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Mirrors > Home > MPE Home > Th. List > Mathboxes > caragenel2d | Structured version Visualization version GIF version |
Description: Membership in the Caratheodory's construction. Similar to carageneld 42804, but here "less then or equal to" is used, instead of equality. This is Remark 113D of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
caragenel2d.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
caragenel2d.x | ⊢ 𝑋 = ∪ dom 𝑂 |
caragenel2d.s | ⊢ 𝑆 = (CaraGen‘𝑂) |
caragenel2d.e | ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑋) |
caragenel2d.a | ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) ≤ (𝑂‘𝑎)) |
Ref | Expression |
---|---|
caragenel2d | ⊢ (𝜑 → 𝐸 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caragenel2d.o | . 2 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
2 | caragenel2d.x | . 2 ⊢ 𝑋 = ∪ dom 𝑂 | |
3 | caragenel2d.s | . 2 ⊢ 𝑆 = (CaraGen‘𝑂) | |
4 | caragenel2d.e | . 2 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑋) | |
5 | 1 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑂 ∈ OutMeas) |
6 | inss1 4205 | . . . . . . 7 ⊢ (𝑎 ∩ 𝐸) ⊆ 𝑎 | |
7 | elpwi 4548 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋) | |
8 | 6, 7 | sstrid 3978 | . . . . . 6 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∩ 𝐸) ⊆ 𝑋) |
9 | 8 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑎 ∩ 𝐸) ⊆ 𝑋) |
10 | 5, 2, 9 | omexrcl 42809 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∩ 𝐸)) ∈ ℝ*) |
11 | 7 | ssdifssd 4119 | . . . . . 6 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∖ 𝐸) ⊆ 𝑋) |
12 | 11 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑎 ∖ 𝐸) ⊆ 𝑋) |
13 | 5, 2, 12 | omexrcl 42809 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∖ 𝐸)) ∈ ℝ*) |
14 | xaddcl 12633 | . . . 4 ⊢ (((𝑂‘(𝑎 ∩ 𝐸)) ∈ ℝ* ∧ (𝑂‘(𝑎 ∖ 𝐸)) ∈ ℝ*) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) ∈ ℝ*) | |
15 | 10, 13, 14 | syl2anc 586 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) ∈ ℝ*) |
16 | 7 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎 ⊆ 𝑋) |
17 | 5, 2, 16 | omexrcl 42809 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝑎) ∈ ℝ*) |
18 | caragenel2d.a | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) ≤ (𝑂‘𝑎)) | |
19 | 5, 2, 16 | omelesplit 42820 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝑎) ≤ ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸)))) |
20 | 15, 17, 18, 19 | xrletrid 12549 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)) |
21 | 1, 2, 3, 4, 20 | carageneld 42804 | 1 ⊢ (𝜑 → 𝐸 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4838 class class class wbr 5066 dom cdm 5555 ‘cfv 6355 (class class class)co 7156 ℝ*cxr 10674 ≤ cle 10676 +𝑒 cxad 12506 OutMeascome 42791 CaraGenccaragen 42793 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-xadd 12509 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 df-sumge0 42665 df-ome 42792 df-caragen 42794 |
This theorem is referenced by: caragencmpl 42837 hspmbl 42931 |
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