caragendifcl - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > caragendifcl		Structured version Visualization version GIF version

Theorem caragendifcl 41252

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 2760	. 2 ⊢ ∪ dom 𝑂 = ∪ dom 𝑂
3		caragendifcl.s	. 2 ⊢ 𝑆 = (CaraGen‘𝑂)
4	3	caragenss 41242	. . . . . 6 ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
5	1, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ dom 𝑂)
6	5	unissd 4614	. . . 4 ⊢ (𝜑 → ∪ 𝑆 ⊆ ∪ dom 𝑂)
7	6	ssdifssd 3891	. . 3 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ⊆ ∪ dom 𝑂)
8		fvex 6363	. . . . . . . 8 ⊢ (CaraGen‘𝑂) ∈ V
9	3, 8	eqeltri 2835	. . . . . . 7 ⊢ 𝑆 ∈ V
10	9	uniex 7119	. . . . . 6 ⊢ ∪ 𝑆 ∈ V
11		difexg 4960	. . . . . 6 ⊢ (∪ 𝑆 ∈ V → (∪ 𝑆 ∖ 𝐸) ∈ V)
12	10, 11	ax-mp 5	. . . . 5 ⊢ (∪ 𝑆 ∖ 𝐸) ∈ V
13	12	a1i 11	. . . 4 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ V)
14		elpwg 4310	. . . 4 ⊢ ((∪ 𝑆 ∖ 𝐸) ∈ V → ((∪ 𝑆 ∖ 𝐸) ∈ 𝒫 ∪ dom 𝑂 ↔ (∪ 𝑆 ∖ 𝐸) ⊆ ∪ dom 𝑂))
15	13, 14	syl 17	. . 3 ⊢ (𝜑 → ((∪ 𝑆 ∖ 𝐸) ∈ 𝒫 ∪ dom 𝑂 ↔ (∪ 𝑆 ∖ 𝐸) ⊆ ∪ dom 𝑂))
16	7, 15	mpbird 247	. 2 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝒫 ∪ dom 𝑂)
17		elpwi 4312	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 → 𝑎 ⊆ ∪ dom 𝑂)
18	17	adantl 473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑎 ⊆ ∪ dom 𝑂)
19	1, 3	caragenuni 41249	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑆 = ∪ dom 𝑂)
20	19	eqcomd 2766	. . . . . . . . 9 ⊢ (𝜑 → ∪ dom 𝑂 = ∪ 𝑆)
21	20	adantr 472	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ∪ dom 𝑂 = ∪ 𝑆)
22	18, 21	sseqtrd 3782	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑎 ⊆ ∪ 𝑆)
23		difin2 4033	. . . . . . 7 ⊢ (𝑎 ⊆ ∪ 𝑆 → (𝑎 ∖ 𝐸) = ((∪ 𝑆 ∖ 𝐸) ∩ 𝑎))
24	22, 23	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ 𝐸) = ((∪ 𝑆 ∖ 𝐸) ∩ 𝑎))
25		incom 3948	. . . . . . 7 ⊢ ((∪ 𝑆 ∖ 𝐸) ∩ 𝑎) = (𝑎 ∩ (∪ 𝑆 ∖ 𝐸))
26	25	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((∪ 𝑆 ∖ 𝐸) ∩ 𝑎) = (𝑎 ∩ (∪ 𝑆 ∖ 𝐸)))
27	24, 26	eqtr2d 2795	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∩ (∪ 𝑆 ∖ 𝐸)) = (𝑎 ∖ 𝐸))
28	27	fveq2d 6357	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∩ (∪ 𝑆 ∖ 𝐸))) = (𝑂‘(𝑎 ∖ 𝐸)))
29	22	ssdifd 3889	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ 𝐸) ⊆ (∪ 𝑆 ∖ 𝐸))
30		sscon 3887	. . . . . . . 8 ⊢ ((𝑎 ∖ 𝐸) ⊆ (∪ 𝑆 ∖ 𝐸) → (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)) ⊆ (𝑎 ∖ (𝑎 ∖ 𝐸)))
31	29, 30	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)) ⊆ (𝑎 ∖ (𝑎 ∖ 𝐸)))
32		dfin4 4010	. . . . . . . . 9 ⊢ (𝑎 ∩ 𝐸) = (𝑎 ∖ (𝑎 ∖ 𝐸))
33	32	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∩ 𝐸) = (𝑎 ∖ (𝑎 ∖ 𝐸)))
34		eqimss2 3799	. . . . . . . 8 ⊢ ((𝑎 ∩ 𝐸) = (𝑎 ∖ (𝑎 ∖ 𝐸)) → (𝑎 ∖ (𝑎 ∖ 𝐸)) ⊆ (𝑎 ∩ 𝐸))
35	33, 34	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ (𝑎 ∖ 𝐸)) ⊆ (𝑎 ∩ 𝐸))
36	31, 35	sstrd 3754	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)) ⊆ (𝑎 ∩ 𝐸))
37		elinel1 3942	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐸) → 𝑥 ∈ 𝑎)
38		elinel2 3943	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐸) → 𝑥 ∈ 𝐸)
39		elndif 3877	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐸 → ¬ 𝑥 ∈ (∪ 𝑆 ∖ 𝐸))
