Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > omeiunlempt | Structured version Visualization version GIF version |
Description: The outer measure of the indexed union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
omeiunlempt.nph | ⊢ Ⅎ𝑛𝜑 |
omeiunlempt.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
omeiunlempt.x | ⊢ 𝑋 = ∪ dom 𝑂 |
omeiunlempt.z | ⊢ 𝑍 = (ℤ≥‘𝑁) |
omeiunlempt.e | ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐸 ⊆ 𝑋) |
Ref | Expression |
---|---|
omeiunlempt | ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 𝐸) ≤ (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘𝐸)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omeiunlempt.nph | . . 3 ⊢ Ⅎ𝑛𝜑 | |
2 | nfmpt1 5166 | . . 3 ⊢ Ⅎ𝑛(𝑛 ∈ 𝑍 ↦ 𝐸) | |
3 | omeiunlempt.o | . . 3 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
4 | omeiunlempt.x | . . 3 ⊢ 𝑋 = ∪ dom 𝑂 | |
5 | omeiunlempt.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑁) | |
6 | omeiunlempt.e | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐸 ⊆ 𝑋) | |
7 | 3, 4 | unidmex 41319 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ V) |
8 | 7 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑋 ∈ V) |
9 | ssexg 5229 | . . . . . . 7 ⊢ ((𝐸 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐸 ∈ V) | |
10 | 6, 8, 9 | syl2anc 586 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐸 ∈ V) |
11 | elpwg 4544 | . . . . . 6 ⊢ (𝐸 ∈ V → (𝐸 ∈ 𝒫 𝑋 ↔ 𝐸 ⊆ 𝑋)) | |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸 ∈ 𝒫 𝑋 ↔ 𝐸 ⊆ 𝑋)) |
13 | 6, 12 | mpbird 259 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐸 ∈ 𝒫 𝑋) |
14 | eqid 2823 | . . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ 𝐸) = (𝑛 ∈ 𝑍 ↦ 𝐸) | |
15 | 1, 13, 14 | fmptdf 6883 | . . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ 𝐸):𝑍⟶𝒫 𝑋) |
16 | 1, 2, 3, 4, 5, 15 | omeiunle 42806 | . 2 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛)) ≤ (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛))))) |
17 | simpr 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍) | |
18 | 14 | fvmpt2 6781 | . . . . . . 7 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝐸 ∈ V) → ((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛) = 𝐸) |
19 | 17, 10, 18 | syl2anc 586 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛) = 𝐸) |
20 | 19 | eqcomd 2829 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐸 = ((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛)) |
21 | 1, 20 | iuneq2df 41315 | . . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 𝐸 = ∪ 𝑛 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛)) |
22 | 21 | fveq2d 6676 | . . 3 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 𝐸) = (𝑂‘∪ 𝑛 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛))) |
23 | 20 | fveq2d 6676 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑂‘𝐸) = (𝑂‘((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛))) |
24 | 1, 23 | mpteq2da 5162 | . . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑂‘𝐸)) = (𝑛 ∈ 𝑍 ↦ (𝑂‘((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛)))) |
25 | 24 | fveq2d 6676 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘𝐸))) = (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛))))) |
26 | 22, 25 | breq12d 5081 | . 2 ⊢ (𝜑 → ((𝑂‘∪ 𝑛 ∈ 𝑍 𝐸) ≤ (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘𝐸))) ↔ (𝑂‘∪ 𝑛 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛)) ≤ (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛)))))) |
27 | 16, 26 | mpbird 259 | 1 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 𝐸) ≤ (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘𝐸)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 𝒫 cpw 4541 ∪ cuni 4840 ∪ ciun 4921 class class class wbr 5068 ↦ cmpt 5148 dom cdm 5557 ‘cfv 6357 ≤ cle 10678 ℤ≥cuz 12246 Σ^csumge0 42651 OutMeascome 42778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-ac2 9887 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-omul 8109 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-oi 8976 df-card 9370 df-acn 9373 df-ac 9544 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-sumge0 42652 df-ome 42779 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |