caragenuncllem - Mathbox for Glauco Siliprandi

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Theorem caragenuncllem 46937

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 4188	. . . . . 6 ⊢ (𝜑 → (𝐴 ∩ (𝐸 ∪ 𝐹)) ⊆ 𝑋)
7	1, 2, 3, 4, 6	caragensplit 46925	. . . . 5 ⊢ (𝜑 → ((𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸))) = (𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))))
8	7	eqcomd 2743	. . . 4 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) = ((𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸))))
9		inass 4169	. . . . . . . 8 ⊢ ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸) = (𝐴 ∩ ((𝐸 ∪ 𝐹) ∩ 𝐸))
10		incom 4150	. . . . . . . . . 10 ⊢ ((𝐸 ∪ 𝐹) ∩ 𝐸) = (𝐸 ∩ (𝐸 ∪ 𝐹))
11		inabs 4207	. . . . . . . . . 10 ⊢ (𝐸 ∩ (𝐸 ∪ 𝐹)) = 𝐸
12	10, 11	eqtri 2760	. . . . . . . . 9 ⊢ ((𝐸 ∪ 𝐹) ∩ 𝐸) = 𝐸
13	12	ineq2i 4158	. . . . . . . 8 ⊢ (𝐴 ∩ ((𝐸 ∪ 𝐹) ∩ 𝐸)) = (𝐴 ∩ 𝐸)
14	9, 13	eqtri 2760	. . . . . . 7 ⊢ ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸) = (𝐴 ∩ 𝐸)
15	14	fveq2i 6841	. . . . . 6 ⊢ (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∩ 𝐸)) = (𝑂‘(𝐴 ∩ 𝐸))
16		incom 4150	. . . . . . . . . 10 ⊢ ((𝐴 ∖ 𝐸) ∩ 𝐹) = (𝐹 ∩ (𝐴 ∖ 𝐸))
17		indifcom 4224	. . . . . . . . . 10 ⊢ (𝐹 ∩ (𝐴 ∖ 𝐸)) = (𝐴 ∩ (𝐹 ∖ 𝐸))
18	16, 17	eqtr2i 2761	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐹 ∖ 𝐸)) = ((𝐴 ∖ 𝐸) ∩ 𝐹)
19	18	eqcomi 2746	. . . . . . . 8 ⊢ ((𝐴 ∖ 𝐸) ∩ 𝐹) = (𝐴 ∩ (𝐹 ∖ 𝐸))
20		difundir 4232	. . . . . . . . . 10 ⊢ ((𝐸 ∪ 𝐹) ∖ 𝐸) = ((𝐸 ∖ 𝐸) ∪ (𝐹 ∖ 𝐸))
21		difid 4317	. . . . . . . . . . 11 ⊢ (𝐸 ∖ 𝐸) = ∅
22	21	uneq1i 4105	. . . . . . . . . 10 ⊢ ((𝐸 ∖ 𝐸) ∪ (𝐹 ∖ 𝐸)) = (∅ ∪ (𝐹 ∖ 𝐸))
23		0un 4337	. . . . . . . . . 10 ⊢ (∅ ∪ (𝐹 ∖ 𝐸)) = (𝐹 ∖ 𝐸)
24	20, 22, 23	3eqtrri 2765	. . . . . . . . 9 ⊢ (𝐹 ∖ 𝐸) = ((𝐸 ∪ 𝐹) ∖ 𝐸)
25	24	ineq2i 4158	. . . . . . . 8 ⊢ (𝐴 ∩ (𝐹 ∖ 𝐸)) = (𝐴 ∩ ((𝐸 ∪ 𝐹) ∖ 𝐸))
26		indif2 4222	. . . . . . . 8 ⊢ (𝐴 ∩ ((𝐸 ∪ 𝐹) ∖ 𝐸)) = ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸)
27	19, 25, 26	3eqtrri 2765	. . . . . . 7 ⊢ ((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸) = ((𝐴 ∖ 𝐸) ∩ 𝐹)
28	27	fveq2i 6841	. . . . . 6 ⊢ (𝑂‘((𝐴 ∩ (𝐸 ∪ 𝐹)) ∖ 𝐸)) = (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))
29	15, 28	oveq12i 7376	. . . . 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 2738	. . . 4 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))) = ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))))
32	8, 30, 31	3eqtrd 2776	. . 3 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) = ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹))))
33		difun1 4240	. . . . 5 ⊢ (𝐴 ∖ (𝐸 ∪ 𝐹)) = ((𝐴 ∖ 𝐸) ∖ 𝐹)
34	33	fveq2i 6841	. . . 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 7382	. 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 4188	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐸) ⊆ 𝑋)
