Theorem caragendifcl 46872

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 2737	. 2 ⊢ ∪ dom 𝑂 = ∪ dom 𝑂
3		caragendifcl.s	. 2 ⊢ 𝑆 = (CaraGen‘𝑂)
4	3	caragenss 46862	. . . . . 6 ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
5	1, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ dom 𝑂)
6	5	unissd 4875	. . . 4 ⊢ (𝜑 → ∪ 𝑆 ⊆ ∪ dom 𝑂)
7	6	ssdifssd 4101	. . 3 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ⊆ ∪ dom 𝑂)
8	3	fvexi 6856	. . . . . . 7 ⊢ 𝑆 ∈ V
9	8	uniex 7696	. . . . . 6 ⊢ ∪ 𝑆 ∈ V
10		difexg 5276	. . . . . 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 4559	. . . 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 257	. 2 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝒫 ∪ dom 𝑂)
16		elpwi 4563	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 → 𝑎 ⊆ ∪ dom 𝑂)
17	16	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑎 ⊆ ∪ dom 𝑂)
18	1, 3	caragenuni 46869	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑆 = ∪ dom 𝑂)
19	18	eqcomd 2743	. . . . . . . . 9 ⊢ (𝜑 → ∪ dom 𝑂 = ∪ 𝑆)
20	19	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ∪ dom 𝑂 = ∪ 𝑆)
21	17, 20	sseqtrd 3972	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑎 ⊆ ∪ 𝑆)
22		difin2 4255	. . . . . . 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 4163	. . . . . . 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 6846	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∩ (∪ 𝑆 ∖ 𝐸))) = (𝑂‘(𝑎 ∖ 𝐸)))
28	21	ssdifd 4099	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ 𝐸) ⊆ (∪ 𝑆 ∖ 𝐸))
29		sscon 4097	. . . . . . . 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 4232	. . . . . . . . 9 ⊢ (𝑎 ∩ 𝐸) = (𝑎 ∖ (𝑎 ∖ 𝐸))
32	31	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∩ 𝐸) = (𝑎 ∖ (𝑎 ∖ 𝐸)))
33		eqimss2 3995	. . . . . . . 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 3946	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)) ⊆ (𝑎 ∩ 𝐸))
36		elinel1 4155	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐸) → 𝑥 ∈ 𝑎)
37		elinel2 4156	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐸) → 𝑥 ∈ 𝐸)
38		elndif 4087	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐸 → ¬ 𝑥 ∈ (∪ 𝑆 ∖ 𝐸))
39	37, 38	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐸) → ¬ 𝑥 ∈ (∪ 𝑆 ∖ 𝐸))
40	36, 39	eldifd 3914	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐸) → 𝑥 ∈ (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)))
41	40	ssriv 3939	. . . . . . 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 3953	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)) = (𝑎 ∩ 𝐸))
44	43	fveq2d 6846	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∖ (∪ 𝑆 ∖ 𝐸))) = (𝑂‘(𝑎 ∩ 𝐸)))
45	27, 44	oveq12d 7386	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ (∪ 𝑆 ∖ 𝐸))) +𝑒 (𝑂‘(𝑎 ∖ (∪ 𝑆 ∖ 𝐸)))) = ((𝑂‘(𝑎 ∖ 𝐸)) +𝑒 (𝑂‘(𝑎 ∩ 𝐸))))
46		iccssxr 13358	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ*
47	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑂 ∈ OutMeas)
48	17	ssdifssd 4101	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ 𝐸) ⊆ ∪ dom 𝑂)
49	47, 2, 48	omecl 46861	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∖ 𝐸)) ∈ (0[,]+∞))
50	46, 49	sselid 3933	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∖ 𝐸)) ∈ ℝ*)
51		ssinss1 4200	. . . . . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∩ 𝐸) ⊆ ∪ dom 𝑂)
54	47, 2, 53	omecl 46861	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∩ 𝐸)) ∈ (0[,]+∞))
55	46, 54	sselid 3933	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∩ 𝐸)) ∈ ℝ*)
56	50, 55	xaddcomd 45683	. . 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 46853	. . . . . 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 232	. . . . 5 ⊢ (𝜑 → (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))
60	59	simprd 495	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
61	60	r19.21bi 3230	. . 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 46860	1 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3442   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4556  ∪ cuni 4865  dom cdm 5632  ‘cfv 6500  (class class class)co 7368  0cc0 11038  +∞cpnf 11175  ℝ^*cxr 11177   +_𝑒 cxad 13036  [,]cicc 13276  OutMeascome 46847  CaraGenccaragen 46849

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-po 5540  df-so 5541  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-xadd 13039  df-icc 13280  df-ome 46848  df-caragen 46850

This theorem is referenced by:  caragensal  46883