caragenuncllem - Mathbox for Glauco Siliprandi

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Theorem caragenuncllem 45228

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 43735	. . . . . 6 ⊢ (𝜑 → (𝐴 ∩ (𝐸 ∪ 𝐹)) ⊆ 𝑋)
7	1, 2, 3, 4, 6	caragensplit 45216	. . . . 5 ⊢ (𝜑 → ((𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸))) = (𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))))
8	7	eqcomd 2739	. . . 4 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) = ((𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸))))
9		inass 4220	. . . . . . . 8 ⊢ ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸) = (𝐴 ∩ ((𝐸 ∪ 𝐹) ∩ 𝐸))
10		incom 4202	. . . . . . . . . 10 ⊢ ((𝐸 ∪ 𝐹) ∩ 𝐸) = (𝐸 ∩ (𝐸 ∪ 𝐹))
11		inabs 4256	. . . . . . . . . 10 ⊢ (𝐸 ∩ (𝐸 ∪ 𝐹)) = 𝐸
12	10, 11	eqtri 2761	. . . . . . . . 9 ⊢ ((𝐸 ∪ 𝐹) ∩ 𝐸) = 𝐸
13	12	ineq2i 4210	. . . . . . . 8 ⊢ (𝐴 ∩ ((𝐸 ∪ 𝐹) ∩ 𝐸)) = (𝐴 ∩ 𝐸)
14	9, 13	eqtri 2761	. . . . . . 7 ⊢ ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸) = (𝐴 ∩ 𝐸)
15	14	fveq2i 6895	. . . . . 6 ⊢ (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) = (𝑂‘(𝐴 ∩ 𝐸))
16		incom 4202	. . . . . . . . . 10 ⊢ ((𝐴 ∖ 𝐸) ∩ 𝐹) = (𝐹 ∩ (𝐴 ∖ 𝐸))
17		indifcom 4273	. . . . . . . . . 10 ⊢ (𝐹 ∩ (𝐴 ∖ 𝐸)) = (𝐴 ∩ (𝐹 ∖ 𝐸))
18	16, 17	eqtr2i 2762	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐹 ∖ 𝐸)) = ((𝐴 ∖ 𝐸) ∩ 𝐹)
19	18	eqcomi 2742	. . . . . . . 8 ⊢ ((𝐴 ∖ 𝐸) ∩ 𝐹) = (𝐴 ∩ (𝐹 ∖ 𝐸))
20		difundir 4281	. . . . . . . . . 10 ⊢ ((𝐸 ∪ 𝐹) ∖ 𝐸) = ((𝐸 ∖ 𝐸) ∪ (𝐹 ∖ 𝐸))
21		difid 4371	. . . . . . . . . . 11 ⊢ (𝐸 ∖ 𝐸) = ∅
22	21	uneq1i 4160	. . . . . . . . . 10 ⊢ ((𝐸 ∖ 𝐸) ∪ (𝐹 ∖ 𝐸)) = (∅ ∪ (𝐹 ∖ 𝐸))
23		0un 4393	. . . . . . . . . 10 ⊢ (∅ ∪ (𝐹 ∖ 𝐸)) = (𝐹 ∖ 𝐸)
24	20, 22, 23	3eqtrri 2766	. . . . . . . . 9 ⊢ (𝐹 ∖ 𝐸) = ((𝐸 ∪ 𝐹) ∖ 𝐸)
25	24	ineq2i 4210	. . . . . . . 8 ⊢ (𝐴 ∩ (𝐹 ∖ 𝐸)) = (𝐴 ∩ ((𝐸 ∪ 𝐹) ∖ 𝐸))
26		indif2 4271	. . . . . . . 8 ⊢ (𝐴 ∩ ((𝐸 ∪ 𝐹) ∖ 𝐸)) = ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸)
27	19, 25, 26	3eqtrri 2766	. . . . . . 7 ⊢ ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸) = ((𝐴 ∖ 𝐸) ∩ 𝐹)
28	27	fveq2i 6895	. . . . . 6 ⊢ (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸)) = (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))
29	15, 28	oveq12i 7421	. . . . 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 2734	. . . 4 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))) = ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))))
32	8, 30, 31	3eqtrd 2777	. . 3 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) = ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))))
33		difun1 4290	. . . . 5 ⊢ (𝐴 ∖ (𝐸 ∪ 𝐹)) = ((𝐴 ∖ 𝐸) ∖ 𝐹)
34	33	fveq2i 6895	. . . 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 7427	. 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 43735	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐸) ⊆ 𝑋)
38	1, 3, 37	omexrcl 45223	. . . 4 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ 𝐸)) ∈ ℝ^*)
39	1, 3, 37	omecl 45219	. . . . 5 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ 𝐸)) ∈ (0[,]+∞))
40	39	xrge0nemnfd 44042	. . . 4 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ 𝐸)) ≠ -∞)
41	38, 40	jca 513	. . 3 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) ∈ ℝ^* ∧ (𝑂‘(𝐴 ∩ 𝐸)) ≠ -∞))
42		caragenuncllem.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
43	1, 2, 42, 3	caragenelss 45217	. . . . . 6 ⊢ (𝜑 → 𝐹 ⊆ 𝑋)
44	43	ssinss2d 43747	. . . . 5 ⊢ (𝜑 → ((𝐴 ∖ 𝐸) ∩ 𝐹) ⊆ 𝑋)
45	1, 3, 44	omexrcl 45223	. . . 4 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ∈ ℝ^*)
46	1, 3, 44	omecl 45219	. . . . 5 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ∈ (0[,]+∞))
47	46	xrge0nemnfd 44042	. . . 4 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ≠ -∞)
48	45, 47	jca 513	. . 3 ⊢ (𝜑 → ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ∈ ℝ^* ∧ (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ≠ -∞))
49	5	ssdifssd 4143	. . . . . 6 ⊢ (𝜑 → (𝐴 ∖ 𝐸) ⊆ 𝑋)
50	49	ssdifssd 4143	. . . . 5 ⊢ (𝜑 → ((𝐴 ∖ 𝐸) ∖ 𝐹) ⊆ 𝑋)
51	1, 3, 50	omexrcl 45223	. . . 4 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ∈ ℝ^*)
52	1, 3, 50	omecl 45219	. . . . 5 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ∈ (0[,]+∞))
53	52	xrge0nemnfd 44042	. . . 4 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ≠ -∞)
54	51, 53	jca 513	. . 3 ⊢ (𝜑 → ((𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ∈ ℝ^* ∧ (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ≠ -∞))
55		xaddass 13228	. . 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 1372	. 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 45216	. . . 4 ⊢ (𝜑 → ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹))) = (𝑂‘(𝐴 ∖ 𝐸)))
58	57	oveq2d 7425	. . 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 45216	. . 3 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴))
60	58, 59	eqtrd 2773	. 2 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)))) = (𝑂‘𝐴))
61	36, 56, 60	3eqtrd 2777	1 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) +_𝑒 (𝑂‘(𝐴 ∖ (𝐸 ∪ 𝐹)))) = (𝑂‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∖ cdif 3946 ∪ cun 3947 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 ∪ cuni 4909 dom cdm 5677 ‘cfv 6544 (class class class)co 7409 -∞cmnf 11246 ℝ^*cxr 11247 +_𝑒 cxad 13090 OutMeascome 45205 CaraGenccaragen 45207

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-addass 11175 ax-i2m1 11178 ax-rnegex 11181 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-xadd 13093 df-icc 13331 df-ome 45206 df-caragen 45208

This theorem is referenced by: caragenuncl 45229

W3C validator