Theorem ismea 42753

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 6989	. . . . 5 ⊢ ((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) → 𝑀 ∈ V)
3		id 22	. . . . . . . . . 10 ⊢ (𝑧 = 𝑀 → 𝑧 = 𝑀)
4		dmeq 5772	. . . . . . . . . 10 ⊢ (𝑧 = 𝑀 → dom 𝑧 = dom 𝑀)
5	3, 4	feq12d 6502	. . . . . . . . 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 6669	. . . . . . . . 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 4558	. . . . . . . 8 ⊢ (𝑧 = 𝑀 → 𝒫 dom 𝑧 = 𝒫 dom 𝑀)
12		fveq1 6669	. . . . . . . . . 10 ⊢ (𝑧 = 𝑀 → (𝑧‘∪ 𝑥) = (𝑀‘∪ 𝑥))
13		reseq1 5847	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑀 → (𝑧 ↾ 𝑥) = (𝑀 ↾ 𝑥))
14	13	fveq2d 6674	. . . . . . . . . 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 42752	. . . . . 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 3668	. . . . 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 3668	. 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 1537   ∈ wcel 2114  ∀wral 3138  Vcvv 3494  ∅c0 4291  𝒫 cpw 4539  ∪ cuni 4838  Disj wdisj 5031   class class class wbr 5066  dom cdm 5555   ↾ cres 5557  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  ωcom 7580   ≼ cdom 8507  0cc0 10537  +∞cpnf 10672  [,]cicc 12742  SAlgcsalg 42613  Σ^{^}csumge0 42664  Meascmea 42751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  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 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-mea 42752

This theorem is referenced by:  dmmeasal  42754  meaf  42755  mea0  42756  meadjuni  42759  ismeannd  42769  psmeasure  42773