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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 5134 | . . . 4 ⊢ (𝜑 → 𝒫 𝑋 ∈ V) |
4 | ssexg 5084 | . . . 4 ⊢ ((𝑌 ⊆ 𝒫 𝑋 ∧ 𝒫 𝑋 ∈ V) → 𝑌 ∈ V) | |
5 | 1, 3, 4 | syl2anc 576 | . . 3 ⊢ (𝜑 → 𝑌 ∈ V) |
6 | simpr 477 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌) | |
7 | uniiun 4849 | . . 3 ⊢ ∪ 𝑌 = ∪ 𝑦 ∈ 𝑌 𝑦 | |
8 | psmeasurelem.h | . . . 4 ⊢ (𝜑 → 𝐻:𝑋⟶(0[,]+∞)) | |
9 | elpwg 4431 | . . . . . . . 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 249 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝒫 𝒫 𝑋) |
12 | pwpwuni 40739 | . . . . . . 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 224 | . . . . 5 ⊢ (𝜑 → ∪ 𝑌 ∈ 𝒫 𝑋) |
15 | 14 | elpwid 4435 | . . . 4 ⊢ (𝜑 → ∪ 𝑌 ⊆ 𝑋) |
16 | 8, 15 | fssresd 6376 | . . 3 ⊢ (𝜑 → (𝐻 ↾ ∪ 𝑌):∪ 𝑌⟶(0[,]+∞)) |
17 | psmeasurelem.dj | . . 3 ⊢ (𝜑 → Disj 𝑦 ∈ 𝑌 𝑦) | |
18 | 5, 6, 7, 16, 17 | sge0iun 42133 | . 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 5691 | . . . 4 ⊢ (𝑥 = ∪ 𝑌 → (𝐻 ↾ 𝑥) = (𝐻 ↾ ∪ 𝑌)) | |
21 | 20 | fveq2d 6505 | . . 3 ⊢ (𝑥 = ∪ 𝑌 → (Σ^‘(𝐻 ↾ 𝑥)) = (Σ^‘(𝐻 ↾ ∪ 𝑌))) |
22 | fvexd 6516 | . . 3 ⊢ (𝜑 → (Σ^‘(𝐻 ↾ ∪ 𝑌)) ∈ V) | |
23 | 19, 21, 14, 22 | fvmptd3 6619 | . 2 ⊢ (𝜑 → (𝑀‘∪ 𝑌) = (Σ^‘(𝐻 ↾ ∪ 𝑌))) |
24 | psmeasurelem.mf | . . . . . 6 ⊢ (𝜑 → 𝑀:𝒫 𝑋⟶(0[,]+∞)) | |
25 | 24, 1 | fssresd 6376 | . . . . 5 ⊢ (𝜑 → (𝑀 ↾ 𝑌):𝑌⟶(0[,]+∞)) |
26 | 25 | feqmptd 6564 | . . . 4 ⊢ (𝜑 → (𝑀 ↾ 𝑌) = (𝑦 ∈ 𝑌 ↦ ((𝑀 ↾ 𝑌)‘𝑦))) |
27 | fvres 6520 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑌 → ((𝑀 ↾ 𝑌)‘𝑦) = (𝑀‘𝑦)) | |
28 | 6, 27 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ((𝑀 ↾ 𝑌)‘𝑦) = (𝑀‘𝑦)) |
29 | reseq2 5691 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐻 ↾ 𝑥) = (𝐻 ↾ 𝑦)) | |
30 | 29 | fveq2d 6505 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (Σ^‘(𝐻 ↾ 𝑥)) = (Σ^‘(𝐻 ↾ 𝑦))) |
31 | 1 | sselda 3860 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝒫 𝑋) |
32 | fvexd 6516 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (Σ^‘(𝐻 ↾ 𝑦)) ∈ V) | |
33 | 19, 30, 31, 32 | fvmptd3 6619 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑀‘𝑦) = (Σ^‘(𝐻 ↾ 𝑦))) |
34 | elssuni 4742 | . . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑌 → 𝑦 ⊆ ∪ 𝑌) | |
35 | resabs1 5730 | . . . . . . . . . 10 ⊢ (𝑦 ⊆ ∪ 𝑌 → ((𝐻 ↾ ∪ 𝑌) ↾ 𝑦) = (𝐻 ↾ 𝑦)) | |
36 | 34, 35 | syl 17 | . . . . . . . . 9 ⊢ (𝑦 ∈ 𝑌 → ((𝐻 ↾ ∪ 𝑌) ↾ 𝑦) = (𝐻 ↾ 𝑦)) |
37 | 36 | eqcomd 2784 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝑌 → (𝐻 ↾ 𝑦) = ((𝐻 ↾ ∪ 𝑌) ↾ 𝑦)) |
38 | 37 | adantl 474 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐻 ↾ 𝑦) = ((𝐻 ↾ ∪ 𝑌) ↾ 𝑦)) |
39 | 38 | fveq2d 6505 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (Σ^‘(𝐻 ↾ 𝑦)) = (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦))) |
40 | 28, 33, 39 | 3eqtrd 2818 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ((𝑀 ↾ 𝑌)‘𝑦) = (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦))) |
41 | 40 | mpteq2dva 5023 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ ((𝑀 ↾ 𝑌)‘𝑦)) = (𝑦 ∈ 𝑌 ↦ (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦)))) |
42 | 26, 41 | eqtrd 2814 | . . 3 ⊢ (𝜑 → (𝑀 ↾ 𝑌) = (𝑦 ∈ 𝑌 ↦ (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦)))) |
43 | 42 | fveq2d 6505 | . 2 ⊢ (𝜑 → (Σ^‘(𝑀 ↾ 𝑌)) = (Σ^‘(𝑦 ∈ 𝑌 ↦ (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦))))) |
44 | 18, 23, 43 | 3eqtr4d 2824 | 1 ⊢ (𝜑 → (𝑀‘∪ 𝑌) = (Σ^‘(𝑀 ↾ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 Vcvv 3415 ⊆ wss 3831 𝒫 cpw 4423 ∪ cuni 4713 Disj wdisj 4898 ↦ cmpt 5009 ↾ cres 5410 ⟶wf 6186 ‘cfv 6190 (class class class)co 6978 0cc0 10337 +∞cpnf 10473 [,]cicc 12560 Σ^csumge0 42076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-inf2 8900 ax-ac2 9685 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-pre-sup 10415 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-disj 4899 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-se 5368 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-isom 6199 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-oadd 7911 df-er 8091 df-map 8210 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-sup 8703 df-oi 8771 df-card 9164 df-acn 9167 df-ac 9338 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-div 11101 df-nn 11442 df-2 11506 df-3 11507 df-n0 11711 df-z 11797 df-uz 12062 df-rp 12208 df-xadd 12328 df-ico 12563 df-icc 12564 df-fz 12712 df-fzo 12853 df-seq 13188 df-exp 13248 df-hash 13509 df-cj 14322 df-re 14323 df-im 14324 df-sqrt 14458 df-abs 14459 df-clim 14709 df-sum 14907 df-sumge0 42077 |
This theorem is referenced by: psmeasure 42185 |
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