pmeasadd - Mathbox for Thierry Arnoux

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Theorem pmeasadd 31234

Description: A premeasure on a ring of sets is additive on disjoint countable collections. This is called sigma-additivity. (Contributed by Thierry Arnoux, 19-Jul-2020.)

**Hypotheses**
Ref	Expression
caraext.1	⊢ (𝜑 → 𝑃:𝑅⟶(0[,]+∞))
caraext.2	⊢ (𝜑 → (𝑃‘∅) = 0)
caraext.3	⊢ ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑃‘∪ 𝑥) = Σ^*𝑦 ∈ 𝑥(𝑃‘𝑦))
pmeassubadd.q	⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}
pmeassubadd.1	⊢ (𝜑 → 𝑅 ∈ 𝑄)
pmeassubadd.2	⊢ (𝜑 → 𝐴 ≼ ω)
pmeassubadd.3	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑅)
pmeasadd.4	⊢ (𝜑 → Disj 𝑘 ∈ 𝐴 𝐵)

**Assertion**
Ref	Expression
pmeasadd	⊢ (𝜑 → (𝑃‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ^*𝑘 ∈ 𝐴(𝑃‘𝐵))

Distinct variable groups: 𝑥,𝑃,𝑦 𝑥,𝑅,𝑦 𝜑,𝑥,𝑦 𝐴,𝑘,𝑥,𝑦 𝑥,𝐵,𝑦 𝑃,𝑘 𝑅,𝑘 𝜑,𝑘 Allowed substitution hints: 𝜑(𝑠) 𝐴(𝑠) 𝐵(𝑘,𝑠) 𝑃(𝑠) 𝑄(𝑥,𝑦,𝑘,𝑠) 𝑅(𝑠) 𝑂(𝑥,𝑦,𝑘,𝑠)

**Proof of Theorem pmeasadd**
Step	Hyp	Ref	Expression
1		pmeassubadd.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑅)
2	1	ralrimiva 3132	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑅)
3		dfiun3g 5677	. . . 4 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑅 → ∪ 𝑘 ∈ 𝐴 𝐵 = ∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵))
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝐵 = ∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵))
5	4	fveq2d 6503	. 2 ⊢ (𝜑 → (𝑃‘∪ 𝑘 ∈ 𝐴 𝐵) = (𝑃‘∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵)))
6		pmeassubadd.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ≼ ω)
7		mptct 9758	. . . . . 6 ⊢ (𝐴 ≼ ω → (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω)
8		rnct 9745	. . . . . 6 ⊢ ((𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω → ran (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω)
9	6, 7, 8	3syl 18	. . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω)
10		eqid 2778	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
11	10	rnmptss 6709	. . . . . 6 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑅 → ran (𝑘 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑅)
12	2, 11	syl 17	. . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑅)
13		pmeasadd.4	. . . . . 6 ⊢ (𝜑 → Disj 𝑘 ∈ 𝐴 𝐵)
14		disjrnmpt 30101	. . . . . 6 ⊢ (Disj 𝑘 ∈ 𝐴 𝐵 → Disj 𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝑦)
15	13, 14	syl 17	. . . . 5 ⊢ (𝜑 → Disj 𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝑦)
16	9, 12, 15	3jca 1108	. . . 4 ⊢ (𝜑 → (ran (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω ∧ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑅 ∧ Disj 𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝑦))
17	16	ancli 541	. . 3 ⊢ (𝜑 → (𝜑 ∧ (ran (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω ∧ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑅 ∧ Disj 𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝑦)))
18		ctex 8321	. . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)
19		mptexg 6810	. . . . 5 ⊢ (𝐴 ∈ V → (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V)
20	6, 18, 19	3syl 18	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V)
21		rnexg 7429	. . . 4 ⊢ ((𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V → ran (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V)
22		breq1 4932	. . . . . . . 8 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → (𝑥 ≼ ω ↔ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω))
23		sseq1 3882	. . . . . . . 8 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → (𝑥 ⊆ 𝑅 ↔ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑅))
24		disjeq1 4904	. . . . . . . 8 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → (Disj 𝑦 ∈ 𝑥 𝑦 ↔ Disj 𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝑦))
25	22, 23, 24	3anbi123d 1415	. . . . . . 7 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → ((𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦) ↔ (ran (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω ∧ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑅 ∧ Disj 𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝑦)))
26	25	anbi2d 619	. . . . . 6 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ↔ (𝜑 ∧ (ran (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω ∧ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑅 ∧ Disj 𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝑦))))
27		unieq 4720	. . . . . . . 8 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → ∪ 𝑥 = ∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵))
28	27	fveq2d 6503	. . . . . . 7 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → (𝑃‘∪ 𝑥) = (𝑃‘∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵)))
29		esumeq1 30943	. . . . . . 7 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → Σ^𝑦 ∈ 𝑥(𝑃‘𝑦) = Σ^𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)(𝑃‘𝑦))
30	28, 29	eqeq12d 2793	. . . . . 6 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → ((𝑃‘∪ 𝑥) = Σ^𝑦 ∈ 𝑥(𝑃‘𝑦) ↔ (𝑃‘∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ^𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)(𝑃‘𝑦)))
31	26, 30	imbi12d 337	. . . . 5 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → (((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑃‘∪ 𝑥) = Σ^𝑦 ∈ 𝑥(𝑃‘𝑦)) ↔ ((𝜑 ∧ (ran (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω ∧ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑅 ∧ Disj 𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝑦)) → (𝑃‘∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ^𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)(𝑃‘𝑦))))
32		caraext.3	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑃‘∪ 𝑥) = Σ^*𝑦 ∈ 𝑥(𝑃‘𝑦))
33	31, 32	vtoclg 3486	. . . 4 ⊢ (ran (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V → ((𝜑 ∧ (ran (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω ∧ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑅 ∧ Disj 𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝑦)) → (𝑃‘∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ^*𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)(𝑃‘𝑦)))
34	20, 21, 33	3syl 18	. . 3 ⊢ (𝜑 → ((𝜑 ∧ (ran (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω ∧ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑅 ∧ Disj 𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝑦)) → (𝑃‘∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ^*𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)(𝑃‘𝑦)))
35	17, 34	mpd 15	. 2 ⊢ (𝜑 → (𝑃‘∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ^*𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)(𝑃‘𝑦))
36		fveq2 6499	. . 3 ⊢ (𝑦 = 𝐵 → (𝑃‘𝑦) = (𝑃‘𝐵))
37	6, 18	syl 17	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
38		caraext.1	. . . . 5 ⊢ (𝜑 → 𝑃:𝑅⟶(0[,]+∞))
39	38	adantr 473	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃:𝑅⟶(0[,]+∞))
40	39, 1	ffvelrnd 6677	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑃‘𝐵) ∈ (0[,]+∞))
41		fveq2 6499	. . . . 5 ⊢ (𝐵 = ∅ → (𝑃‘𝐵) = (𝑃‘∅))
42	41	adantl 474	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → (𝑃‘𝐵) = (𝑃‘∅))
43		caraext.2	. . . . 5 ⊢ (𝜑 → (𝑃‘∅) = 0)
44	43	ad2antrr 713	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → (𝑃‘∅) = 0)
45	42, 44	eqtrd 2814	. . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → (𝑃‘𝐵) = 0)
46	36, 37, 40, 1, 45, 13	esumrnmpt2 30977	. 2 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)(𝑃‘𝑦) = Σ^𝑘 ∈ 𝐴(𝑃‘𝐵))
47	5, 35, 46	3eqtrd 2818	1 ⊢ (𝜑 → (𝑃‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ^*𝑘 ∈ 𝐴(𝑃‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ∀wral 3088 {crab 3092 Vcvv 3415 ∖ cdif 3826 ∪ cun 3827 ⊆ wss 3829 ∅c0 4178 𝒫 cpw 4422 ∪ cuni 4712 ∪ ciun 4792 Disj wdisj 4897 class class class wbr 4929 ↦ cmpt 5008 ran crn 5408 ⟶wf 6184 ‘cfv 6188 (class class class)co 6976 ωcom 7396 ≼ cdom 8304 0cc0 10335 +∞cpnf 10471 [,]cicc 12557 Σ^*cesum 30936

