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Theorem uniioombllem5 25568

Description: Lemma for uniioombl 25570. (Contributed by Mario Carneiro, 25-Aug-2014.)

**Hypotheses**
Ref	Expression
uniioombl.1	⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
uniioombl.2	⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))
uniioombl.3	⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
uniioombl.a	⊢ 𝐴 = ∪ ran ((,) ∘ 𝐹)
uniioombl.e	⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)
uniioombl.c	⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
uniioombl.g	⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
uniioombl.s	⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐺))
uniioombl.t	⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
uniioombl.v	⊢ (𝜑 → sup(ran 𝑇, ℝ^, < ) ≤ ((vol‘𝐸) + 𝐶))
uniioombl.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
uniioombl.m2	⊢ (𝜑 → (abs‘((𝑇‘𝑀) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)
uniioombl.k	⊢ 𝐾 = ∪ (((,) ∘ 𝐺) “ (1...𝑀))
uniioombl.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
uniioombl.n2	⊢ (𝜑 → ∀𝑗 ∈ (1...𝑀)(abs‘(Σ𝑖 ∈ (1...𝑁)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑀))
uniioombl.l	⊢ 𝐿 = ∪ (((,) ∘ 𝐹) “ (1...𝑁))

**Assertion**
Ref	Expression
uniioombllem5	⊢ (𝜑 → ((vol‘(𝐸 ∩ 𝐴)) + (vol‘(𝐸 ∖ 𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))

Distinct variable groups: 𝑖,𝑗,𝑥,𝐹 𝑖,𝐺,𝑗,𝑥 𝑗,𝐾,𝑥 𝐴,𝑗,𝑥 𝐶,𝑖,𝑗,𝑥 𝑖,𝑀,𝑗,𝑥 𝑖,𝑁,𝑗 𝜑,𝑖,𝑗,𝑥 𝑇,𝑖,𝑗,𝑥 Allowed substitution hints: 𝐴(𝑖) 𝑆(𝑥,𝑖,𝑗) 𝐸(𝑥,𝑖,𝑗) 𝐾(𝑖) 𝐿(𝑥,𝑖,𝑗) 𝑁(𝑥)

Proof of Theorem uniioombllem5 Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 4178	. . . 4 ⊢ (𝐸 ∩ 𝐴) ⊆ 𝐸
2		uniioombl.s	. . . . 5 ⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐺))
3		uniioombl.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
4	3	uniiccdif 25559	. . . . . . 7 ⊢ (𝜑 → (∪ ran ((,) ∘ 𝐺) ⊆ ∪ ran ([,] ∘ 𝐺) ∧ (vol*‘(∪ ran ([,] ∘ 𝐺) ∖ ∪ ran ((,) ∘ 𝐺))) = 0))
5	4	simpld 494	. . . . . 6 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐺) ⊆ ∪ ran ([,] ∘ 𝐺))
6		ovolficcss 25450	. . . . . . 7 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐺) ⊆ ℝ)
7	3, 6	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐺) ⊆ ℝ)
8	5, 7	sstrd 3933	. . . . 5 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐺) ⊆ ℝ)
9	2, 8	sstrd 3933	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ ℝ)
10		uniioombl.e	. . . 4 ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)
11		ovolsscl 25467	. . . 4 ⊢ (((𝐸 ∩ 𝐴) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol‘𝐸) ∈ ℝ) → (vol‘(𝐸 ∩ 𝐴)) ∈ ℝ)
12	1, 9, 10, 11	mp3an2i 1469	. . 3 ⊢ (𝜑 → (vol*‘(𝐸 ∩ 𝐴)) ∈ ℝ)
13		difssd 4078	. . . 4 ⊢ (𝜑 → (𝐸 ∖ 𝐴) ⊆ 𝐸)
