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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 5245 | . . . 4 ⊢ (𝜑 → 𝒫 𝑋 ∈ V) |
4 | ssexg 5191 | . . . 4 ⊢ ((𝑌 ⊆ 𝒫 𝑋 ∧ 𝒫 𝑋 ∈ V) → 𝑌 ∈ V) | |
5 | 1, 3, 4 | syl2anc 587 | . . 3 ⊢ (𝜑 → 𝑌 ∈ V) |
6 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌) | |
7 | uniiun 4945 | . . 3 ⊢ ∪ 𝑌 = ∪ 𝑦 ∈ 𝑌 𝑦 | |
8 | psmeasurelem.h | . . . 4 ⊢ (𝜑 → 𝐻:𝑋⟶(0[,]+∞)) | |
9 | elpwg 4500 | . . . . . . . 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 260 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝒫 𝒫 𝑋) |
12 | pwpwuni 41691 | . . . . . . 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 235 | . . . . 5 ⊢ (𝜑 → ∪ 𝑌 ∈ 𝒫 𝑋) |
15 | 14 | elpwid 4508 | . . . 4 ⊢ (𝜑 → ∪ 𝑌 ⊆ 𝑋) |
16 | 8, 15 | fssresd 6519 | . . 3 ⊢ (𝜑 → (𝐻 ↾ ∪ 𝑌):∪ 𝑌⟶(0[,]+∞)) |
17 | psmeasurelem.dj | . . 3 ⊢ (𝜑 → Disj 𝑦 ∈ 𝑌 𝑦) | |
18 | 5, 6, 7, 16, 17 | sge0iun 43058 | . 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 5813 | . . . 4 ⊢ (𝑥 = ∪ 𝑌 → (𝐻 ↾ 𝑥) = (𝐻 ↾ ∪ 𝑌)) | |
21 | 20 | fveq2d 6649 | . . 3 ⊢ (𝑥 = ∪ 𝑌 → (Σ^‘(𝐻 ↾ 𝑥)) = (Σ^‘(𝐻 ↾ ∪ 𝑌))) |
22 | fvexd 6660 | . . 3 ⊢ (𝜑 → (Σ^‘(𝐻 ↾ ∪ 𝑌)) ∈ V) | |
23 | 19, 21, 14, 22 | fvmptd3 6768 | . 2 ⊢ (𝜑 → (𝑀‘∪ 𝑌) = (Σ^‘(𝐻 ↾ ∪ 𝑌))) |
24 | psmeasurelem.mf | . . . . . 6 ⊢ (𝜑 → 𝑀:𝒫 𝑋⟶(0[,]+∞)) | |
25 | 24, 1 | fssresd 6519 | . . . . 5 ⊢ (𝜑 → (𝑀 ↾ 𝑌):𝑌⟶(0[,]+∞)) |
26 | 25 | feqmptd 6708 | . . . 4 ⊢ (𝜑 → (𝑀 ↾ 𝑌) = (𝑦 ∈ 𝑌 ↦ ((𝑀 ↾ 𝑌)‘𝑦))) |
27 | fvres 6664 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑌 → ((𝑀 ↾ 𝑌)‘𝑦) = (𝑀‘𝑦)) | |
28 | 6, 27 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ((𝑀 ↾ 𝑌)‘𝑦) = (𝑀‘𝑦)) |
29 | reseq2 5813 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐻 ↾ 𝑥) = (𝐻 ↾ 𝑦)) | |
30 | 29 | fveq2d 6649 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (Σ^‘(𝐻 ↾ 𝑥)) = (Σ^‘(𝐻 ↾ 𝑦))) |
31 | 1 | sselda 3915 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝒫 𝑋) |
32 | fvexd 6660 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (Σ^‘(𝐻 ↾ 𝑦)) ∈ V) | |
33 | 19, 30, 31, 32 | fvmptd3 6768 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑀‘𝑦) = (Σ^‘(𝐻 ↾ 𝑦))) |
34 | elssuni 4830 | . . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑌 → 𝑦 ⊆ ∪ 𝑌) | |
35 | resabs1 5848 | . . . . . . . . . 10 ⊢ (𝑦 ⊆ ∪ 𝑌 → ((𝐻 ↾ ∪ 𝑌) ↾ 𝑦) = (𝐻 ↾ 𝑦)) | |
36 | 34, 35 | syl 17 | . . . . . . . . 9 ⊢ (𝑦 ∈ 𝑌 → ((𝐻 ↾ ∪ 𝑌) ↾ 𝑦) = (𝐻 ↾ 𝑦)) |
37 | 36 | eqcomd 2804 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝑌 → (𝐻 ↾ 𝑦) = ((𝐻 ↾ ∪ 𝑌) ↾ 𝑦)) |
38 | 37 | adantl 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐻 ↾ 𝑦) = ((𝐻 ↾ ∪ 𝑌) ↾ 𝑦)) |
39 | 38 | fveq2d 6649 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (Σ^‘(𝐻 ↾ 𝑦)) = (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦))) |
40 | 28, 33, 39 | 3eqtrd 2837 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ((𝑀 ↾ 𝑌)‘𝑦) = (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦))) |
41 | 40 | mpteq2dva 5125 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ ((𝑀 ↾ 𝑌)‘𝑦)) = (𝑦 ∈ 𝑌 ↦ (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦)))) |
42 | 26, 41 | eqtrd 2833 | . . 3 ⊢ (𝜑 → (𝑀 ↾ 𝑌) = (𝑦 ∈ 𝑌 ↦ (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦)))) |
43 | 42 | fveq2d 6649 | . 2 ⊢ (𝜑 → (Σ^‘(𝑀 ↾ 𝑌)) = (Σ^‘(𝑦 ∈ 𝑌 ↦ (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦))))) |
44 | 18, 23, 43 | 3eqtr4d 2843 | 1 ⊢ (𝜑 → (𝑀‘∪ 𝑌) = (Σ^‘(𝑀 ↾ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 𝒫 cpw 4497 ∪ cuni 4800 Disj wdisj 4995 ↦ cmpt 5110 ↾ cres 5521 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 0cc0 10526 +∞cpnf 10661 [,]cicc 12729 Σ^csumge0 43001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-ac2 9874 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-oi 8958 df-card 9352 df-acn 9355 df-ac 9527 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 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 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-sumge0 43002 |
This theorem is referenced by: psmeasure 43110 |
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