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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 44990 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ V) |
6 | 5, 4 | ssexd 5330 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ V) |
7 | elpwg 4608 | . . . 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 4246 | . . . . . . 7 ⊢ (𝑎 ∩ 𝐸) ⊆ 𝐸 | |
15 | 14 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑎 ∩ 𝐸) ⊆ 𝐸) |
16 | 10, 2, 11, 13, 15 | omess0 46490 | . . . . 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 4147 | . . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∖ 𝐸) ⊆ 𝑎) | |
19 | elpwi 4612 | . . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋) | |
20 | 18, 19 | sstrd 4006 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∖ 𝐸) ⊆ 𝑋) |
21 | 20 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑎 ∖ 𝐸) ⊆ 𝑋) |
22 | 10, 2, 21 | omexrcl 46463 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∖ 𝐸)) ∈ ℝ*) |
23 | xaddlid 13281 | . . . . 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 2775 | . . 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 46454 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∖ 𝐸)) ≤ (𝑂‘𝑎)) |
29 | 25, 28 | eqbrtrd 5170 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) ≤ (𝑂‘𝑎)) |
30 | 1, 2, 3, 9, 29 | caragenel2d 46488 | 1 ⊢ (𝜑 → 𝐸 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∖ cdif 3960 ∩ cin 3962 ⊆ wss 3963 𝒫 cpw 4605 ∪ cuni 4912 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 0cc0 11153 ℝ*cxr 11292 ≤ cle 11294 +𝑒 cxad 13150 OutMeascome 46445 CaraGenccaragen 46447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-xadd 13153 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-sumge0 46319 df-ome 46446 df-caragen 46448 |
This theorem is referenced by: voncmpl 46577 |
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