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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 | pwexg 4955 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ V) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝜑 → 𝒫 𝑋 ∈ V) |
5 | ssexg 4912 | . . . 4 ⊢ ((𝑌 ⊆ 𝒫 𝑋 ∧ 𝒫 𝑋 ∈ V) → 𝑌 ∈ V) | |
6 | 1, 4, 5 | syl2anc 696 | . . 3 ⊢ (𝜑 → 𝑌 ∈ V) |
7 | simpr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌) | |
8 | uniiun 4681 | . . 3 ⊢ ∪ 𝑌 = ∪ 𝑦 ∈ 𝑌 𝑦 | |
9 | psmeasurelem.h | . . . 4 ⊢ (𝜑 → 𝐻:𝑋⟶(0[,]+∞)) | |
10 | elpwg 4274 | . . . . . . . 8 ⊢ (𝑌 ∈ V → (𝑌 ∈ 𝒫 𝒫 𝑋 ↔ 𝑌 ⊆ 𝒫 𝑋)) | |
11 | 6, 10 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑌 ∈ 𝒫 𝒫 𝑋 ↔ 𝑌 ⊆ 𝒫 𝑋)) |
12 | 1, 11 | mpbird 247 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝒫 𝒫 𝑋) |
13 | pwpwuni 39641 | . . . . . . 7 ⊢ (𝑌 ∈ V → (𝑌 ∈ 𝒫 𝒫 𝑋 ↔ ∪ 𝑌 ∈ 𝒫 𝑋)) | |
14 | 6, 13 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑌 ∈ 𝒫 𝒫 𝑋 ↔ ∪ 𝑌 ∈ 𝒫 𝑋)) |
15 | 12, 14 | mpbid 222 | . . . . 5 ⊢ (𝜑 → ∪ 𝑌 ∈ 𝒫 𝑋) |
16 | 15 | elpwid 4278 | . . . 4 ⊢ (𝜑 → ∪ 𝑌 ⊆ 𝑋) |
17 | 9, 16 | fssresd 6184 | . . 3 ⊢ (𝜑 → (𝐻 ↾ ∪ 𝑌):∪ 𝑌⟶(0[,]+∞)) |
18 | psmeasurelem.dj | . . 3 ⊢ (𝜑 → Disj 𝑦 ∈ 𝑌 𝑦) | |
19 | 6, 7, 8, 17, 18 | sge0iun 41056 | . 2 ⊢ (𝜑 → (Σ^‘(𝐻 ↾ ∪ 𝑌)) = (Σ^‘(𝑦 ∈ 𝑌 ↦ (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦))))) |
20 | psmeasurelem.m | . . . 4 ⊢ 𝑀 = (𝑥 ∈ 𝒫 𝑋 ↦ (Σ^‘(𝐻 ↾ 𝑥))) | |
21 | 20 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑀 = (𝑥 ∈ 𝒫 𝑋 ↦ (Σ^‘(𝐻 ↾ 𝑥)))) |
22 | reseq2 5498 | . . . . 5 ⊢ (𝑥 = ∪ 𝑌 → (𝐻 ↾ 𝑥) = (𝐻 ↾ ∪ 𝑌)) | |
23 | 22 | fveq2d 6308 | . . . 4 ⊢ (𝑥 = ∪ 𝑌 → (Σ^‘(𝐻 ↾ 𝑥)) = (Σ^‘(𝐻 ↾ ∪ 𝑌))) |
24 | 23 | adantl 473 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = ∪ 𝑌) → (Σ^‘(𝐻 ↾ 𝑥)) = (Σ^‘(𝐻 ↾ ∪ 𝑌))) |
25 | fvexd 6316 | . . 3 ⊢ (𝜑 → (Σ^‘(𝐻 ↾ ∪ 𝑌)) ∈ V) | |
26 | 21, 24, 15, 25 | fvmptd 6402 | . 2 ⊢ (𝜑 → (𝑀‘∪ 𝑌) = (Σ^‘(𝐻 ↾ ∪ 𝑌))) |
27 | psmeasurelem.mf | . . . . . 6 ⊢ (𝜑 → 𝑀:𝒫 𝑋⟶(0[,]+∞)) | |
28 | 27, 1 | fssresd 6184 | . . . . 5 ⊢ (𝜑 → (𝑀 ↾ 𝑌):𝑌⟶(0[,]+∞)) |
29 | 28 | feqmptd 6363 | . . . 4 ⊢ (𝜑 → (𝑀 ↾ 𝑌) = (𝑦 ∈ 𝑌 ↦ ((𝑀 ↾ 𝑌)‘𝑦))) |
30 | fvres 6320 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑌 → ((𝑀 ↾ 𝑌)‘𝑦) = (𝑀‘𝑦)) | |
31 | 7, 30 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ((𝑀 ↾ 𝑌)‘𝑦) = (𝑀‘𝑦)) |
