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Mathbox for Glauco Siliprandi |
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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 4675 | . . 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 38242 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ V) |
8 | 7 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑋 ∈ V) |
9 | ssexg 4732 | . . . . . . 7 ⊢ ((𝐸 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐸 ∈ V) | |
10 | 6, 8, 9 | syl2anc 691 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐸 ∈ V) |
11 | elpwg 4116 | . . . . . 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 246 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐸 ∈ 𝒫 𝑋) |
14 | eqid 2610 | . . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ 𝐸) = (𝑛 ∈ 𝑍 ↦ 𝐸) | |
15 | 1, 13, 14 | fmptdf 6294 | . . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ 𝐸):𝑍⟶𝒫 𝑋) |
16 | 1, 2, 3, 4, 5, 15 | omeiunle 39407 | . 2 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛)) ≤ (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛))))) |
17 | simpr 476 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍) | |
18 | 14 | fvmpt2 6200 | . . . . . . 7 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝐸 ∈ V) → ((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛) = 𝐸) |
19 | 17, 10, 18 | syl2anc 691 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛) = 𝐸) |
20 | 19 | eqcomd 2616 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐸 = ((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛)) |
21 | 1, 20 | iuneq2df 38237 | . . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 𝐸 = ∪ 𝑛 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛)) |
22 | 21 | fveq2d 6107 | . . 3 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 𝐸) = (𝑂‘∪ 𝑛 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛))) |
23 | 20 | fveq2d 6107 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑂‘𝐸) = (𝑂‘((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛))) |
24 | 1, 23 | mpteq2da 4671 | . . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑂‘𝐸)) = (𝑛 ∈ 𝑍 ↦ (𝑂‘((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛)))) |
25 | 24 | fveq2d 6107 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘𝐸))) = (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘((𝑛 ∈ 𝑍 ↦ 𝐸)‘𝑛))))) |
26 | 22, 25 | breq12d 4596 | . 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 246 | 1 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 𝐸) ≤ (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘𝐸)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 ∪ ciun 4455 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ‘cfv 5804 ≤ cle 9954 ℤ≥cuz 11563 Σ^csumge0 39255 OutMeascome 39379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-ac2 9168 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-acn 8651 df-ac 8822 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-sumge0 39256 df-ome 39380 |
This theorem is referenced by: (None) |
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