pmeasadd - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > pmeasadd		Structured version Visualization version GIF version

Theorem pmeasadd 34327

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 3146	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑅)
3		dfiun3g 5978	. . . 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 6910	. 2 ⊢ (𝜑 → (𝑃‘∪ 𝑘 ∈ 𝐴 𝐵) = (𝑃‘∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵)))
6		pmeassubadd.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ≼ ω)
7		mptct 10578	. . . . . 6 ⊢ (𝐴 ≼ ω → (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω)
8		rnct 10565	. . . . . 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 7143	. . . . . 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 32598	. . . . . 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 1129	. . . 4 ⊢ (𝜑 → (ran (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω ∧ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑅 ∧ Disj 𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝑦))
17	16	ancli 548	. . 3 ⊢ (𝜑 → (𝜑 ∧ (ran (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω ∧ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑅 ∧ Disj 𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝑦)))
18		ctex 9004	. . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)
19		mptexg 7241	. . . . 5 ⊢ (𝐴 ∈ V → (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V)
20	6, 18, 19	3syl 18	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V)
21		rnexg 7924	. . . 4 ⊢ ((𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V → ran (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V)
22		breq1 5146	. . . . . . . 8 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → (𝑥 ≼ ω ↔ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω))
23		sseq1 4009	. . . . . . . 8 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → (𝑥 ⊆ 𝑅 ↔ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑅))
24		disjeq1 5117	. . . . . . . 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 630	. . . . . 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 4918	. . . . . . . 8 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → ∪ 𝑥 = ∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵))
28	27	fveq2d 6910	. . . . . . 7 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → (𝑃‘∪ 𝑥) = (𝑃‘∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵)))
29		esumeq1 34035	. . . . . . 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 344	. . . . 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 3554	. . . 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 6906	. . 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 480	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃:𝑅⟶(0[,]+∞))
40	39, 1	ffvelcdmd 7105	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑃‘𝐵) ∈ (0[,]+∞))
41		fveq2 6906	. . . . 5 ⊢ (𝐵 = ∅ → (𝑃‘𝐵) = (𝑃‘∅))
42	41	adantl 481	. . . 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 34069	. 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 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 ∪ cuni 4907 ∪ ciun 4991 Disj wdisj 5110 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ωcom 7887 ≼ cdom 8983 0cc0 11155 +∞cpnf 11292 [,]cicc 13390 Σ^*cesum 34028

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-ac2 10503 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-acn 9982 df-ac 10156 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-ordt 17546 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-ps 18611 df-tsr 18612 df-plusf 18652 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrng 20546 df-subrg 20570 df-abv 20810 df-lmod 20860 df-scaf 20861 df-sra 21172 df-rgmod 21173 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-tmd 24080 df-tgp 24081 df-tsms 24135 df-trg 24168 df-xms 24330 df-ms 24331 df-tms 24332 df-nm 24595 df-ngp 24596 df-nrg 24598 df-nlm 24599 df-ii 24903 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598 df-esum 34029

This theorem is referenced by: (None)

W3C validator