caragendifcl - Mathbox for Glauco Siliprandi

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Theorem caragendifcl 45216

Description: The Caratheodory's construction is closed under the complement operation. Second part of Step (b) in the proof of Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Hypotheses**
Ref	Expression
caragendifcl.o	⊢ (𝜑 → 𝑂 ∈ OutMeas)
caragendifcl.s	⊢ 𝑆 = (CaraGen‘𝑂)
caragendifcl.e	⊢ (𝜑 → 𝐸 ∈ 𝑆)

**Assertion**
Ref	Expression
caragendifcl	⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)

Proof of Theorem caragendifcl Dummy variables 𝑎 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caragendifcl.o	. 2 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
2		eqid 2732	. 2 ⊢ ∪ dom 𝑂 = ∪ dom 𝑂
3		caragendifcl.s	. 2 ⊢ 𝑆 = (CaraGen‘𝑂)
4	3	caragenss 45206	. . . . . 6 ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
5	1, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ dom 𝑂)
6	5	unissd 4917	. . . 4 ⊢ (𝜑 → ∪ 𝑆 ⊆ ∪ dom 𝑂)
7	6	ssdifssd 4141	. . 3 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ⊆ ∪ dom 𝑂)
8	3	fvexi 6902	. . . . . . 7 ⊢ 𝑆 ∈ V
9	8	uniex 7727	. . . . . 6 ⊢ ∪ 𝑆 ∈ V
10		difexg 5326	. . . . . 6 ⊢ (∪ 𝑆 ∈ V → (∪ 𝑆 ∖ 𝐸) ∈ V)
11	9, 10	ax-mp 5	. . . . 5 ⊢ (∪ 𝑆 ∖ 𝐸) ∈ V
12	11	a1i 11	. . . 4 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ V)
13		elpwg 4604	. . . 4 ⊢ ((∪ 𝑆 ∖ 𝐸) ∈ V → ((∪ 𝑆 ∖ 𝐸) ∈ 𝒫 ∪ dom 𝑂 ↔ (∪ 𝑆 ∖ 𝐸) ⊆ ∪ dom 𝑂))
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → ((∪ 𝑆 ∖ 𝐸) ∈ 𝒫 ∪ dom 𝑂 ↔ (∪ 𝑆 ∖ 𝐸) ⊆ ∪ dom 𝑂))
15	7, 14	mpbird 256	. 2 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝒫 ∪ dom 𝑂)
16		elpwi 4608	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 → 𝑎 ⊆ ∪ dom 𝑂)
17	16	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑎 ⊆ ∪ dom 𝑂)
18	1, 3	caragenuni 45213	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑆 = ∪ dom 𝑂)
19	18	eqcomd 2738	. . . . . . . . 9 ⊢ (𝜑 → ∪ dom 𝑂 = ∪ 𝑆)
20	19	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ∪ dom 𝑂 = ∪ 𝑆)
21	17, 20	sseqtrd 4021	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑎 ⊆ ∪ 𝑆)
22		difin2 4290	. . . . . . 7 ⊢ (𝑎 ⊆ ∪ 𝑆 → (𝑎 ∖ 𝐸) = ((∪ 𝑆 ∖ 𝐸) ∩ 𝑎))
23	21, 22	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ 𝐸) = ((∪ 𝑆 ∖ 𝐸) ∩ 𝑎))
24		incom 4200	. . . . . . 7 ⊢ ((∪ 𝑆 ∖ 𝐸) ∩ 𝑎) = (𝑎 ∩ (∪ 𝑆 ∖ 𝐸))
25	24	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((∪ 𝑆 ∖ 𝐸) ∩ 𝑎) = (𝑎 ∩ (∪ 𝑆 ∖ 𝐸)))
26	23, 25	eqtr2d 2773	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∩ (∪ 𝑆 ∖ 𝐸)) = (𝑎 ∖ 𝐸))
27	26	fveq2d 6892	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∩ (∪ 𝑆 ∖ 𝐸))) = (𝑂‘(𝑎 ∖ 𝐸)))
28	21	ssdifd 4139	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ 𝐸) ⊆ (∪ 𝑆 ∖ 𝐸))
29		sscon 4137	. . . . . . . 8 ⊢ ((𝑎 ∖ 𝐸) ⊆ (∪ 𝑆 ∖ 𝐸) → (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)) ⊆ (𝑎 ∖ (𝑎 ∖ 𝐸)))
30	28, 29	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)) ⊆ (𝑎 ∖ (𝑎 ∖ 𝐸)))
31		dfin4 4266	. . . . . . . . 9 ⊢ (𝑎 ∩ 𝐸) = (𝑎 ∖ (𝑎 ∖ 𝐸))
32	31	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∩ 𝐸) = (𝑎 ∖ (𝑎 ∖ 𝐸)))
33		eqimss2 4040	. . . . . . . 8 ⊢ ((𝑎 ∩ 𝐸) = (𝑎 ∖ (𝑎 ∖ 𝐸)) → (𝑎 ∖ (𝑎 ∖ 𝐸)) ⊆ (𝑎 ∩ 𝐸))
34	32, 33	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ (𝑎 ∖ 𝐸)) ⊆ (𝑎 ∩ 𝐸))
35	30, 34	sstrd 3991	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)) ⊆ (𝑎 ∩ 𝐸))
