Theorem caragendifcl 46543

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 2735	. 2 ⊢ ∪ dom 𝑂 = ∪ dom 𝑂
3		caragendifcl.s	. 2 ⊢ 𝑆 = (CaraGen‘𝑂)
4	3	caragenss 46533	. . . . . 6 ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
5	1, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ dom 𝑂)
6	5	unissd 4893	. . . 4 ⊢ (𝜑 → ∪ 𝑆 ⊆ ∪ dom 𝑂)
7	6	ssdifssd 4122	. . 3 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ⊆ ∪ dom 𝑂)
8	3	fvexi 6890	. . . . . . 7 ⊢ 𝑆 ∈ V
9	8	uniex 7735	. . . . . 6 ⊢ ∪ 𝑆 ∈ V
10		difexg 5299	. . . . . 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 4578	. . . 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 4582	. . . . . . . . 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 46540	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑆 = ∪ dom 𝑂)
19	18	eqcomd 2741	. . . . . . . . 9 ⊢ (𝜑 → ∪ dom 𝑂 = ∪ 𝑆)
20	19	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ∪ dom 𝑂 = ∪ 𝑆)
21	17, 20	sseqtrd 3995	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑎 ⊆ ∪ 𝑆)
22		difin2 4276	. . . . . . 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 4184	. . . . . . 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 2771	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∩ (∪ 𝑆 ∖ 𝐸)) = (𝑎 ∖ 𝐸))
27	26	fveq2d 6880	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∩ (∪ 𝑆 ∖ 𝐸))) = (𝑂‘(𝑎 ∖ 𝐸)))
28	21	ssdifd 4120	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ 𝐸) ⊆ (∪ 𝑆 ∖ 𝐸))
29		sscon 4118	. . . . . . . 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 4253	. . . . . . . . 9 ⊢ (𝑎 ∩ 𝐸) = (𝑎 ∖ (𝑎 ∖ 𝐸))
32	31	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∩ 𝐸) = (𝑎 ∖ (𝑎 ∖ 𝐸)))
33		eqimss2 4018	. . . . . . . 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 3969	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)) ⊆ (𝑎 ∩ 𝐸))
36		elinel1 4176	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐸) → 𝑥 ∈ 𝑎)
37		elinel2 4177	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐸) → 𝑥 ∈ 𝐸)
38		elndif 4108	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐸 → ¬ 𝑥 ∈ (∪ 𝑆 ∖ 𝐸))
39	37, 38	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐸) → ¬ 𝑥 ∈ (∪ 𝑆 ∖ 𝐸))
40	36, 39	eldifd 3937	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐸) → 𝑥 ∈ (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)))
41	40	ssriv 3962	. . . . . . 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 3976	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)) = (𝑎 ∩ 𝐸))
44	43	fveq2d 6880	. . . 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 13447	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ*
47	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑂 ∈ OutMeas)
48	17	ssdifssd 4122	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ 𝐸) ⊆ ∪ dom 𝑂)
49	47, 2, 48	omecl 46532	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∖ 𝐸)) ∈ (0[,]+∞))
50	46, 49	sselid 3956	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∖ 𝐸)) ∈ ℝ*)
51		ssinss1 4221	. . . . . . . 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 46532	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∩ 𝐸)) ∈ (0[,]+∞))
55	46, 54	sselid 3956	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∩ 𝐸)) ∈ ℝ*)
56	50, 55	xaddcomd 45351	. . 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 46524	. . . . . 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 3234	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
62	45, 56, 61	3eqtrd 2774	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ (∪ 𝑆 ∖ 𝐸))) +𝑒 (𝑂‘(𝑎 ∖ (∪ 𝑆 ∖ 𝐸)))) = (𝑂‘𝑎))
63	1, 2, 3, 15, 62	carageneld 46531	1 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  Vcvv 3459   ∖ cdif 3923   ∩ cin 3925   ⊆ wss 3926  𝒫 cpw 4575  ∪ cuni 4883  dom cdm 5654  ‘cfv 6531  (class class class)co 7405  0cc0 11129  +∞cpnf 11266  ℝ^*cxr 11268   +_𝑒 cxad 13126  [,]cicc 13365  OutMeascome 46518  CaraGenccaragen 46520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-po 5561  df-so 5562  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-xadd 13129  df-icc 13369  df-ome 46519  df-caragen 46521

This theorem is referenced by:  caragensal  46554