14		ovolsscl 25467	. . . 4 ⊢ (((𝐸 ∖ 𝐴) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol‘𝐸) ∈ ℝ) → (vol‘(𝐸 ∖ 𝐴)) ∈ ℝ)
15	13, 9, 10, 14	syl3anc 1374	. . 3 ⊢ (𝜑 → (vol*‘(𝐸 ∖ 𝐴)) ∈ ℝ)
16	12, 15	readdcld 11169	. 2 ⊢ (𝜑 → ((vol‘(𝐸 ∩ 𝐴)) + (vol‘(𝐸 ∖ 𝐴))) ∈ ℝ)
17		inss1 4178	. . . . 5 ⊢ (𝐾 ∩ 𝐴) ⊆ 𝐾
18		uniioombl.k	. . . . . . 7 ⊢ 𝐾 = ∪ (((,) ∘ 𝐺) “ (1...𝑀))
19		imassrn 6032	. . . . . . . 8 ⊢ (((,) ∘ 𝐺) “ (1...𝑀)) ⊆ ran ((,) ∘ 𝐺)
20	19	unissi 4860	. . . . . . 7 ⊢ ∪ (((,) ∘ 𝐺) “ (1...𝑀)) ⊆ ∪ ran ((,) ∘ 𝐺)
21	18, 20	eqsstri 3969	. . . . . 6 ⊢ 𝐾 ⊆ ∪ ran ((,) ∘ 𝐺)
22	21, 8	sstrid 3934	. . . . 5 ⊢ (𝜑 → 𝐾 ⊆ ℝ)
23		uniioombl.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
24		uniioombl.2	. . . . . . . 8 ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))
25		uniioombl.3	. . . . . . . 8 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
26		uniioombl.a	. . . . . . . 8 ⊢ 𝐴 = ∪ ran ((,) ∘ 𝐹)
27		uniioombl.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
28		uniioombl.t	. . . . . . . 8 ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
29		uniioombl.v	. . . . . . . 8 ⊢ (𝜑 → sup(ran 𝑇, ℝ^, < ) ≤ ((vol‘𝐸) + 𝐶))
30	23, 24, 25, 26, 10, 27, 3, 2, 28, 29	uniioombllem1 25562	. . . . . . 7 ⊢ (𝜑 → sup(ran 𝑇, ℝ^*, < ) ∈ ℝ)
31		ssid 3945	. . . . . . . 8 ⊢ ∪ ran ((,) ∘ 𝐺) ⊆ ∪ ran ((,) ∘ 𝐺)
32	28	ovollb 25460	. . . . . . . 8 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ∪ ran ((,) ∘ 𝐺) ⊆ ∪ ran ((,) ∘ 𝐺)) → (vol‘∪ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ^, < ))
33	3, 31, 32	sylancl 587	. . . . . . 7 ⊢ (𝜑 → (vol‘∪ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ^, < ))
34		ovollecl 25464	. . . . . . 7 ⊢ ((∪ ran ((,) ∘ 𝐺) ⊆ ℝ ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ ∧ (vol‘∪ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ^, < )) → (vol‘∪ ran ((,) ∘ 𝐺)) ∈ ℝ)
35	8, 30, 33, 34	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐺)) ∈ ℝ)
36		ovolsscl 25467	. . . . . 6 ⊢ ((𝐾 ⊆ ∪ ran ((,) ∘ 𝐺) ∧ ∪ ran ((,) ∘ 𝐺) ⊆ ℝ ∧ (vol‘∪ ran ((,) ∘ 𝐺)) ∈ ℝ) → (vol‘𝐾) ∈ ℝ)
37	21, 8, 35, 36	mp3an2i 1469	. . . . 5 ⊢ (𝜑 → (vol*‘𝐾) ∈ ℝ)
38		ovolsscl 25467	. . . . 5 ⊢ (((𝐾 ∩ 𝐴) ⊆ 𝐾 ∧ 𝐾 ⊆ ℝ ∧ (vol‘𝐾) ∈ ℝ) → (vol‘(𝐾 ∩ 𝐴)) ∈ ℝ)
39	17, 22, 37, 38	mp3an2i 1469	. . . 4 ⊢ (𝜑 → (vol*‘(𝐾 ∩ 𝐴)) ∈ ℝ)
40		difssd 4078	. . . . 5 ⊢ (𝜑 → (𝐾 ∖ 𝐴) ⊆ 𝐾)
41		ovolsscl 25467	. . . . 5 ⊢ (((𝐾 ∖ 𝐴) ⊆ 𝐾 ∧ 𝐾 ⊆ ℝ ∧ (vol‘𝐾) ∈ ℝ) → (vol‘(𝐾 ∖ 𝐴)) ∈ ℝ)
42	40, 22, 37, 41	syl3anc 1374	. . . 4 ⊢ (𝜑 → (vol*‘(𝐾 ∖ 𝐴)) ∈ ℝ)
43	39, 42	readdcld 11169	. . 3 ⊢ (𝜑 → ((vol‘(𝐾 ∩ 𝐴)) + (vol‘(𝐾 ∖ 𝐴))) ∈ ℝ)
44	27	rpred 12981	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
45	44, 44	readdcld 11169	. . 3 ⊢ (𝜑 → (𝐶 + 𝐶) ∈ ℝ)
