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 5283 | . . . 4 ⊢ (𝜑 → 𝒫 𝑋 ∈ V) |
4 | ssexg 5230 | . . . 4 ⊢ ((𝑌 ⊆ 𝒫 𝑋 ∧ 𝒫 𝑋 ∈ V) → 𝑌 ∈ V) | |
5 | 1, 3, 4 | syl2anc 586 | . . 3 ⊢ (𝜑 → 𝑌 ∈ V) |
6 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌) | |
7 | uniiun 4985 | . . 3 ⊢ ∪ 𝑌 = ∪ 𝑦 ∈ 𝑌 𝑦 | |
8 | psmeasurelem.h | . . . 4 ⊢ (𝜑 → 𝐻:𝑋⟶(0[,]+∞)) | |
9 | elpwg 4545 | . . . . . . . 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 259 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝒫 𝒫 𝑋) |
12 | pwpwuni 41325 | . . . . . . 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 234 | . . . . 5 ⊢ (𝜑 → ∪ 𝑌 ∈ 𝒫 𝑋) |
15 | 14 | elpwid 4553 | . . . 4 ⊢ (𝜑 → ∪ 𝑌 ⊆ 𝑋) |
16 | 8, 15 | fssresd 6548 | . . 3 ⊢ (𝜑 → (𝐻 ↾ ∪ 𝑌):∪ 𝑌⟶(0[,]+∞)) |
17 | psmeasurelem.dj | . . 3 ⊢ (𝜑 → Disj 𝑦 ∈ 𝑌 𝑦) | |
18 | 5, 6, 7, 16, 17 | sge0iun 42708 | . 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 5851 | . . . 4 ⊢ (𝑥 = ∪ 𝑌 → (𝐻 ↾ 𝑥) = (𝐻 ↾ ∪ 𝑌)) | |
21 | 20 | fveq2d 6677 | . . 3 ⊢ (𝑥 = ∪ 𝑌 → (Σ^‘(𝐻 ↾ 𝑥)) = (Σ^‘(𝐻 ↾ ∪ 𝑌))) |
22 | fvexd 6688 | . . 3 ⊢ (𝜑 → (Σ^‘(𝐻 ↾ ∪ 𝑌)) ∈ V) | |
23 | 19, 21, 14, 22 | fvmptd3 6794 | . 2 ⊢ (𝜑 → (𝑀‘∪ 𝑌) = (Σ^‘(𝐻 ↾ ∪ 𝑌))) |
24 | psmeasurelem.mf | . . . . . 6 ⊢ (𝜑 → 𝑀:𝒫 𝑋⟶(0[,]+∞)) | |
25 | 24, 1 | fssresd 6548 | . . . . 5 ⊢ (𝜑 → (𝑀 ↾ 𝑌):𝑌⟶(0[,]+∞)) |
26 | 25 | feqmptd 6736 | . . . 4 ⊢ (𝜑 → (𝑀 ↾ 𝑌) = (𝑦 ∈ 𝑌 ↦ ((𝑀 ↾ 𝑌)‘𝑦))) |
27 | fvres 6692 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑌 → ((𝑀 ↾ 𝑌)‘𝑦) = (𝑀‘𝑦)) | |
28 | 6, 27 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ((𝑀 ↾ 𝑌)‘𝑦) = (𝑀‘𝑦)) |
29 | reseq2 5851 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐻 ↾ 𝑥) = (𝐻 ↾ 𝑦)) | |
30 | 29 | fveq2d 6677 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (Σ^‘(𝐻 ↾ 𝑥)) = (Σ^‘(𝐻 ↾ 𝑦))) |
31 | 1 | sselda 3970 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝒫 𝑋) |
32 | fvexd 6688 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (Σ^‘(𝐻 ↾ 𝑦)) ∈ V) | |
33 | 19, 30, 31, 32 | fvmptd3 6794 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑀‘𝑦) = (Σ^‘(𝐻 ↾ 𝑦))) |
34 | elssuni 4871 | . . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑌 → 𝑦 ⊆ ∪ 𝑌) | |
35 | resabs1 5886 | . . . . . . . . . 10 ⊢ (𝑦 ⊆ ∪ 𝑌 → ((𝐻 ↾ ∪ 𝑌) ↾ 𝑦) = (𝐻 ↾ 𝑦)) | |
36 | 34, 35 | syl 17 | . . . . . . . . 9 ⊢ (𝑦 ∈ 𝑌 → ((𝐻 ↾ ∪ 𝑌) ↾ 𝑦) = (𝐻 ↾ 𝑦)) |
37 | 36 | eqcomd 2830 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝑌 → (𝐻 ↾ 𝑦) = ((𝐻 ↾ ∪ 𝑌) ↾ 𝑦)) |
38 | 37 | adantl 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐻 ↾ 𝑦) = ((𝐻 ↾ ∪ 𝑌) ↾ 𝑦)) |
39 | 38 | fveq2d 6677 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (Σ^‘(𝐻 ↾ 𝑦)) = (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦))) |
40 | 28, 33, 39 | 3eqtrd 2863 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ((𝑀 ↾ 𝑌)‘𝑦) = (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦))) |
41 | 40 | mpteq2dva 5164 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ ((𝑀 ↾ 𝑌)‘𝑦)) = (𝑦 ∈ 𝑌 ↦ (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦)))) |
42 | 26, 41 | eqtrd 2859 | . . 3 ⊢ (𝜑 → (𝑀 ↾ 𝑌) = (𝑦 ∈ 𝑌 ↦ (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦)))) |
43 | 42 | fveq2d 6677 | . 2 ⊢ (𝜑 → (Σ^‘(𝑀 ↾ 𝑌)) = (Σ^‘(𝑦 ∈ 𝑌 ↦ (Σ^‘((𝐻 ↾ ∪ 𝑌) ↾ 𝑦))))) |
44 | 18, 23, 43 | 3eqtr4d 2869 | 1 ⊢ (𝜑 → (𝑀‘∪ 𝑌) = (Σ^‘(𝑀 ↾ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ⊆ wss 3939 𝒫 cpw 4542 ∪ cuni 4841 Disj wdisj 5034 ↦ cmpt 5149 ↾ cres 5560 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 0cc0 10540 +∞cpnf 10675 [,]cicc 12744 Σ^csumge0 42651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-ac2 9888 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-disj 5035 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-oi 8977 df-card 9371 df-acn 9374 df-ac 9545 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-xadd 12511 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-sum 15046 df-sumge0 42652 |
This theorem is referenced by: psmeasure 42760 |
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