Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > psmeasurelem | Structured version Visualization version GIF version |
Description: 𝑀 applied to a disjoint union of subsets of its domain is the sum of 𝑀 applied to such subset. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
psmeasurelem.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
psmeasurelem.h | ⊢ (𝜑 → 𝐻:𝑋⟶(0[,]+∞)) |
psmeasurelem.m | ⊢ 𝑀 = (𝑥 ∈ 𝒫 𝑋 ↦ (Σ^‘(𝐻 ↾ 𝑥))) |
psmeasurelem.mf | ⊢ (𝜑 → 𝑀:𝒫 𝑋⟶(0[,]+∞)) |
psmeasurelem.y | ⊢ (𝜑 → 𝑌 ⊆ 𝒫 𝑋) |
psmeasurelem.dj | ⊢ (𝜑 → Disj 𝑦 ∈ 𝑌 𝑦) |
Ref | Expression |
---|---|
psmeasurelem | ⊢ (𝜑 → (𝑀‘∪ 𝑌) = (Σ^‘(𝑀 ↾ 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psmeasurelem.y | . . . 4 ⊢ (𝜑 → 𝑌 ⊆ 𝒫 𝑋) | |
2 | psmeasurelem.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
3 | 2 | pwexd 5271 | . . . 4 ⊢ (𝜑 → 𝒫 𝑋 ∈ V) |
4 | ssexg 5218 | . . . 4 ⊢ ((𝑌 ⊆ 𝒫 𝑋 ∧ 𝒫 𝑋 ∈ V) → 𝑌 ∈ V) | |
5 | 1, 3, 4 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝑌 ∈ V) |
6 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌) | |
7 | uniiun 4973 | . . 3 ⊢ ∪ 𝑌 = ∪ 𝑦 ∈ 𝑌 𝑦 | |
8 | psmeasurelem.h | . . . 4 ⊢ (𝜑 → 𝐻:𝑋⟶(0[,]+∞)) | |
9 | elpwg 4541 | . . . . . . . 8 ⊢ (𝑌 ∈ V → (𝑌 ∈ 𝒫 𝒫 𝑋 ↔ 𝑌 ⊆ 𝒫 𝑋)) | |
10 | 5, 9 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑌 ∈ 𝒫 𝒫 𝑋 ↔ 𝑌 ⊆ 𝒫 𝑋)) |
11 | 1, 10 | mpbird 258 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝒫 𝒫 𝑋) |
12 | pwpwuni 41196 | . . . . . . 7 ⊢ (𝑌 ∈ V → (𝑌 ∈ 𝒫 𝒫 𝑋 ↔ ∪ 𝑌 ∈ 𝒫 𝑋)) | |
13 | 5, 12 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑌 ∈ 𝒫 𝒫 𝑋 ↔ ∪ 𝑌 ∈ 𝒫 𝑋)) |
14 | 11, 13 | mpbid 233 | . . . . 5 ⊢ (𝜑 → ∪ 𝑌 ∈ 𝒫 𝑋) |
15 | 14 | elpwid 4549 | . . . 4 ⊢ (𝜑 → ∪ 𝑌 ⊆ 𝑋) |
16 | 8, 15 | fssresd 6538 | . . 3 ⊢ (𝜑 → (𝐻 ↾ ∪ 𝑌):∪ 𝑌⟶(0[,]+∞)) |
17 | psmeasurelem.dj | . . 3 ⊢ (𝜑 → Disj 𝑦 ∈ 𝑌 𝑦) | |
18 | 5, 6, 7, 16, 17 | sge0iun 42578 | . 2 ⊢ (𝜑 → (Σ^‘(𝐻 ↾ ∪ 𝑌)) = (Σ^‘(𝑦 ∈ 𝑌 ↦ (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦))))) |
19 | psmeasurelem.m | . . 3 ⊢ 𝑀 = (𝑥 ∈ 𝒫 𝑋 ↦ (Σ^‘(𝐻 ↾ 𝑥))) | |
20 | reseq2 5841 | . . . 4 ⊢ (𝑥 = ∪ 𝑌 → (𝐻 ↾ 𝑥) = (𝐻 ↾ ∪ 𝑌)) | |
21 | 20 | fveq2d 6667 | . . 3 ⊢ (𝑥 = ∪ 𝑌 → (Σ^‘(𝐻 ↾ 𝑥)) = (Σ^‘(𝐻 ↾ ∪ 𝑌))) |
22 | fvexd 6678 | . . 3 ⊢ (𝜑 → (Σ^‘(𝐻 ↾ ∪ 𝑌)) ∈ V) | |
23 | 19, 21, 14, 22 | fvmptd3 6783 | . 2 ⊢ (𝜑 → (𝑀‘∪ 𝑌) = (Σ^‘(𝐻 ↾ ∪ 𝑌))) |
24 | psmeasurelem.mf | . . . . . 6 ⊢ (𝜑 → 𝑀:𝒫 𝑋⟶(0[,]+∞)) | |
25 | 24, 1 | fssresd 6538 | . . . . 5 ⊢ (𝜑 → (𝑀 ↾ 𝑌):𝑌⟶(0[,]+∞)) |
26 | 25 | feqmptd 6726 | . . . 4 ⊢ (𝜑 → (𝑀 ↾ 𝑌) = (𝑦 ∈ 𝑌 ↦ ((𝑀 ↾ 𝑌)‘𝑦))) |
27 | fvres 6682 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑌 → ((𝑀 ↾ 𝑌)‘𝑦) = (𝑀‘𝑦)) | |