46	43, 45	readdcld 11169	. 2 ⊢ (𝜑 → (((vol‘(𝐾 ∩ 𝐴)) + (vol‘(𝐾 ∖ 𝐴))) + (𝐶 + 𝐶)) ∈ ℝ)
47		4re 12260	. . . 4 ⊢ 4 ∈ ℝ
48		remulcl 11118	. . . 4 ⊢ ((4 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (4 · 𝐶) ∈ ℝ)
49	47, 44, 48	sylancr 588	. . 3 ⊢ (𝜑 → (4 · 𝐶) ∈ ℝ)
50	10, 49	readdcld 11169	. 2 ⊢ (𝜑 → ((vol*‘𝐸) + (4 · 𝐶)) ∈ ℝ)
51		uniioombl.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
52		uniioombl.m2	. . . 4 ⊢ (𝜑 → (abs‘((𝑇‘𝑀) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)
53	23, 24, 25, 26, 10, 27, 3, 2, 28, 29, 51, 52, 18	uniioombllem3 25566	. . 3 ⊢ (𝜑 → ((vol‘(𝐸 ∩ 𝐴)) + (vol‘(𝐸 ∖ 𝐴))) < (((vol‘(𝐾 ∩ 𝐴)) + (vol‘(𝐾 ∖ 𝐴))) + (𝐶 + 𝐶)))
54	16, 46, 53	ltled 11289	. 2 ⊢ (𝜑 → ((vol‘(𝐸 ∩ 𝐴)) + (vol‘(𝐸 ∖ 𝐴))) ≤ (((vol‘(𝐾 ∩ 𝐴)) + (vol‘(𝐾 ∖ 𝐴))) + (𝐶 + 𝐶)))
55	10, 45	readdcld 11169	. . . 4 ⊢ (𝜑 → ((vol*‘𝐸) + (𝐶 + 𝐶)) ∈ ℝ)
56	37, 44	readdcld 11169	. . . . 5 ⊢ (𝜑 → ((vol*‘𝐾) + 𝐶) ∈ ℝ)
57		inss1 4178	. . . . . . . . 9 ⊢ (𝐾 ∩ 𝐿) ⊆ 𝐾
58		ovolsscl 25467	. . . . . . . . 9 ⊢ (((𝐾 ∩ 𝐿) ⊆ 𝐾 ∧ 𝐾 ⊆ ℝ ∧ (vol‘𝐾) ∈ ℝ) → (vol‘(𝐾 ∩ 𝐿)) ∈ ℝ)
59	57, 22, 37, 58	mp3an2i 1469	. . . . . . . 8 ⊢ (𝜑 → (vol*‘(𝐾 ∩ 𝐿)) ∈ ℝ)
60	59, 44	readdcld 11169	. . . . . . 7 ⊢ (𝜑 → ((vol*‘(𝐾 ∩ 𝐿)) + 𝐶) ∈ ℝ)
61		difssd 4078	. . . . . . . 8 ⊢ (𝜑 → (𝐾 ∖ 𝐿) ⊆ 𝐾)
62		ovolsscl 25467	. . . . . . . 8 ⊢ (((𝐾 ∖ 𝐿) ⊆ 𝐾 ∧ 𝐾 ⊆ ℝ ∧ (vol‘𝐾) ∈ ℝ) → (vol‘(𝐾 ∖ 𝐿)) ∈ ℝ)
63	61, 22, 37, 62	syl3anc 1374	. . . . . . 7 ⊢ (𝜑 → (vol*‘(𝐾 ∖ 𝐿)) ∈ ℝ)
64		uniioombl.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
65		uniioombl.n2	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ (1...𝑀)(abs‘(Σ𝑖 ∈ (1...𝑁)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑀))
66		uniioombl.l	. . . . . . . 8 ⊢ 𝐿 = ∪ (((,) ∘ 𝐹) “ (1...𝑁))
67	23, 24, 25, 26, 10, 27, 3, 2, 28, 29, 51, 52, 18, 64, 65, 66	uniioombllem4 25567	. . . . . . 7 ⊢ (𝜑 → (vol‘(𝐾 ∩ 𝐴)) ≤ ((vol‘(𝐾 ∩ 𝐿)) + 𝐶))
68		imassrn 6032	. . . . . . . . . . 11 ⊢ (((,) ∘ 𝐹) “ (1...𝑁)) ⊆ ran ((,) ∘ 𝐹)
69	68	unissi 4860	. . . . . . . . . 10 ⊢ ∪ (((,) ∘ 𝐹) “ (1...𝑁)) ⊆ ∪ ran ((,) ∘ 𝐹)
70	69, 66, 26	3sstr4i 3974	. . . . . . . . 9 ⊢ 𝐿 ⊆ 𝐴
71		sscon 4084	. . . . . . . . 9 ⊢ (𝐿 ⊆ 𝐴 → (𝐾 ∖ 𝐴) ⊆ (𝐾 ∖ 𝐿))
72	70, 71	mp1i 13	. . . . . . . 8 ⊢ (𝜑 → (𝐾 ∖ 𝐴) ⊆ (𝐾 ∖ 𝐿))
73	61, 22	sstrd 3933	. . . . . . . 8 ⊢ (𝜑 → (𝐾 ∖ 𝐿) ⊆ ℝ)
74		ovolss 25466	. . . . . . . 8 ⊢ (((𝐾 ∖ 𝐴) ⊆ (𝐾 ∖ 𝐿) ∧ (𝐾 ∖ 𝐿) ⊆ ℝ) → (vol‘(𝐾 ∖ 𝐴)) ≤ (vol‘(𝐾 ∖ 𝐿)))
75	72, 73, 74	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (vol‘(𝐾 ∖ 𝐴)) ≤ (vol‘(𝐾 ∖ 𝐿)))
76	39, 42, 60, 63, 67, 75	le2addd 11764	. . . . . 6 ⊢ (𝜑 → ((vol‘(𝐾 ∩ 𝐴)) + (vol‘(𝐾 ∖ 𝐴))) ≤ (((vol‘(𝐾 ∩ 𝐿)) + 𝐶) + (vol‘(𝐾 ∖ 𝐿))))
77	59	recnd 11168	. . . . . . . 8 ⊢ (𝜑 → (vol*‘(𝐾 ∩ 𝐿)) ∈ ℂ)
78	44	recnd 11168	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℂ)
