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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pmeasadd | Structured version Visualization version GIF version |
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.) |
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) |
Ref | Expression |
---|---|
pmeasadd | ⊢ (𝜑 → (𝑃‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ*𝑘 ∈ 𝐴(𝑃‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmeassubadd.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑅) | |
2 | 1 | ralrimiva 3145 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑅) |
3 | dfiun3g 5580 | . . . 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 6413 | . 2 ⊢ (𝜑 → (𝑃‘∪ 𝑘 ∈ 𝐴 𝐵) = (𝑃‘∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵))) |
6 | pmeassubadd.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ≼ ω) | |
7 | mptct 9646 | . . . . . 6 ⊢ (𝐴 ≼ ω → (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω) | |
8 | rnct 9633 | . . . . . 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 2797 | . . . . . . 7 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
11 | 10 | rnmptss 6616 | . . . . . 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 29907 | . . . . . 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 1159 | . . . 4 ⊢ (𝜑 → (ran (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω ∧ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑅 ∧ Disj 𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝑦)) |
17 | 16 | ancli 545 | . . 3 ⊢ (𝜑 → (𝜑 ∧ (ran (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω ∧ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑅 ∧ Disj 𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝑦))) |
18 | ctex 8208 | . . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | |
19 | mptexg 6711 | . . . . 5 ⊢ (𝐴 ∈ V → (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V) | |
20 | 6, 18, 19 | 3syl 18 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V) |
21 | rnexg 7330 | . . . 4 ⊢ ((𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V → ran (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V) | |
22 | breq1 4844 | . . . . . . . 8 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → (𝑥 ≼ ω ↔ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ≼ ω)) | |
23 | sseq1 3820 | . . . . . . . 8 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → (𝑥 ⊆ 𝑅 ↔ ran (𝑘 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑅)) | |
24 | disjeq1 4816 | . . . . . . . 8 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → (Disj 𝑦 ∈ 𝑥 𝑦 ↔ Disj 𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝑦)) | |
25 | 22, 23, 24 | 3anbi123d 1561 | . . . . . . 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 623 | . . . . . 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 4634 | . . . . . . . 8 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → ∪ 𝑥 = ∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
28 | 27 | fveq2d 6413 | . . . . . . 7 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → (𝑃‘∪ 𝑥) = (𝑃‘∪ ran (𝑘 ∈ 𝐴 ↦ 𝐵))) |
29 | esumeq1 30604 | . . . . . . 7 ⊢ (𝑥 = ran (𝑘 ∈ 𝐴 ↦ 𝐵) → Σ*𝑦 ∈ 𝑥(𝑃‘𝑦) = Σ*𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)(𝑃‘𝑦)) | |
30 | 28, 29 | eqeq12d 2812 | . . . . . 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 336 | . . . . 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 3451 | . . . 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 6409 | . . 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 6584 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑃‘𝐵) ∈ (0[,]+∞)) |
41 | fveq2 6409 | . . . . 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 718 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → (𝑃‘∅) = 0) |
45 | 42, 44 | eqtrd 2831 | . . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = ∅) → (𝑃‘𝐵) = 0) |
46 | 36, 37, 40, 1, 45, 13 | esumrnmpt2 30638 | . 2 ⊢ (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)(𝑃‘𝑦) = Σ*𝑘 ∈ 𝐴(𝑃‘𝐵)) |
47 | 5, 35, 46 | 3eqtrd 2835 | 1 ⊢ (𝜑 → (𝑃‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ*𝑘 ∈ 𝐴(𝑃‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ∀wral 3087 {crab 3091 Vcvv 3383 ∖ cdif 3764 ∪ cun 3765 ⊆ wss 3767 ∅c0 4113 𝒫 cpw 4347 ∪ cuni 4626 ∪ ciun 4708 Disj wdisj 4809 class class class wbr 4841 ↦ cmpt 4920 ran crn 5311 ⟶wf 6095 ‘cfv 6099 (class class class)co 6876 ωcom 7297 ≼ cdom 8191 0cc0 10222 +∞cpnf 10358 [,]cicc 12423 Σ*cesum 30597 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-inf2 8786 ax-ac2 9571 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 ax-pre-sup 10300 ax-addf 10301 ax-mulf 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-iin 4711 df-disj 4810 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-se 5270 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-isom 6108 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-of 7129 df-om 7298 df-1st 7399 df-2nd 7400 df-supp 7531 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-2o 7798 df-oadd 7801 df-er 7980 df-map 8095 df-pm 8096 df-ixp 8147 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-fsupp 8516 df-fi 8557 df-sup 8588 df-inf 8589 df-oi 8655 df-card 9049 df-acn 9052 df-ac 9223 df-cda 9276 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-div 10975 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-7 11377 df-8 11378 df-9 11379 df-n0 11577 df-z 11663 df-dec 11780 df-uz 11927 df-q 12030 df-rp 12071 df-xneg 12189 df-xadd 12190 df-xmul 12191 df-ioo 12424 df-ioc 12425 df-ico 12426 df-icc 12427 df-fz 12577 df-fzo 12717 df-fl 12844 df-mod 12920 df-seq 13052 df-exp 13111 df-fac 13310 df-bc 13339 df-hash 13367 df-shft 14145 df-cj 14177 df-re 14178 df-im 14179 df-sqrt 14313 df-abs 14314 df-limsup 14540 df-clim 14557 df-rlim 14558 df-sum 14755 df-ef 15131 df-sin 15133 df-cos 15134 df-pi 15136 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-sets 16188 df-ress 16189 df-plusg 16277 df-mulr 16278 df-starv 16279 df-sca 16280 df-vsca 16281 df-ip 16282 df-tset 16283 df-ple 16284 df-ds 16286 df-unif 16287 df-hom 16288 df-cco 16289 df-rest 16395 df-topn 16396 df-0g 16414 df-gsum 16415 df-topgen 16416 df-pt 16417 df-prds 16420 df-ordt 16473 df-xrs 16474 df-qtop 16479 df-imas 16480 df-xps 16482 df-mre 16558 df-mrc 16559 df-acs 16561 df-ps 17512 df-tsr 17513 df-plusf 17553 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-mhm 17647 df-submnd 17648 df-grp 17738 df-minusg 17739 df-sbg 17740 df-mulg 17854 df-subg 17901 df-cntz 18059 df-cmn 18507 df-abl 18508 df-mgp 18803 df-ur 18815 df-ring 18862 df-cring 18863 df-subrg 19093 df-abv 19132 df-lmod 19180 df-scaf 19181 df-sra 19492 df-rgmod 19493 df-psmet 20057 df-xmet 20058 df-met 20059 df-bl 20060 df-mopn 20061 df-fbas 20062 df-fg 20063 df-cnfld 20066 df-top 21024 df-topon 21041 df-topsp 21063 df-bases 21076 df-cld 21149 df-ntr 21150 df-cls 21151 df-nei 21228 df-lp 21266 df-perf 21267 df-cn 21357 df-cnp 21358 df-haus 21445 df-tx 21691 df-hmeo 21884 df-fil 21975 df-fm 22067 df-flim 22068 df-flf 22069 df-tmd 22201 df-tgp 22202 df-tsms 22255 df-trg 22288 df-xms 22450 df-ms 22451 df-tms 22452 df-nm 22712 df-ngp 22713 df-nrg 22715 df-nlm 22716 df-ii 23005 df-cncf 23006 df-limc 23968 df-dv 23969 df-log 24641 df-esum 30598 |
This theorem is referenced by: (None) |
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