36		elinel1 4194	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐸) → 𝑥 ∈ 𝑎)
37		elinel2 4195	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐸) → 𝑥 ∈ 𝐸)
38		elndif 4127	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐸 → ¬ 𝑥 ∈ (∪ 𝑆 ∖ 𝐸))
39	37, 38	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐸) → ¬ 𝑥 ∈ (∪ 𝑆 ∖ 𝐸))
40	36, 39	eldifd 3958	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐸) → 𝑥 ∈ (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)))
41	40	ssriv 3985	. . . . . . 7 ⊢ (𝑎 ∩ 𝐸) ⊆ (𝑎 ∖ (∪ 𝑆 ∖ 𝐸))
42	41	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∩ 𝐸) ⊆ (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)))
43	35, 42	eqssd 3998	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)) = (𝑎 ∩ 𝐸))
44	43	fveq2d 6892	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∖ (∪ 𝑆 ∖ 𝐸))) = (𝑂‘(𝑎 ∩ 𝐸)))
45	27, 44	oveq12d 7423	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ (∪ 𝑆 ∖ 𝐸))) +_𝑒 (𝑂‘(𝑎 ∖ (∪ 𝑆 ∖ 𝐸)))) = ((𝑂‘(𝑎 ∖ 𝐸)) +_𝑒 (𝑂‘(𝑎 ∩ 𝐸))))
46		iccssxr 13403	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
47	1	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑂 ∈ OutMeas)
48	17	ssdifssd 4141	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ 𝐸) ⊆ ∪ dom 𝑂)
49	47, 2, 48	omecl 45205	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∖ 𝐸)) ∈ (0[,]+∞))
50	46, 49	sselid 3979	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∖ 𝐸)) ∈ ℝ^*)
51		ssinss1 4236	. . . . . . . 8 ⊢ (𝑎 ⊆ ∪ dom 𝑂 → (𝑎 ∩ 𝐸) ⊆ ∪ dom 𝑂)
52	16, 51	syl 17	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 → (𝑎 ∩ 𝐸) ⊆ ∪ dom 𝑂)
53	52	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∩ 𝐸) ⊆ ∪ dom 𝑂)
54	47, 2, 53	omecl 45205	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∩ 𝐸)) ∈ (0[,]+∞))
55	46, 54	sselid 3979	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∩ 𝐸)) ∈ ℝ^*)
56	50, 55	xaddcomd 44020	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∖ 𝐸)) +_𝑒 (𝑂‘(𝑎 ∩ 𝐸))) = ((𝑂‘(𝑎 ∩ 𝐸)) +_𝑒 (𝑂‘(𝑎 ∖ 𝐸))))
57		caragendifcl.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑆)
58	1, 3	caragenel 45197	. . . . . 6 ⊢ (𝜑 → (𝐸 ∈ 𝑆 ↔ (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +_𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))))
59	57, 58	mpbid 231	. . . . 5 ⊢ (𝜑 → (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +_𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))
60	59	simprd 496	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +_𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
61	60	r19.21bi 3248	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ 𝐸)) +_𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
62	45, 56, 61	3eqtrd 2776	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ (∪ 𝑆 ∖ 𝐸))) +_𝑒 (𝑂‘(𝑎 ∖ (∪ 𝑆 ∖ 𝐸)))) = (𝑂‘𝑎))
63	1, 2, 3, 15, 62	carageneld 45204	1 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ∖ cdif 3944 ∩ cin 3946 ⊆ wss 3947 𝒫 cpw 4601 ∪ cuni 4907 dom cdm 5675 ‘cfv 6540 (class class class)co 7405 0cc0 11106 +∞cpnf 11241 ℝ^*cxr 11243 +_𝑒 cxad 13086 [,]cicc 13323 OutMeascome 45191 CaraGenccaragen 45193

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-xadd 13089 df-icc 13327 df-ome 45192 df-caragen 45194

This theorem is referenced by: caragensal 45227

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