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| Mirrors > Home > MPE Home > Th. List > Mathboxes > caragencmpl | Structured version Visualization version GIF version | ||
| Description: A measure built with the Caratheodory's construction is complete. See Definition 112Df of [Fremlin1] p. 19. This is Exercise 113Xa of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 24-Dec-2020.) | 
| Ref | Expression | 
|---|---|
| caragencmpl.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) | 
| caragencmpl.x | ⊢ 𝑋 = ∪ dom 𝑂 | 
| caragencmpl.e | ⊢ (𝜑 → 𝐸 ⊆ 𝑋) | 
| caragencmpl.z | ⊢ (𝜑 → (𝑂‘𝐸) = 0) | 
| caragencmpl.s | ⊢ 𝑆 = (CaraGen‘𝑂) | 
| Ref | Expression | 
|---|---|
| caragencmpl | ⊢ (𝜑 → 𝐸 ∈ 𝑆) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | caragencmpl.o | . 2 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
| 2 | caragencmpl.x | . 2 ⊢ 𝑋 = ∪ dom 𝑂 | |
| 3 | caragencmpl.s | . 2 ⊢ 𝑆 = (CaraGen‘𝑂) | |
| 4 | caragencmpl.e | . . 3 ⊢ (𝜑 → 𝐸 ⊆ 𝑋) | |
| 5 | 1, 2 | unidmex 45055 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ V) | 
| 6 | 5, 4 | ssexd 5324 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ V) | 
| 7 | elpwg 4603 | . . . 4 ⊢ (𝐸 ∈ V → (𝐸 ∈ 𝒫 𝑋 ↔ 𝐸 ⊆ 𝑋)) | |
| 8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝐸 ∈ 𝒫 𝑋 ↔ 𝐸 ⊆ 𝑋)) | 
| 9 | 4, 8 | mpbird 257 | . 2 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑋) | 
| 10 | 1 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑂 ∈ OutMeas) | 
| 11 | 4 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝐸 ⊆ 𝑋) | 
| 12 | caragencmpl.z | . . . . . . 7 ⊢ (𝜑 → (𝑂‘𝐸) = 0) | |
| 13 | 12 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝐸) = 0) | 
| 14 | inss2 4238 | . . . . . . 7 ⊢ (𝑎 ∩ 𝐸) ⊆ 𝐸 | |
| 15 | 14 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑎 ∩ 𝐸) ⊆ 𝐸) | 
| 16 | 10, 2, 11, 13, 15 | omess0 46549 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∩ 𝐸)) = 0) | 
| 17 | 16 | oveq1d 7446 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (0 +𝑒 (𝑂‘(𝑎 ∖ 𝐸)))) | 
| 18 | difssd 4137 | . . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∖ 𝐸) ⊆ 𝑎) | |
| 19 | elpwi 4607 | . . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋) | |
| 20 | 18, 19 | sstrd 3994 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∖ 𝐸) ⊆ 𝑋) | 
| 21 | 20 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑎 ∖ 𝐸) ⊆ 𝑋) | 
| 22 | 10, 2, 21 | omexrcl 46522 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∖ 𝐸)) ∈ ℝ*) | 
| 23 | xaddlid 13284 | . . . . 5 ⊢ ((𝑂‘(𝑎 ∖ 𝐸)) ∈ ℝ* → (0 +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘(𝑎 ∖ 𝐸))) | |
| 24 | 22, 23 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (0 +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘(𝑎 ∖ 𝐸))) | 
| 25 | 17, 24 | eqtrd 2777 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘(𝑎 ∖ 𝐸))) | 
| 26 | 19 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎 ⊆ 𝑋) | 
| 27 | 18 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑎 ∖ 𝐸) ⊆ 𝑎) | 
| 28 | 10, 2, 26, 27 | omessle 46513 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∖ 𝐸)) ≤ (𝑂‘𝑎)) | 
| 29 | 25, 28 | eqbrtrd 5165 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) ≤ (𝑂‘𝑎)) | 
| 30 | 1, 2, 3, 9, 29 | caragenel2d 46547 | 1 ⊢ (𝜑 → 𝐸 ∈ 𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 0cc0 11155 ℝ*cxr 11294 ≤ cle 11296 +𝑒 cxad 13152 OutMeascome 46504 CaraGenccaragen 46506 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-xadd 13155 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-sumge0 46378 df-ome 46505 df-caragen 46507 | 
| This theorem is referenced by: voncmpl 46636 | 
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