝑥)) = (Σ^‘(𝑀 ↾ 𝑥)))
15	12, 14	eqeq12d 2777	. . . . . . . . 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 3335	. . . . . . 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 644	. . . . . 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 47429	. . . . . 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 3634	. . . . 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 18	. . . 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 739	. . 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 271	. 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 3634	. 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 381	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)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451  ∅c0 4279  𝒫 cpw 4557  ∪ cuni 4867  Disj wdisj 5070   class class class wbr 5103  dom cdm 5651   ↾ cres 5653  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418  ωcom 7875   ≼ cdom 8964  0cc0 11193  +∞cpnf 11333  [,]cicc 13472  SAlgcsalg 47287  Σ^{^}csumge0 47341  Meascmea 47428

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-mea 47429

This theorem is used by:  dmmeasal  47431  meaf  47432  mea0  47433  meadjuni  47436  ismeannd  47446  psmeasure  47450