40	38, 39	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐸) → ¬ 𝑥 ∈ (∪ 𝑆 ∖ 𝐸))
41	37, 40	eldifd 3726	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐸) → 𝑥 ∈ (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)))
42	41	ssriv 3748	. . . . . . 7 ⊢ (𝑎 ∩ 𝐸) ⊆ (𝑎 ∖ (∪ 𝑆 ∖ 𝐸))
43	42	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∩ 𝐸) ⊆ (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)))
44	36, 43	eqssd 3761	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)) = (𝑎 ∩ 𝐸))
45	44	fveq2d 6357	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∖ (∪ 𝑆 ∖ 𝐸))) = (𝑂‘(𝑎 ∩ 𝐸)))
46	28, 45	oveq12d 6832	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ (∪ 𝑆 ∖ 𝐸))) +_𝑒 (𝑂‘(𝑎 ∖ (∪ 𝑆 ∖ 𝐸)))) = ((𝑂‘(𝑎 ∖ 𝐸)) +_𝑒 (𝑂‘(𝑎 ∩ 𝐸))))
47		iccssxr 12469	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
48	1	adantr 472	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑂 ∈ OutMeas)
49	18	ssdifssd 3891	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ 𝐸) ⊆ ∪ dom 𝑂)
50	48, 2, 49	omecl 41241	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∖ 𝐸)) ∈ (0[,]+∞))
51	47, 50	sseldi 3742	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∖ 𝐸)) ∈ ℝ^*)
52		ssinss1 3984	. . . . . . . 8 ⊢ (𝑎 ⊆ ∪ dom 𝑂 → (𝑎 ∩ 𝐸) ⊆ ∪ dom 𝑂)
53	17, 52	syl 17	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 → (𝑎 ∩ 𝐸) ⊆ ∪ dom 𝑂)
54	53	adantl 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∩ 𝐸) ⊆ ∪ dom 𝑂)
55	48, 2, 54	omecl 41241	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∩ 𝐸)) ∈ (0[,]+∞))
56	47, 55	sseldi 3742	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∩ 𝐸)) ∈ ℝ^*)
57	51, 56	xaddcomd 40056	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∖ 𝐸)) +_𝑒 (𝑂‘(𝑎 ∩ 𝐸))) = ((𝑂‘(𝑎 ∩ 𝐸)) +_𝑒 (𝑂‘(𝑎 ∖ 𝐸))))
58		caragendifcl.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑆)
59	1, 3	caragenel 41233	. . . . . 6 ⊢ (𝜑 → (𝐸 ∈ 𝑆 ↔ (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +_𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))))
60	58, 59	mpbid 222	. . . . 5 ⊢ (𝜑 → (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +_𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))
61	60	simprd 482	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +_𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
62	61	r19.21bi 3070	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ 𝐸)) +_𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
63	46, 57, 62	3eqtrd 2798	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ (∪ 𝑆 ∖ 𝐸))) +_𝑒 (𝑂‘(𝑎 ∖ (∪ 𝑆 ∖ 𝐸)))) = (𝑂‘𝑎))
64	1, 2, 3, 16, 63	carageneld 41240	1 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ∀wral 3050 Vcvv 3340 ∖ cdif 3712 ∩ cin 3714 ⊆ wss 3715 𝒫 cpw 4302 ∪ cuni 4588 dom cdm 5266 ‘cfv 6049 (class class class)co 6814 0cc0 10148 +∞cpnf 10283 ℝ^*cxr 10285 +_𝑒 cxad 12157 [,]cicc 12391 OutMeascome 41227 CaraGenccaragen 41229

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-po 5187 df-so 5188 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-1st 7334 df-2nd 7335 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-xadd 12160 df-icc 12395 df-ome 41228 df-caragen 41230

This theorem is referenced by: caragensal 41263

W3C validator