caragenuncllem - Mathbox for Glauco Siliprandi

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Theorem caragenuncllem 46609

Description: The Caratheodory's construction is closed under the union. Step (c) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Hypotheses**
Ref	Expression
caragenuncllem.o	⊢ (𝜑 → 𝑂 ∈ OutMeas)
caragenuncllem.s	⊢ 𝑆 = (CaraGen‘𝑂)
caragenuncllem.e	⊢ (𝜑 → 𝐸 ∈ 𝑆)
caragenuncllem.f	⊢ (𝜑 → 𝐹 ∈ 𝑆)
caragenuncllem.x	⊢ 𝑋 = ∪ dom 𝑂
caragenuncllem.a	⊢ (𝜑 → 𝐴 ⊆ 𝑋)

**Assertion**
Ref	Expression
caragenuncllem	⊢ (𝜑 → ((𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) +_𝑒 (𝑂‘(𝐴 ∖ (𝐸 ∪ 𝐹)))) = (𝑂‘𝐴))

**Proof of Theorem caragenuncllem**
Step	Hyp	Ref	Expression
1		caragenuncllem.o	. . . . . 6 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
2		caragenuncllem.s	. . . . . 6 ⊢ 𝑆 = (CaraGen‘𝑂)
3		caragenuncllem.x	. . . . . 6 ⊢ 𝑋 = ∪ dom 𝑂
4		caragenuncllem.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑆)
5		caragenuncllem.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
6	5	ssinss1d 4194	. . . . . 6 ⊢ (𝜑 → (𝐴 ∩ (𝐸 ∪ 𝐹)) ⊆ 𝑋)
7	1, 2, 3, 4, 6	caragensplit 46597	. . . . 5 ⊢ (𝜑 → ((𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸))) = (𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))))
8	7	eqcomd 2737	. . . 4 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) = ((𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸))))
9		inass 4175	. . . . . . . 8 ⊢ ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸) = (𝐴 ∩ ((𝐸 ∪ 𝐹) ∩ 𝐸))
10		incom 4156	. . . . . . . . . 10 ⊢ ((𝐸 ∪ 𝐹) ∩ 𝐸) = (𝐸 ∩ (𝐸 ∪ 𝐹))
11		inabs 4213	. . . . . . . . . 10 ⊢ (𝐸 ∩ (𝐸 ∪ 𝐹)) = 𝐸
12	10, 11	eqtri 2754	. . . . . . . . 9 ⊢ ((𝐸 ∪ 𝐹) ∩ 𝐸) = 𝐸
13	12	ineq2i 4164	. . . . . . . 8 ⊢ (𝐴 ∩ ((𝐸 ∪ 𝐹) ∩ 𝐸)) = (𝐴 ∩ 𝐸)
14	9, 13	eqtri 2754	. . . . . . 7 ⊢ ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸) = (𝐴 ∩ 𝐸)
15	14	fveq2i 6825	. . . . . 6 ⊢ (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) = (𝑂‘(𝐴 ∩ 𝐸))
16		incom 4156	. . . . . . . . . 10 ⊢ ((𝐴 ∖ 𝐸) ∩ 𝐹) = (𝐹 ∩ (𝐴 ∖ 𝐸))
17		indifcom 4230	. . . . . . . . . 10 ⊢ (𝐹 ∩ (𝐴 ∖ 𝐸)) = (𝐴 ∩ (𝐹 ∖ 𝐸))
18	16, 17	eqtr2i 2755	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐹 ∖ 𝐸)) = ((𝐴 ∖ 𝐸) ∩ 𝐹)
19	18	eqcomi 2740	. . . . . . . 8 ⊢ ((𝐴 ∖ 𝐸) ∩ 𝐹) = (𝐴 ∩ (𝐹 ∖ 𝐸))
20		difundir 4238	. . . . . . . . . 10 ⊢ ((𝐸 ∪ 𝐹) ∖ 𝐸) = ((𝐸 ∖ 𝐸) ∪ (𝐹 ∖ 𝐸))
21		difid 4323	. . . . . . . . . . 11 ⊢ (𝐸 ∖ 𝐸) = ∅
22	21	uneq1i 4111	. . . . . . . . . 10 ⊢ ((𝐸 ∖ 𝐸) ∪ (𝐹 ∖ 𝐸)) = (∅ ∪ (𝐹 ∖ 𝐸))
23		0un 4343	. . . . . . . . . 10 ⊢ (∅ ∪ (𝐹 ∖ 𝐸)) = (𝐹 ∖ 𝐸)
24	20, 22, 23	3eqtrri 2759	. . . . . . . . 9 ⊢ (𝐹 ∖ 𝐸) = ((𝐸 ∪ 𝐹) ∖ 𝐸)
25	24	ineq2i 4164	. . . . . . . 8 ⊢ (𝐴 ∩ (𝐹 ∖ 𝐸)) = (𝐴 ∩ ((𝐸 ∪ 𝐹) ∖ 𝐸))
26		indif2 4228	. . . . . . . 8 ⊢ (𝐴 ∩ ((𝐸 ∪ 𝐹) ∖ 𝐸)) = ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸)
27	19, 25, 26	3eqtrri 2759	. . . . . . 7 ⊢ ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸) = ((𝐴 ∖ 𝐸) ∩ 𝐹)
28	27	fveq2i 6825	. . . . . 6 ⊢ (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸)) = (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))
29	15, 28	oveq12i 7358	. . . . 5 ⊢ ((𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸))) = ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)))
30	29	a1i 11	. . . 4 ⊢ (𝜑 → ((𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸))) = ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))))
31		eqidd 2732	. . . 4 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))) = ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))))
32	8, 30, 31	3eqtrd 2770	. . 3 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) = ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))))
33		difun1 4246	. . . . 5 ⊢ (𝐴 ∖ (𝐸 ∪ 𝐹)) = ((𝐴 ∖ 𝐸) ∖ 𝐹)
34	33	fveq2i 6825	. . . 4 ⊢ (𝑂‘(𝐴 ∖ (𝐸 ∪ 𝐹))) = (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹))
