caragenuncllem - Mathbox for Glauco Siliprandi

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Theorem caragenuncllem 40726

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 39214	. . . . . 6 ⊢ (𝜑 → (𝐴 ∩ (𝐸 ∪ 𝐹)) ⊆ 𝑋)
7	1, 2, 3, 4, 6	caragensplit 40714	. . . . 5 ⊢ (𝜑 → ((𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸))) = (𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))))
8	7	eqcomd 2628	. . . 4 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) = ((𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸))))
9		inass 3823	. . . . . . . 8 ⊢ ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸) = (𝐴 ∩ ((𝐸 ∪ 𝐹) ∩ 𝐸))
10		incom 3805	. . . . . . . . . 10 ⊢ ((𝐸 ∪ 𝐹) ∩ 𝐸) = (𝐸 ∩ (𝐸 ∪ 𝐹))
11		inabs 3855	. . . . . . . . . 10 ⊢ (𝐸 ∩ (𝐸 ∪ 𝐹)) = 𝐸
12	10, 11	eqtri 2644	. . . . . . . . 9 ⊢ ((𝐸 ∪ 𝐹) ∩ 𝐸) = 𝐸
13	12	ineq2i 3811	. . . . . . . 8 ⊢ (𝐴 ∩ ((𝐸 ∪ 𝐹) ∩ 𝐸)) = (𝐴 ∩ 𝐸)
14	9, 13	eqtri 2644	. . . . . . 7 ⊢ ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸) = (𝐴 ∩ 𝐸)
15	14	fveq2i 6194	. . . . . 6 ⊢ (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) = (𝑂‘(𝐴 ∩ 𝐸))
16		incom 3805	. . . . . . . . . 10 ⊢ ((𝐴 ∖ 𝐸) ∩ 𝐹) = (𝐹 ∩ (𝐴 ∖ 𝐸))
17		indifcom 3872	. . . . . . . . . 10 ⊢ (𝐹 ∩ (𝐴 ∖ 𝐸)) = (𝐴 ∩ (𝐹 ∖ 𝐸))
18	16, 17	eqtr2i 2645	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐹 ∖ 𝐸)) = ((𝐴 ∖ 𝐸) ∩ 𝐹)
19	18	eqcomi 2631	. . . . . . . 8 ⊢ ((𝐴 ∖ 𝐸) ∩ 𝐹) = (𝐴 ∩ (𝐹 ∖ 𝐸))
20		difundir 3880	. . . . . . . . . 10 ⊢ ((𝐸 ∪ 𝐹) ∖ 𝐸) = ((𝐸 ∖ 𝐸) ∪ (𝐹 ∖ 𝐸))
21		difid 3948	. . . . . . . . . . 11 ⊢ (𝐸 ∖ 𝐸) = ∅
22	21	uneq1i 3763	. . . . . . . . . 10 ⊢ ((𝐸 ∖ 𝐸) ∪ (𝐹 ∖ 𝐸)) = (∅ ∪ (𝐹 ∖ 𝐸))
23		0un 39215	. . . . . . . . . 10 ⊢ (∅ ∪ (𝐹 ∖ 𝐸)) = (𝐹 ∖ 𝐸)
24	20, 22, 23	3eqtrri 2649	. . . . . . . . 9 ⊢ (𝐹 ∖ 𝐸) = ((𝐸 ∪ 𝐹) ∖ 𝐸)
25	24	ineq2i 3811	. . . . . . . 8 ⊢ (𝐴 ∩ (𝐹 ∖ 𝐸)) = (𝐴 ∩ ((𝐸 ∪ 𝐹) ∖ 𝐸))
26		indif2 3870	. . . . . . . 8 ⊢ (𝐴 ∩ ((𝐸 ∪ 𝐹) ∖ 𝐸)) = ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸)
27	19, 25, 26	3eqtrri 2649	. . . . . . 7 ⊢ ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸) = ((𝐴 ∖ 𝐸) ∩ 𝐹)
28	27	fveq2i 6194	. . . . . 6 ⊢ (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸)) = (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))
29	15, 28	oveq12i 6662	. . . . 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 2623	. . . 4 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))) = ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))))
32	8, 30, 31	3eqtrd 2660	. . 3 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) = ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))))
33		difun1 3887	. . . . 5 ⊢ (𝐴 ∖ (𝐸 ∪ 𝐹)) = ((𝐴 ∖ 𝐸) ∖ 𝐹)
34	33	fveq2i 6194	. . . 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 6668	. 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 39214	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐸) ⊆ 𝑋)
38	1, 3, 37	omexrcl 40721	. . . 4 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ 𝐸)) ∈ ℝ^*)
39	1, 3, 37	omecl 40717	. . . . 5 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ 𝐸)) ∈ (0[,]+∞))
40	39	xrge0nemnfd 39548	. . . 4 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ 𝐸)) ≠ -∞)
41	38, 40	jca 554	. . 3 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) ∈ ℝ^* ∧ (𝑂‘(𝐴 ∩ 𝐸)) ≠ -∞))
42		caragenuncllem.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
43	1, 2, 42, 3	caragenelss 40715	. . . . . 6 ⊢ (𝜑 → 𝐹 ⊆ 𝑋)
44	43	ssinss2d 39228	. . . . 5 ⊢ (𝜑 → ((𝐴 ∖ 𝐸) ∩ 𝐹) ⊆ 𝑋)
45	1, 3, 44	omexrcl 40721	. . . 4 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ∈ ℝ^*)
46	1, 3, 44	omecl 40717	. . . . 5 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ∈ (0[,]+∞))
47	46	xrge0nemnfd 39548	. . . 4 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ≠ -∞)
48	45, 47	jca 554	. . 3 ⊢ (𝜑 → ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ∈ ℝ^* ∧ (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ≠ -∞))
49	5	ssdifssd 3748	. . . . . 6 ⊢ (𝜑 → (𝐴 ∖ 𝐸) ⊆ 𝑋)
50	49	ssdifssd 3748	. . . . 5 ⊢ (𝜑 → ((𝐴 ∖ 𝐸) ∖ 𝐹) ⊆ 𝑋)
51	1, 3, 50	omexrcl 40721	. . . 4 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ∈ ℝ^*)
52	1, 3, 50	omecl 40717	. . . . 5 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ∈ (0[,]+∞))
53	52	xrge0nemnfd 39548	. . . 4 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ≠ -∞)
54	51, 53	jca 554	. . 3 ⊢ (𝜑 → ((𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ∈ ℝ^* ∧ (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ≠ -∞))
55		xaddass 12079	. . 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 1326	. 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 40714	. . . 4 ⊢ (𝜑 → ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹))) = (𝑂‘(𝐴 ∖ 𝐸)))
58	57	oveq2d 6666	. . 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 40714	. . 3 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴))
60	58, 59	eqtrd 2656	. 2 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)))) = (𝑂‘𝐴))
61	36, 56, 60	3eqtrd 2660	1 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) +_𝑒 (𝑂‘(𝐴 ∖ (𝐸 ∪ 𝐹)))) = (𝑂‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 ∪ cuni 4436 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 -∞cmnf 10072 ℝ^*cxr 10073 +_𝑒 cxad 11944 OutMeascome 40703 CaraGenccaragen 40705

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-addass 10001 ax-i2m1 10004 ax-1ne0 10005 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-xadd 11947 df-icc 12182 df-ome 40704 df-caragen 40706

This theorem is referenced by: caragenuncl 40727

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