caragenuncllem - Mathbox for Glauco Siliprandi

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Theorem caragenuncllem 46468

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 44988	. . . . . 6 ⊢ (𝜑 → (𝐴 ∩ (𝐸 ∪ 𝐹)) ⊆ 𝑋)
7	1, 2, 3, 4, 6	caragensplit 46456	. . . . 5 ⊢ (𝜑 → ((𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸))) = (𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))))
8	7	eqcomd 2741	. . . 4 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) = ((𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸))))
9		inass 4236	. . . . . . . 8 ⊢ ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸) = (𝐴 ∩ ((𝐸 ∪ 𝐹) ∩ 𝐸))
10		incom 4217	. . . . . . . . . 10 ⊢ ((𝐸 ∪ 𝐹) ∩ 𝐸) = (𝐸 ∩ (𝐸 ∪ 𝐹))
11		inabs 4272	. . . . . . . . . 10 ⊢ (𝐸 ∩ (𝐸 ∪ 𝐹)) = 𝐸
12	10, 11	eqtri 2763	. . . . . . . . 9 ⊢ ((𝐸 ∪ 𝐹) ∩ 𝐸) = 𝐸
13	12	ineq2i 4225	. . . . . . . 8 ⊢ (𝐴 ∩ ((𝐸 ∪ 𝐹) ∩ 𝐸)) = (𝐴 ∩ 𝐸)
14	9, 13	eqtri 2763	. . . . . . 7 ⊢ ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸) = (𝐴 ∩ 𝐸)
15	14	fveq2i 6910	. . . . . 6 ⊢ (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) = (𝑂‘(𝐴 ∩ 𝐸))
16		incom 4217	. . . . . . . . . 10 ⊢ ((𝐴 ∖ 𝐸) ∩ 𝐹) = (𝐹 ∩ (𝐴 ∖ 𝐸))
17		indifcom 4289	. . . . . . . . . 10 ⊢ (𝐹 ∩ (𝐴 ∖ 𝐸)) = (𝐴 ∩ (𝐹 ∖ 𝐸))
18	16, 17	eqtr2i 2764	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐹 ∖ 𝐸)) = ((𝐴 ∖ 𝐸) ∩ 𝐹)
19	18	eqcomi 2744	. . . . . . . 8 ⊢ ((𝐴 ∖ 𝐸) ∩ 𝐹) = (𝐴 ∩ (𝐹 ∖ 𝐸))
20		difundir 4297	. . . . . . . . . 10 ⊢ ((𝐸 ∪ 𝐹) ∖ 𝐸) = ((𝐸 ∖ 𝐸) ∪ (𝐹 ∖ 𝐸))
21		difid 4382	. . . . . . . . . . 11 ⊢ (𝐸 ∖ 𝐸) = ∅
22	21	uneq1i 4174	. . . . . . . . . 10 ⊢ ((𝐸 ∖ 𝐸) ∪ (𝐹 ∖ 𝐸)) = (∅ ∪ (𝐹 ∖ 𝐸))
23		0un 4402	. . . . . . . . . 10 ⊢ (∅ ∪ (𝐹 ∖ 𝐸)) = (𝐹 ∖ 𝐸)
24	20, 22, 23	3eqtrri 2768	. . . . . . . . 9 ⊢ (𝐹 ∖ 𝐸) = ((𝐸 ∪ 𝐹) ∖ 𝐸)
25	24	ineq2i 4225	. . . . . . . 8 ⊢ (𝐴 ∩ (𝐹 ∖ 𝐸)) = (𝐴 ∩ ((𝐸 ∪ 𝐹) ∖ 𝐸))
26		indif2 4287	. . . . . . . 8 ⊢ (𝐴 ∩ ((𝐸 ∪ 𝐹) ∖ 𝐸)) = ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸)
27	19, 25, 26	3eqtrri 2768	. . . . . . 7 ⊢ ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸) = ((𝐴 ∖ 𝐸) ∩ 𝐹)
28	27	fveq2i 6910	. . . . . 6 ⊢ (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸)) = (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))
29	15, 28	oveq12i 7443	. . . . 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 2736	. . . 4 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))) = ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))))
32	8, 30, 31	3eqtrd 2779	. . 3 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) = ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))))
33		difun1 4305	. . . . 5 ⊢ (𝐴 ∖ (𝐸 ∪ 𝐹)) = ((𝐴 ∖ 𝐸) ∖ 𝐹)
34	33	fveq2i 6910	. . . 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 7449	. 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 44988	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐸) ⊆ 𝑋)
38	1, 3, 37	omexrcl 46463	. . . 4 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ 𝐸)) ∈ ℝ^*)
39	1, 3, 37	omecl 46459	. . . . 5 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ 𝐸)) ∈ (0[,]+∞))
40	39	xrge0nemnfd 45282	. . . 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 46457	. . . . . 6 ⊢ (𝜑 → 𝐹 ⊆ 𝑋)
44	43	ssinss2d 45000	. . . . 5 ⊢ (𝜑 → ((𝐴 ∖ 𝐸) ∩ 𝐹) ⊆ 𝑋)
45	1, 3, 44	omexrcl 46463	. . . 4 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ∈ ℝ^*)
46	1, 3, 44	omecl 46459	. . . . 5 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ∈ (0[,]+∞))
47	46	xrge0nemnfd 45282	. . . 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 4157	. . . . . 6 ⊢ (𝜑 → (𝐴 ∖ 𝐸) ⊆ 𝑋)
50	49	ssdifssd 4157	. . . . 5 ⊢ (𝜑 → ((𝐴 ∖ 𝐸) ∖ 𝐹) ⊆ 𝑋)
51	1, 3, 50	omexrcl 46463	. . . 4 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ∈ ℝ^*)
52	1, 3, 50	omecl 46459	. . . . 5 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ∈ (0[,]+∞))
53	52	xrge0nemnfd 45282	. . . 4 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ≠ -∞)
54	51, 53	jca 511	. . 3 ⊢ (𝜑 → ((𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ∈ ℝ^* ∧ (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ≠ -∞))
55		xaddass 13288	. . 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 1370	. 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 46456	. . . 4 ⊢ (𝜑 → ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹))) = (𝑂‘(𝐴 ∖ 𝐸)))
58	57	oveq2d 7447	. . 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 46456	. . 3 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴))
60	58, 59	eqtrd 2775	. 2 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)))) = (𝑂‘𝐴))
61	36, 56, 60	3eqtrd 2779	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 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 ∪ cun 3961 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 ∪ cuni 4912 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 -∞cmnf 11291 ℝ^*cxr 11292 +_𝑒 cxad 13150 OutMeascome 46445 CaraGenccaragen 46447

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-addass 11218 ax-i2m1 11221 ax-rnegex 11224 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-xadd 13153 df-icc 13391 df-ome 46446 df-caragen 46448

This theorem is referenced by: caragenuncl 46469

W3C validator