35	34	a1i 11	. . 3 ⊢ (𝜑 → (𝑂‘(𝐴 ∖ (𝐸 ∪ 𝐹))) = (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)))
36	32, 35	oveq12d 7364	. 2 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) +_𝑒 (𝑂‘(𝐴 ∖ (𝐸 ∪ 𝐹)))) = (((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹))))
37	5	ssinss1d 4194	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐸) ⊆ 𝑋)
38	1, 3, 37	omexrcl 46604	. . . 4 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ 𝐸)) ∈ ℝ^*)
39	1, 3, 37	omecl 46600	. . . . 5 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ 𝐸)) ∈ (0[,]+∞))
40	39	xrge0nemnfd 45430	. . . 4 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ 𝐸)) ≠ -∞)
41	38, 40	jca 511	. . 3 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) ∈ ℝ^* ∧ (𝑂‘(𝐴 ∩ 𝐸)) ≠ -∞))
42		caragenuncllem.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
43	1, 2, 42, 3	caragenelss 46598	. . . . . 6 ⊢ (𝜑 → 𝐹 ⊆ 𝑋)
44	43	ssinss2d 45156	. . . . 5 ⊢ (𝜑 → ((𝐴 ∖ 𝐸) ∩ 𝐹) ⊆ 𝑋)
45	1, 3, 44	omexrcl 46604	. . . 4 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ∈ ℝ^*)
46	1, 3, 44	omecl 46600	. . . . 5 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ∈ (0[,]+∞))
47	46	xrge0nemnfd 45430	. . . 4 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ≠ -∞)
48	45, 47	jca 511	. . 3 ⊢ (𝜑 → ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ∈ ℝ^* ∧ (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ≠ -∞))
49	5	ssdifssd 4094	. . . . . 6 ⊢ (𝜑 → (𝐴 ∖ 𝐸) ⊆ 𝑋)
50	49	ssdifssd 4094	. . . . 5 ⊢ (𝜑 → ((𝐴 ∖ 𝐸) ∖ 𝐹) ⊆ 𝑋)
51	1, 3, 50	omexrcl 46604	. . . 4 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ∈ ℝ^*)
52	1, 3, 50	omecl 46600	. . . . 5 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ∈ (0[,]+∞))
53	52	xrge0nemnfd 45430	. . . 4 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ≠ -∞)
54	51, 53	jca 511	. . 3 ⊢ (𝜑 → ((𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ∈ ℝ^* ∧ (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ≠ -∞))
55		xaddass 13148	. . 3 ⊢ ((((𝑂‘(𝐴 ∩ 𝐸)) ∈ ℝ^* ∧ (𝑂‘(𝐴 ∩ 𝐸)) ≠ -∞) ∧ ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ∈ ℝ^* ∧ (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ≠ -∞) ∧ ((𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ∈ ℝ^* ∧ (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ≠ -∞)) → (((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹))) = ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)))))
56	41, 48, 54, 55	syl3anc 1373	. 2 ⊢ (𝜑 → (((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹))) = ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)))))
57	1, 2, 3, 42, 49	caragensplit 46597	. . . 4 ⊢ (𝜑 → ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹))) = (𝑂‘(𝐴 ∖ 𝐸)))
58	57	oveq2d 7362	. . 3 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)))) = ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘(𝐴 ∖ 𝐸))))
59	1, 2, 3, 4, 5	caragensplit 46597	. . 3 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴))
60	58, 59	eqtrd 2766	. 2 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)))) = (𝑂‘𝐴))
61	36, 56, 60	3eqtrd 2770	1 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) +_𝑒 (𝑂‘(𝐴 ∖ (𝐸 ∪ 𝐹)))) = (𝑂‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∖ cdif 3894 ∪ cun 3895 ∩ cin 3896 ⊆ wss 3897 ∅c0 4280 ∪ cuni 4856 dom cdm 5614 ‘cfv 6481 (class class class)co 7346 -∞cmnf 11144 ℝ^*cxr 11145 +_𝑒 cxad 13009 OutMeascome 46586 CaraGenccaragen 46588

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-addass 11071 ax-i2m1 11074 ax-rnegex 11077 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-po 5522 df-so 5523 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-xadd 13012 df-icc 13252 df-ome 46587 df-caragen 46589

This theorem is referenced by: caragenuncl 46610

W3C validator