79	63	recnd 11168	. . . . . . . 8 ⊢ (𝜑 → (vol*‘(𝐾 ∖ 𝐿)) ∈ ℂ)
80	77, 78, 79	add32d 11369	. . . . . . 7 ⊢ (𝜑 → (((vol‘(𝐾 ∩ 𝐿)) + 𝐶) + (vol‘(𝐾 ∖ 𝐿))) = (((vol‘(𝐾 ∩ 𝐿)) + (vol‘(𝐾 ∖ 𝐿))) + 𝐶))
81		ioof 13395	. . . . . . . . . . . . 13 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
82		inss2 4179	. . . . . . . . . . . . . . 15 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
83		rexpssxrxp 11185	. . . . . . . . . . . . . . 15 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
84	82, 83	sstri 3932	. . . . . . . . . . . . . 14 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
85		fss 6680	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^)) → 𝐹:ℕ⟶(ℝ^ × ℝ^*))
86	23, 84, 85	sylancl 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ^* × ℝ^*))
87		fco 6688	. . . . . . . . . . . . 13 ⊢ (((,):(ℝ^* × ℝ^)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ^ × ℝ^*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
88	81, 86, 87	sylancr 588	. . . . . . . . . . . 12 ⊢ (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
89		ffun 6667	. . . . . . . . . . . 12 ⊢ (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → Fun ((,) ∘ 𝐹))
90		funiunfv 7198	. . . . . . . . . . . 12 ⊢ (Fun ((,) ∘ 𝐹) → ∪ 𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) = ∪ (((,) ∘ 𝐹) “ (1...𝑁)))
91	88, 89, 90	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) = ∪ (((,) ∘ 𝐹) “ (1...𝑁)))
92	91, 66	eqtr4di 2790	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) = 𝐿)
93		fzfid 13930	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
94		elfznn 13502	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
95		fvco3 6935	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹‘𝑛)))
96	23, 94, 95	syl2an 597	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹‘𝑛)))
97		ffvelcdm 7029	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
98	23, 94, 97	syl2an 597	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝐹‘𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
99	98	elin2d 4146	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝐹‘𝑛) ∈ (ℝ × ℝ))
100		1st2nd2 7976	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑛) ∈ (ℝ × ℝ) → (𝐹‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
101	99, 100	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝐹‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
102	101	fveq2d 6840	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((,)‘(𝐹‘𝑛)) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩))
103		df-ov 7365	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
104	102, 103	eqtr4di 2790	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((,)‘(𝐹‘𝑛)) = ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))))
