|   | 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 5249 | . . 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 45060 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ V) | 
| 8 | 7 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑋 ∈ V) | 
| 9 | ssexg 5322 | . . . . . . 7 ⊢ ((𝐸 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐸 ∈ V) | |
| 10 | 6, 8, 9 | syl2anc 584 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐸 ∈ V) | 
| 11 | elpwg 4602 | . . . . . 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 257 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐸 ∈ 𝒫 𝑋) | 
| 14 | eqid 2736 | . . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ 𝐸) = (𝑛 ∈ 𝑍 ↦ 𝐸) | |
| 15 | 1, 13, 14 | fmptdf 7136 | . . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ 𝐸):𝑍⟶𝒫 𝑋) | 
| 16 | 1, 2, 3, 4, 5, 15 | omeiunle 46537 | . 2 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛)) ≤ (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛))))) | 
| 17 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍) | |
| 18 | 14 | fvmpt2 7026 | . . . . . . 7 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝐸 ∈ V) → ((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛) = 𝐸) | 
| 19 | 17, 10, 18 | syl2anc 584 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛) = 𝐸) | 
| 20 | 19 | eqcomd 2742 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐸 = ((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛)) | 
| 21 | 1, 20 | iuneq2df 45057 | . . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 𝐸 = ∪ 𝑛 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛)) | 
| 22 | 21 | fveq2d 6909 | . . 3 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 𝐸) = (𝑂‘∪ 𝑛 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛))) | 
| 23 | 20 | fveq2d 6909 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑂‘𝐸) = (𝑂‘((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛))) | 
| 24 | 1, 23 | mpteq2da 5239 | . . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑂‘𝐸)) = (𝑛 ∈ 𝑍 ↦ (𝑂‘((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛)))) | 
| 25 | 24 | fveq2d 6909 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘𝐸))) = (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛))))) | 
| 26 | 22, 25 | breq12d 5155 | . 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 257 | 1 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 𝐸) ≤ (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘𝐸)))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 Ⅎwnf 1782 ∈ wcel 2107 Vcvv 3479 ⊆ wss 3950 𝒫 cpw 4599 ∪ cuni 4906 ∪ ciun 4990 class class class wbr 5142 ↦ cmpt 5224 dom cdm 5684 ‘cfv 6560 ≤ cle 11297 ℤ≥cuz 12879 Σ^csumge0 46382 OutMeascome 46509 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-ac2 10504 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-oadd 8511 df-omul 8512 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-oi 9551 df-card 9980 df-acn 9983 df-ac 10157 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-z 12616 df-uz 12880 df-rp 13036 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-clim 15525 df-sum 15724 df-sumge0 46383 df-ome 46510 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |