caragenel2d - Mathbox for Glauco Siliprandi

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Theorem caragenel2d 45922

Description: Membership in the Caratheodory's construction. Similar to carageneld 45892, 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.)

**Hypotheses**
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))

**Assertion**
Ref	Expression
caragenel2d	⊢ (𝜑 → 𝐸 ∈ 𝑆)

Distinct variable groups: 𝐸,𝑎 𝑂,𝑎 𝜑,𝑎 Allowed substitution hints: 𝑆(𝑎) 𝑋(𝑎)

**Proof of Theorem caragenel2d**
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 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑂 ∈ OutMeas)
6		inss1 4229	. . . . . . 7 ⊢ (𝑎 ∩ 𝐸) ⊆ 𝑎
7		elpwi 4611	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋)
8	6, 7	sstrid 3991	. . . . . 6 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∩ 𝐸) ⊆ 𝑋)
9	8	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑎 ∩ 𝐸) ⊆ 𝑋)
10	5, 2, 9	omexrcl 45897	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∩ 𝐸)) ∈ ℝ^*)
11	7	ssdifssd 4141	. . . . . 6 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∖ 𝐸) ⊆ 𝑋)
12	11	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑎 ∖ 𝐸) ⊆ 𝑋)
13	5, 2, 12	omexrcl 45897	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘(𝑎 ∖ 𝐸)) ∈ ℝ^*)
14		xaddcl 13256	. . . 4 ⊢ (((𝑂‘(𝑎 ∩ 𝐸)) ∈ ℝ^* ∧ (𝑂‘(𝑎 ∖ 𝐸)) ∈ ℝ^) → ((𝑂‘(𝑎 ∩ 𝐸)) +_𝑒 (𝑂‘(𝑎 ∖ 𝐸))) ∈ ℝ^)
15	10, 13, 14	syl2anc 582	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +_𝑒 (𝑂‘(𝑎 ∖ 𝐸))) ∈ ℝ^*)
16	7	adantl 480	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎 ⊆ 𝑋)
17	5, 2, 16	omexrcl 45897	. . 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 45908	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑂‘𝑎) ≤ ((𝑂‘(𝑎 ∩ 𝐸)) +_𝑒 (𝑂‘(𝑎 ∖ 𝐸))))
20	15, 17, 18, 19	xrletrid 13172	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +_𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
21	1, 2, 3, 4, 20	carageneld 45892	1 ⊢ (𝜑 → 𝐸 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∖ cdif 3944 ∩ cin 3946 ⊆ wss 3947 𝒫 cpw 4604 ∪ cuni 4910 class class class wbr 5150 dom cdm 5680 ‘cfv 6551 (class class class)co 7424 ℝ^*cxr 11283 ≤ cle 11285 +_𝑒 cxad 13128 OutMeascome 45879 CaraGenccaragen 45881

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-inf2 9670 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-sup 9471 df-oi 9539 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-n0 12509 df-z 12595 df-uz 12859 df-rp 13013 df-xadd 13131 df-ico 13368 df-icc 13369 df-fz 13523 df-fzo 13666 df-seq 14005 df-exp 14065 df-hash 14328 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-clim 15470 df-sum 15671 df-sumge0 45753 df-ome 45880 df-caragen 45882

This theorem is referenced by: caragencmpl 45925 hspmbl 46019

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