28 | 6, 27 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ((𝑀 ↾ 𝑌)‘𝑦) = (𝑀‘𝑦)) |
29 | reseq2 5841 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐻 ↾ 𝑥) = (𝐻 ↾ 𝑦)) | |
30 | 29 | fveq2d 6667 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (Σ^‘(𝐻 ↾ 𝑥)) = (Σ^‘(𝐻 ↾ 𝑦))) |
31 | 1 | sselda 3964 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝒫 𝑋) |
32 | fvexd 6678 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (Σ^‘(𝐻 ↾ 𝑦)) ∈ V) | |
33 | 19, 30, 31, 32 | fvmptd3 6783 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑀‘𝑦) = (Σ^‘(𝐻 ↾ 𝑦))) |
34 | elssuni 4859 | . . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑌 → 𝑦 ⊆ ∪ 𝑌) | |
35 | resabs1 5876 | . . . . . . . . . 10 ⊢ (𝑦 ⊆ ∪ 𝑌 → ((𝐻 ↾ ∪ 𝑌) ↾ 𝑦) = (𝐻 ↾ 𝑦)) | |
36 | 34, 35 | syl 17 | . . . . . . . . 9 ⊢ (𝑦 ∈ 𝑌 → ((𝐻 ↾ ∪ 𝑌) ↾ 𝑦) = (𝐻 ↾ 𝑦)) |
37 | 36 | eqcomd 2824 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝑌 → (𝐻 ↾ 𝑦) = ((𝐻 ↾ ∪ 𝑌) ↾ 𝑦)) |
38 | 37 | adantl 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐻 ↾ 𝑦) = ((𝐻 ↾ ∪ 𝑌) ↾ 𝑦)) |
39 | 38 | fveq2d 6667 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (Σ^‘(𝐻 ↾ 𝑦)) = (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦))) |
40 | 28, 33, 39 | 3eqtrd 2857 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ((𝑀 ↾ 𝑌)‘𝑦) = (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦))) |
41 | 40 | mpteq2dva 5152 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ ((𝑀 ↾ 𝑌)‘𝑦)) = (𝑦 ∈ 𝑌 ↦ (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦)))) |
42 | 26, 41 | eqtrd 2853 | . . 3 ⊢ (𝜑 → (𝑀 ↾ 𝑌) = (𝑦 ∈ 𝑌 ↦ (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦)))) |
43 | 42 | fveq2d 6667 | . 2 ⊢ (𝜑 → (Σ^‘(𝑀 ↾ 𝑌)) = (Σ^‘(𝑦 ∈ 𝑌 ↦ (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦))))) |
44 | 18, 23, 43 | 3eqtr4d 2863 | 1 ⊢ (𝜑 → (𝑀‘∪ 𝑌) = (Σ^‘(𝑀 ↾ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 𝒫 cpw 4535 ∪ cuni 4830 Disj wdisj 5022 ↦ cmpt 5137 ↾ cres 5550 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 0cc0 10525 +∞cpnf 10660 [,]cicc 12729 Σ^csumge0 42521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-ac2 9873 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-disj 5023 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-oi 8962 df-card 9356 df-acn 9359 df-ac 9530 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-xadd 12496 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-sum 15031 df-sumge0 42522 |
This theorem is referenced by: psmeasure 42630 |
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