Theorem caragendifcl 42816

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 2821	. 2 ⊢ ∪ dom 𝑂 = ∪ dom 𝑂
3		caragendifcl.s	. 2 ⊢ 𝑆 = (CaraGen‘𝑂)
4	3	caragenss 42806	. . . . . 6 ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
5	1, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ dom 𝑂)
6	5	unissd 4848	. . . 4 ⊢ (𝜑 → ∪ 𝑆 ⊆ ∪ dom 𝑂)
7	6	ssdifssd 4119	. . 3 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ⊆ ∪ dom 𝑂)
8	3	fvexi 6684	. . . . . . 7 ⊢ 𝑆 ∈ V
9	8	uniex 7467	. . . . . 6 ⊢ ∪ 𝑆 ∈ V
10		difexg 5231	. . . . . 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 4542	. . . 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 259	. 2 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝒫 ∪ dom 𝑂)
16		elpwi 4548	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 → 𝑎 ⊆ ∪ dom 𝑂)
17	16	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑎 ⊆ ∪ dom 𝑂)
18	1, 3	caragenuni 42813	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑆 = ∪ dom 𝑂)
19	18	eqcomd 2827	. . . . . . . . 9 ⊢ (𝜑 → ∪ dom 𝑂 = ∪ 𝑆)
20	19	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ∪ dom 𝑂 = ∪ 𝑆)
21	17, 20	sseqtrd 4007	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑎 ⊆ ∪ 𝑆)
22		difin2 4266	. . . . . . 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 4178	. . . . . . 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 2857	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∩ (∪ 𝑆 ∖ 𝐸)) = (𝑎 ∖ 𝐸))
27	26	fveq2d 6674	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∩ (∪ 𝑆 ∖ 𝐸))) = (𝑂‘(𝑎 ∖ 𝐸)))
28	21	ssdifd 4117	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ 𝐸) ⊆ (∪ 𝑆 ∖ 𝐸))
29		sscon 4115	. . . . . . . 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 4244	. . . . . . . . 9 ⊢ (𝑎 ∩ 𝐸) = (𝑎 ∖ (𝑎 ∖ 𝐸))
32	31	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∩ 𝐸) = (𝑎 ∖ (𝑎 ∖ 𝐸)))
33		eqimss2 4024	. . . . . . . 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 3977	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)) ⊆ (𝑎 ∩ 𝐸))
36		elinel1 4172	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐸) → 𝑥 ∈ 𝑎)
37		elinel2 4173	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐸) → 𝑥 ∈ 𝐸)
38		elndif 4105	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐸 → ¬ 𝑥 ∈ (∪ 𝑆 ∖ 𝐸))
39	37, 38	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐸) → ¬ 𝑥 ∈ (∪ 𝑆 ∖ 𝐸))
40	36, 39	eldifd 3947	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐸) → 𝑥 ∈ (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)))
41	40	ssriv 3971	. . . . . . 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 3984	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ (∪ 𝑆 ∖ 𝐸)) = (𝑎 ∩ 𝐸))
44	43	fveq2d 6674	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∖ (∪ 𝑆 ∖ 𝐸))) = (𝑂‘(𝑎 ∩ 𝐸)))
45	27, 44	oveq12d 7174	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ (∪ 𝑆 ∖ 𝐸))) +𝑒 (𝑂‘(𝑎 ∖ (∪ 𝑆 ∖ 𝐸)))) = ((𝑂‘(𝑎 ∖ 𝐸)) +𝑒 (𝑂‘(𝑎 ∩ 𝐸))))
46		iccssxr 12820	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ*
47	1	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑂 ∈ OutMeas)
48	17	ssdifssd 4119	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ 𝐸) ⊆ ∪ dom 𝑂)
49	47, 2, 48	omecl 42805	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∖ 𝐸)) ∈ (0[,]+∞))
50	46, 49	sseldi 3965	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∖ 𝐸)) ∈ ℝ*)
51		ssinss1 4214	. . . . . . . 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 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∩ 𝐸) ⊆ ∪ dom 𝑂)
54	47, 2, 53	omecl 42805	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∩ 𝐸)) ∈ (0[,]+∞))
55	46, 54	sseldi 3965	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∩ 𝐸)) ∈ ℝ*)
56	50, 55	xaddcomd 41612	. . 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 42797	. . . . . 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 234	. . . . 5 ⊢ (𝜑 → (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))
60	59	simprd 498	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
61	60	r19.21bi 3208	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
62	45, 56, 61	3eqtrd 2860	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ (∪ 𝑆 ∖ 𝐸))) +𝑒 (𝑂‘(𝑎 ∖ (∪ 𝑆 ∖ 𝐸)))) = (𝑂‘𝑎))
63	1, 2, 3, 15, 62	carageneld 42804	1 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  Vcvv 3494   ∖ cdif 3933   ∩ cin 3935   ⊆ wss 3936  𝒫 cpw 4539  ∪ cuni 4838  dom cdm 5555  ‘cfv 6355  (class class class)co 7156  0cc0 10537  +∞cpnf 10672  ℝ^*cxr 10674   +_𝑒 cxad 12506  [,]cicc 12742  OutMeascome 42791  CaraGenccaragen 42793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-po 5474  df-so 5475  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-xadd 12509  df-icc 12746  df-ome 42792  df-caragen 42794

This theorem is referenced by:  caragensal  42827