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 ax-ac2 9683 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 ax-addf 10414 ax-mulf 10415

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-disj 4898 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-of 7227 df-om 7397 df-1st 7501 df-2nd 7502 df-supp 7634 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-2o 7906 df-oadd 7909 df-er 8089 df-map 8208 df-pm 8209 df-ixp 8260 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-fsupp 8629 df-fi 8670 df-sup 8701 df-inf 8702 df-oi 8769 df-card 9162 df-acn 9165 df-ac 9336 df-cda 9388 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-z 11794 df-dec 11912 df-uz 12059 df-q 12163 df-rp 12205 df-xneg 12324 df-xadd 12325 df-xmul 12326 df-ioo 12558 df-ioc 12559 df-ico 12560 df-icc 12561 df-fz 12709 df-fzo 12850 df-fl 12977 df-mod 13053 df-seq 13185 df-exp 13245 df-fac 13449 df-bc 13478 df-hash 13506 df-shft 14287 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 df-limsup 14689 df-clim 14706 df-rlim 14707 df-sum 14904 df-ef 15281 df-sin 15283 df-cos 15284 df-pi 15286 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-starv 16436 df-sca 16437 df-vsca 16438 df-ip 16439 df-tset 16440 df-ple 16441 df-ds 16443 df-unif 16444 df-hom 16445 df-cco 16446 df-rest 16552 df-topn 16553 df-0g 16571 df-gsum 16572 df-topgen 16573 df-pt 16574 df-prds 16577 df-ordt 16630 df-xrs 16631 df-qtop 16636 df-imas 16637 df-xps 16639 df-mre 16715 df-mrc 16716 df-acs 16718 df-ps 17668 df-tsr 17669 df-plusf 17709 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-mhm 17803 df-submnd 17804 df-grp 17894 df-minusg 17895 df-sbg 17896 df-mulg 18012 df-subg 18060 df-cntz 18218 df-cmn 18668 df-abl 18669 df-mgp 18963 df-ur 18975 df-ring 19022 df-cring 19023 df-subrg 19256 df-abv 19310 df-lmod 19358 df-scaf 19359 df-sra 19666 df-rgmod 19667 df-psmet 20239 df-xmet 20240 df-met 20241 df-bl 20242 df-mopn 20243 df-fbas 20244 df-fg 20245 df-cnfld 20248 df-top 21206 df-topon 21223 df-topsp 21245 df-bases 21258 df-cld 21331 df-ntr 21332 df-cls 21333 df-nei 21410 df-lp 21448 df-perf 21449 df-cn 21539 df-cnp 21540 df-haus 21627 df-tx 21874 df-hmeo 22067 df-fil 22158 df-fm 22250 df-flim 22251 df-flf 22252 df-tmd 22384 df-tgp 22385 df-tsms 22438 df-trg 22471 df-xms 22633 df-ms 22634 df-tms 22635 df-nm 22895 df-ngp 22896 df-nrg 22898 df-nlm 22899 df-ii 23188 df-cncf 23189 df-limc 24167 df-dv 24168 df-log 24841 df-esum 30937

This theorem is referenced by: (None)

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