pmeasadd - Mathbox for Thierry Arnoux

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Theorem pmeasadd 32004

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 3105	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑅)
3		dfiun3g 5833	. . . 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 6721	. 2 ⊢ (𝜑 → (𝑃‘∪ 𝑘 ∈ 𝐴 𝐵) = (𝑃‘∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵)))
6		pmeassubadd.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ≼ ω)
7		mptct 10152	. . . . . 6 ⊢ (𝐴 ≼ ω → (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω)
8		rnct 10139	. . . . . 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 2737	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
11	10	rnmptss 6939	. . . . . 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 30643	. . . . . 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 1130	. . . 4 ⊢ (𝜑 → (ran (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω ∧ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑅 ∧ Disj 𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝑦))
17	16	ancli 552	. . 3 ⊢ (𝜑 → (𝜑 ∧ (ran (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω ∧ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑅 ∧ Disj 𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝑦)))
18		ctex 8643	. . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)
19		mptexg 7037	. . . . 5 ⊢ (𝐴 ∈ V → (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V)
20	6, 18, 19	3syl 18	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V)
21		rnexg 7682	. . . 4 ⊢ ((𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V → ran (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V)
22		breq1 5056	. . . . . . . 8 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → (𝑥 ≼ ω ↔ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω))
23		sseq1 3926	. . . . . . . 8 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → (𝑥 ⊆ 𝑅 ↔ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑅))
24		disjeq1 5025	. . . . . . . 8 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → (Disj 𝑦 ∈ 𝑥 𝑦 ↔ Disj 𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝑦))
25	22, 23, 24	3anbi123d 1438	. . . . . . 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 632	. . . . . 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 4830	. . . . . . . 8 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → ∪ 𝑥 = ∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵))
28	27	fveq2d 6721	. . . . . . 7 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → (𝑃‘∪ 𝑥) = (𝑃‘∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵)))
29		esumeq1 31714	. . . . . . 7 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → Σ^𝑦 ∈ 𝑥(𝑃‘𝑦) = Σ^𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)(𝑃‘𝑦))
30	28, 29	eqeq12d 2753	. . . . . 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 348	. . . . 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 3481	. . . 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 6717	. . 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 484	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃:𝑅⟶(0[,]+∞))
40	39, 1	ffvelrnd 6905	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑃‘𝐵) ∈ (0[,]+∞))
41		fveq2 6717	. . . . 5 ⊢ (𝐵 = ∅ → (𝑃‘𝐵) = (𝑃‘∅))
42	41	adantl 485	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → (𝑃‘𝐵) = (𝑃‘∅))
43		caraext.2	. . . . 5 ⊢ (𝜑 → (𝑃‘∅) = 0)
44	43	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → (𝑃‘∅) = 0)
45	42, 44	eqtrd 2777	. . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → (𝑃‘𝐵) = 0)
46	36, 37, 40, 1, 45, 13	esumrnmpt2 31748	. 2 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)(𝑃‘𝑦) = Σ^𝑘 ∈ 𝐴(𝑃‘𝐵))
47	5, 35, 46	3eqtrd 2781	1 ⊢ (𝜑 → (𝑃‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ^*𝑘 ∈ 𝐴(𝑃‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∀wral 3061 {crab 3065 Vcvv 3408 ∖ cdif 3863 ∪ cun 3864 ⊆ wss 3866 ∅c0 4237 𝒫 cpw 4513 ∪ cuni 4819 ∪ ciun 4904 Disj wdisj 5018 class class class wbr 5053 ↦ cmpt 5135 ran crn 5552 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ωcom 7644 ≼ cdom 8624 0cc0 10729 +∞cpnf 10864 [,]cicc 12938 Σ^*cesum 31707

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-ac2 10077 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-disj 5019 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-acn 9558 df-ac 9730 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ioc 12940 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-mod 13443 df-seq 13575 df-exp 13636 df-fac 13840 df-bc 13869 df-hash 13897 df-shft 14630 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-limsup 15032 df-clim 15049 df-rlim 15050 df-sum 15250 df-ef 15629 df-sin 15631 df-cos 15632 df-pi 15634 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-ordt 17006 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-ps 18072 df-tsr 18073 df-plusf 18113 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-mhm 18218 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-mulg 18489 df-subg 18540 df-cntz 18711 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-cring 19565 df-subrg 19798 df-abv 19853 df-lmod 19901 df-scaf 19902 df-sra 20209 df-rgmod 20210 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-fbas 20360 df-fg 20361 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cld 21916 df-ntr 21917 df-cls 21918 df-nei 21995 df-lp 22033 df-perf 22034 df-cn 22124 df-cnp 22125 df-haus 22212 df-tx 22459 df-hmeo 22652 df-fil 22743 df-fm 22835 df-flim 22836 df-flf 22837 df-tmd 22969 df-tgp 22970 df-tsms 23024 df-trg 23057 df-xms 23218 df-ms 23219 df-tms 23220 df-nm 23480 df-ngp 23481 df-nrg 23483 df-nlm 23484 df-ii 23774 df-cncf 23775 df-limc 24763 df-dv 24764 df-log 25445 df-esum 31708

This theorem is referenced by: (None)

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