38	1, 3, 37	omexrcl 46932	. . . 4 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ 𝐸)) ∈ ℝ^*)
39	1, 3, 37	omecl 46928	. . . . 5 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ 𝐸)) ∈ (0[,]+∞))
40	39	xrge0nemnfd 45759	. . . 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 46926	. . . . . 6 ⊢ (𝜑 → 𝐹 ⊆ 𝑋)
44	43	ssinss2d 45488	. . . . 5 ⊢ (𝜑 → ((𝐴 ∖ 𝐸) ∩ 𝐹) ⊆ 𝑋)
45	1, 3, 44	omexrcl 46932	. . . 4 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ∈ ℝ^*)
46	1, 3, 44	omecl 46928	. . . . 5 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) ∈ (0[,]+∞))
47	46	xrge0nemnfd 45759	. . . 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 4088	. . . . . 6 ⊢ (𝜑 → (𝐴 ∖ 𝐸) ⊆ 𝑋)
50	49	ssdifssd 4088	. . . . 5 ⊢ (𝜑 → ((𝐴 ∖ 𝐸) ∖ 𝐹) ⊆ 𝑋)
51	1, 3, 50	omexrcl 46932	. . . 4 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ∈ ℝ^*)
52	1, 3, 50	omecl 46928	. . . . 5 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ∈ (0[,]+∞))
53	52	xrge0nemnfd 45759	. . . 4 ⊢ (𝜑 → (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ≠ -∞)
54	51, 53	jca 511	. . 3 ⊢ (𝜑 → ((𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ∈ ℝ^* ∧ (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)) ≠ -∞))
55		xaddass 13198	. . 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 1374	. 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 46925	. . . 4 ⊢ (𝜑 → ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹))) = (𝑂‘(𝐴 ∖ 𝐸)))
58	57	oveq2d 7380	. . 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 46925	. . 3 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴))
60	58, 59	eqtrd 2772	. 2 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +_𝑒 ((𝑂‘((𝐴 ∖ 𝐸) ∩ 𝐹)) +_𝑒 (𝑂‘((𝐴 ∖ 𝐸) ∖ 𝐹)))) = (𝑂‘𝐴))
61	36, 56, 60	3eqtrd 2776	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 1542 ∈ wcel 2114 ≠ wne 2933 ∖ cdif 3887 ∪ cun 3888 ∩ cin 3889 ⊆ wss 3890 ∅c0 4274 ∪ cuni 4851 dom cdm 5628 ‘cfv 6496 (class class class)co 7364 -∞cmnf 11174 ℝ^*cxr 11175 +_𝑒 cxad 13058 OutMeascome 46914 CaraGenccaragen 46916

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5306 ax-pr 5374 ax-un 7686 ax-cnex 11091 ax-resscn 11092 ax-1cn 11093 ax-icn 11094 ax-addcl 11095 ax-addrcl 11096 ax-mulcl 11097 ax-addass 11100 ax-i2m1 11103 ax-rnegex 11106 ax-cnre 11108 ax-pre-lttri 11109 ax-pre-lttrn 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5523 df-po 5536 df-so 5537 df-xp 5634 df-rel 5635 df-cnv 5636 df-co 5637 df-dm 5638 df-rn 5639 df-res 5640 df-ima 5641 df-iota 6452 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7367 df-oprab 7368 df-mpo 7369 df-1st 7939 df-2nd 7940 df-er 8640 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11178 df-mnf 11179 df-xr 11180 df-ltxr 11181 df-le 11182 df-xadd 13061 df-icc 13302 df-ome 46915 df-caragen 46917

This theorem is referenced by: caragenuncl 46938

W3C validator