pmeasadd - Mathbox for Thierry Arnoux

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Theorem pmeasadd 34328

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 3122	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑅)
3		dfiun3g 5904	. . . 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 6821	. 2 ⊢ (𝜑 → (𝑃‘∪ 𝑘 ∈ 𝐴 𝐵) = (𝑃‘∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵)))
6		pmeassubadd.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ≼ ω)
7		mptct 10421	. . . . . 6 ⊢ (𝐴 ≼ ω → (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω)
8		rnct 10408	. . . . . 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 2730	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
11	10	rnmptss 7051	. . . . . 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 32555	. . . . . 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 1128	. . . 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 8881	. . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)
19		mptexg 7150	. . . . 5 ⊢ (𝐴 ∈ V → (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V)
20	6, 18, 19	3syl 18	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V)
21		rnexg 7827	. . . 4 ⊢ ((𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V → ran (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V)
22		breq1 5092	. . . . . . . 8 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → (𝑥 ≼ ω ↔ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω))
23		sseq1 3958	. . . . . . . 8 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → (𝑥 ⊆ 𝑅 ↔ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑅))
24		disjeq1 5063	. . . . . . . 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 4868	. . . . . . . 8 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → ∪ 𝑥 = ∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵))
28	27	fveq2d 6821	. . . . . . 7 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → (𝑃‘∪ 𝑥) = (𝑃‘∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵)))
29		esumeq1 34037	. . . . . . 7 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → Σ^𝑦 ∈ 𝑥(𝑃‘𝑦) = Σ^𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)(𝑃‘𝑦))
30	28, 29	eqeq12d 2746	. . . . . 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 3507	. . . 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 6817	. . 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 7013	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑃‘𝐵) ∈ (0[,]+∞))
41		fveq2 6817	. . . . 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 2765	. . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → (𝑃‘𝐵) = 0)
46	36, 37, 40, 1, 45, 13	esumrnmpt2 34071	. 2 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)(𝑃‘𝑦) = Σ^𝑘 ∈ 𝐴(𝑃‘𝐵))
47	5, 35, 46	3eqtrd 2769	1 ⊢ (𝜑 → (𝑃‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ^*𝑘 ∈ 𝐴(𝑃‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2110 ∀wral 3045 {crab 3393 Vcvv 3434 ∖ cdif 3897 ∪ cun 3898 ⊆ wss 3900 ∅c0 4281 𝒫 cpw 4548 ∪ cuni 4857 ∪ ciun 4939 Disj wdisj 5056 class class class wbr 5089 ↦ cmpt 5170 ran crn 5615 ⟶wf 6473 ‘cfv 6477 (class class class)co 7341 ωcom 7791 ≼ cdom 8862 0cc0 10998 +∞cpnf 11135 [,]cicc 13240 Σ^*cesum 34030

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 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-inf2 9526 ax-ac2 10346 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075 ax-pre-sup 11076 ax-addf 11077 ax-mulf 11078

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-uni 4858 df-int 4896 df-iun 4941 df-iin 4942 df-disj 5057 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-isom 6486 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-card 9824 df-acn 9827 df-ac 9999 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-div 11767 df-nn 12118 df-2 12180 df-3 12181 df-4 12182 df-5 12183 df-6 12184 df-7 12185 df-8 12186 df-9 12187 df-n0 12374 df-z 12461 df-dec 12581 df-uz 12725 df-q 12839 df-rp 12883 df-xneg 13003 df-xadd 13004 df-xmul 13005 df-ioo 13241 df-ioc 13242 df-ico 13243 df-icc 13244 df-fz 13400 df-fzo 13547 df-fl 13688 df-mod 13766 df-seq 13901 df-exp 13961 df-fac 14173 df-bc 14202 df-hash 14230 df-shft 14966 df-cj 14998 df-re 14999 df-im 15000 df-sqrt 15134 df-abs 15135 df-limsup 15370 df-clim 15387 df-rlim 15388 df-sum 15586 df-ef 15966 df-sin 15968 df-cos 15969 df-pi 15971 df-struct 17050 df-sets 17067 df-slot 17085 df-ndx 17097 df-base 17113 df-ress 17134 df-plusg 17166 df-mulr 17167 df-starv 17168 df-sca 17169 df-vsca 17170 df-ip 17171 df-tset 17172 df-ple 17173 df-ds 17175 df-unif 17176 df-hom 17177 df-cco 17178 df-rest 17318 df-topn 17319 df-0g 17337 df-gsum 17338 df-topgen 17339 df-pt 17340 df-prds 17343 df-ordt 17397 df-xrs 17398 df-qtop 17403 df-imas 17404 df-xps 17406 df-mre 17480 df-mrc 17481 df-acs 17483 df-ps 18464 df-tsr 18465 df-plusf 18539 df-mgm 18540 df-sgrp 18619 df-mnd 18635 df-mhm 18683 df-submnd 18684 df-grp 18841 df-minusg 18842 df-sbg 18843 df-mulg 18973 df-subg 19028 df-cntz 19222 df-cmn 19687 df-abl 19688 df-mgp 20052 df-rng 20064 df-ur 20093 df-ring 20146 df-cring 20147 df-subrng 20454 df-subrg 20478 df-abv 20717 df-lmod 20788 df-scaf 20789 df-sra 21100 df-rgmod 21101 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-fbas 21281 df-fg 21282 df-cnfld 21285 df-top 22802 df-topon 22819 df-topsp 22841 df-bases 22854 df-cld 22927 df-ntr 22928 df-cls 22929 df-nei 23006 df-lp 23044 df-perf 23045 df-cn 23135 df-cnp 23136 df-haus 23223 df-tx 23470 df-hmeo 23663 df-fil 23754 df-fm 23846 df-flim 23847 df-flf 23848 df-tmd 23980 df-tgp 23981 df-tsms 24035 df-trg 24068 df-xms 24228 df-ms 24229 df-tms 24230 df-nm 24490 df-ngp 24491 df-nrg 24493 df-nlm 24494 df-ii 24790 df-cncf 24791 df-limc 25787 df-dv 25788 df-log 26485 df-esum 34031

This theorem is referenced by: (None)

W3C validator