32 | 20 | a1i 11 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑀 = (𝑥 ∈ 𝒫 𝑋 ↦ (Σ^‘(𝐻 ↾ 𝑥)))) |
33 | reseq2 5498 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐻 ↾ 𝑥) = (𝐻 ↾ 𝑦)) | |
34 | 33 | fveq2d 6308 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (Σ^‘(𝐻 ↾ 𝑥)) = (Σ^‘(𝐻 ↾ 𝑦))) |
35 | 34 | adantl 473 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑥 = 𝑦) → (Σ^‘(𝐻 ↾ 𝑥)) = (Σ^‘(𝐻 ↾ 𝑦))) |
36 | 1 | sselda 3709 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝒫 𝑋) |
37 | fvexd 6316 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (Σ^‘(𝐻 ↾ 𝑦)) ∈ V) | |
38 | 32, 35, 36, 37 | fvmptd 6402 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑀‘𝑦) = (Σ^‘(𝐻 ↾ 𝑦))) |
39 | elssuni 4575 | . . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑌 → 𝑦 ⊆ ∪ 𝑌) | |
40 | resabs1 5537 | . . . . . . . . . 10 ⊢ (𝑦 ⊆ ∪ 𝑌 → ((𝐻 ↾ ∪ 𝑌) ↾ 𝑦) = (𝐻 ↾ 𝑦)) | |
41 | 39, 40 | syl 17 | . . . . . . . . 9 ⊢ (𝑦 ∈ 𝑌 → ((𝐻 ↾ ∪ 𝑌) ↾ 𝑦) = (𝐻 ↾ 𝑦)) |
42 | 41 | eqcomd 2730 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝑌 → (𝐻 ↾ 𝑦) = ((𝐻 ↾ ∪ 𝑌) ↾ 𝑦)) |
43 | 42 | adantl 473 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐻 ↾ 𝑦) = ((𝐻 ↾ ∪ 𝑌) ↾ 𝑦)) |
44 | 43 | fveq2d 6308 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (Σ^‘(𝐻 ↾ 𝑦)) = (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦))) |
45 | 31, 38, 44 | 3eqtrd 2762 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ((𝑀 ↾ 𝑌)‘𝑦) = (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦))) |
46 | 45 | mpteq2dva 4852 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ ((𝑀 ↾ 𝑌)‘𝑦)) = (𝑦 ∈ 𝑌 ↦ (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦)))) |
47 | 29, 46 | eqtrd 2758 | . . 3 ⊢ (𝜑 → (𝑀 ↾ 𝑌) = (𝑦 ∈ 𝑌 ↦ (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦)))) |
48 | 47 | fveq2d 6308 | . 2 ⊢ (𝜑 → (Σ^‘(𝑀 ↾ 𝑌)) = (Σ^‘(𝑦 ∈ 𝑌 ↦ (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦))))) |
49 | 19, 26, 48 | 3eqtr4d 2768 | 1 ⊢ (𝜑 → (𝑀‘∪ 𝑌) = (Σ^‘(𝑀 ↾ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1596 ∈ wcel 2103 Vcvv 3304 ⊆ wss 3680 𝒫 cpw 4266 ∪ cuni 4544 Disj wdisj 4728 ↦ cmpt 4837 ↾ cres 5220 ⟶wf 5997 ‘cfv 6001 (class class class)co 6765 0cc0 10049 +∞cpnf 10184 [,]cicc 12292 Σ^csumge0 40999 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-8 2105 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 ax-rep 4879 ax-sep 4889 ax-nul 4897 ax-pow 4948 ax-pr 5011 ax-un 7066 ax-inf2 8651 ax-ac2 9398 ax-cnex 10105 ax-resscn 10106 ax-1cn 10107 ax-icn 10108 ax-addcl 10109 ax-addrcl 10110 ax-mulcl 10111 ax-mulrcl 10112 ax-mulcom 10113 ax-addass 10114 ax-mulass 10115 ax-distr 10116 ax-i2m1 10117 ax-1ne0 10118 ax-1rid 10119 ax-rnegex 10120 ax-rrecex 10121 ax-cnre 10122 ax-pre-lttri 10123 ax-pre-lttrn 10124 ax-pre-ltadd 10125 ax-pre-mulgt0 10126 ax-pre-sup 10127 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1599 df-fal 1602 df-ex 1818 df-nf 1823 df-sb 2011 df-eu 2575 df-mo 2576 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-ne 2897 df-nel 3000 df-ral 3019 df-rex 3020 df-reu 3021 df-rmo 3022 df-rab 3023 df-v 3306 df-sbc 3542 df-csb 3640 df-dif 3683 df-un 3685 df-in 3687 df-ss 3694 df-pss 3696 df-nul 4024 df-if 4195 df-pw 4268 df-sn 4286 df-pr 4288 df-tp 4290 df-op 4292 df-uni 4545 df-int 4584 df-iun 4630 df-disj 4729 df-br 4761 df-opab 4821 df-mpt 4838 df-tr 4861 df-id 5128 df-eprel 5133 df-po 5139 df-so 5140 df-fr 5177 df-se 5178 df-we 5179 df-xp 5224 df-rel 5225 df-cnv 5226 df-co 5227 df-dm 5228 df-rn 5229 df-res 5230 df-ima 5231 df-pred 5793 df-ord 5839 df-on 5840 df-lim 5841 df-suc 5842 df-iota 5964 df-fun 6003 df-fn 6004 df-f 6005 df-f1 6006 df-fo 6007 df-f1o 6008 df-fv 6009 df-isom 6010 df-riota 6726 df-ov 6768 df-oprab 6769 df-mpt2 6770 df-om 7183 df-1st 7285 df-2nd 7286 df-wrecs 7527 df-recs 7588 df-rdg 7626 df-1o 7680 df-oadd 7684 df-er 7862 df-map 7976 df-en 8073 df-dom 8074 df-sdom 8075 df-fin 8076 df-sup 8464 df-oi 8531 df-card 8878 df-acn 8881 df-ac 9052 df-pnf 10189 df-mnf 10190 df-xr 10191 df-ltxr 10192 df-le 10193 df-sub 10381 df-neg 10382 df-div 10798 df-nn 11134 df-2 11192 df-3 11193 df-n0 11406 df-z 11491 df-uz 11801 df-rp 11947 df-xadd 12061 df-ico 12295 df-icc 12296 df-fz 12441 df-fzo 12581 df-seq 12917 df-exp 12976 df-hash 13233 df-cj 13959 df-re 13960 df-im 13961 df-sqrt 14095 df-abs 14096 df-clim 14339 df-sum 14537 df-sumge0 41000 |
This theorem is referenced by: psmeasure 41108 |
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