105	96, 104	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((,) ∘ 𝐹)‘𝑛) = ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))))
106		ioombl 25546	. . . . . . . . . . . . 13 ⊢ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) ∈ dom vol
107	105, 106	eqeltrdi 2845	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
108	107	ralrimiva 3130	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
109		finiunmbl 25525	. . . . . . . . . . 11 ⊢ (((1...𝑁) ∈ Fin ∧ ∀𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) ∈ dom vol) → ∪ 𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
110	93, 108, 109	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
111	92, 110	eqeltrrd 2838	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ dom vol)
112		mblsplit 25513	. . . . . . . . 9 ⊢ ((𝐿 ∈ dom vol ∧ 𝐾 ⊆ ℝ ∧ (vol‘𝐾) ∈ ℝ) → (vol‘𝐾) = ((vol‘(𝐾 ∩ 𝐿)) + (vol‘(𝐾 ∖ 𝐿))))
113	111, 22, 37, 112	syl3anc 1374	. . . . . . . 8 ⊢ (𝜑 → (vol‘𝐾) = ((vol‘(𝐾 ∩ 𝐿)) + (vol*‘(𝐾 ∖ 𝐿))))
114	113	oveq1d 7377	. . . . . . 7 ⊢ (𝜑 → ((vol‘𝐾) + 𝐶) = (((vol‘(𝐾 ∩ 𝐿)) + (vol*‘(𝐾 ∖ 𝐿))) + 𝐶))
115	80, 114	eqtr4d 2775	. . . . . 6 ⊢ (𝜑 → (((vol‘(𝐾 ∩ 𝐿)) + 𝐶) + (vol‘(𝐾 ∖ 𝐿))) = ((vol*‘𝐾) + 𝐶))
116	76, 115	breqtrd 5112	. . . . 5 ⊢ (𝜑 → ((vol‘(𝐾 ∩ 𝐴)) + (vol‘(𝐾 ∖ 𝐴))) ≤ ((vol*‘𝐾) + 𝐶))
117	10, 44	readdcld 11169	. . . . . . 7 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ)
118	28	ovollb 25460	. . . . . . . . 9 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐾 ⊆ ∪ ran ((,) ∘ 𝐺)) → (vol‘𝐾) ≤ sup(ran 𝑇, ℝ^, < ))
119	3, 21, 118	sylancl 587	. . . . . . . 8 ⊢ (𝜑 → (vol‘𝐾) ≤ sup(ran 𝑇, ℝ^, < ))
120	37, 30, 117, 119, 29	letrd 11298	. . . . . . 7 ⊢ (𝜑 → (vol‘𝐾) ≤ ((vol‘𝐸) + 𝐶))
121	37, 117, 44, 120	leadd1dd 11759	. . . . . 6 ⊢ (𝜑 → ((vol‘𝐾) + 𝐶) ≤ (((vol‘𝐸) + 𝐶) + 𝐶))
122	10	recnd 11168	. . . . . . 7 ⊢ (𝜑 → (vol*‘𝐸) ∈ ℂ)
123	122, 78, 78	addassd 11162	. . . . . 6 ⊢ (𝜑 → (((vol‘𝐸) + 𝐶) + 𝐶) = ((vol‘𝐸) + (𝐶 + 𝐶)))
124	121, 123	breqtrd 5112	. . . . 5 ⊢ (𝜑 → ((vol‘𝐾) + 𝐶) ≤ ((vol‘𝐸) + (𝐶 + 𝐶)))
125	43, 56, 55, 116, 124	letrd 11298	. . . 4 ⊢ (𝜑 → ((vol‘(𝐾 ∩ 𝐴)) + (vol‘(𝐾 ∖ 𝐴))) ≤ ((vol*‘𝐸) + (𝐶 + 𝐶)))
126	43, 55, 45, 125	leadd1dd 11759	. . 3 ⊢ (𝜑 → (((vol‘(𝐾 ∩ 𝐴)) + (vol‘(𝐾 ∖ 𝐴))) + (𝐶 + 𝐶)) ≤ (((vol*‘𝐸) + (𝐶 + 𝐶)) + (𝐶 + 𝐶)))
127	45	recnd 11168	. . . . 5 ⊢ (𝜑 → (𝐶 + 𝐶) ∈ ℂ)
128	122, 127, 127	addassd 11162	. . . 4 ⊢ (𝜑 → (((vol‘𝐸) + (𝐶 + 𝐶)) + (𝐶 + 𝐶)) = ((vol‘𝐸) + ((𝐶 + 𝐶) + (𝐶 + 𝐶))))
129		2t2e4 12335	. . . . . . 7 ⊢ (2 · 2) = 4
130	129	oveq1i 7372	. . . . . 6 ⊢ ((2 · 2) · 𝐶) = (4 · 𝐶)
131		2cnd 12254	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
132	131, 131, 78	mulassd 11163	. . . . . . 7 ⊢ (𝜑 → ((2 · 2) · 𝐶) = (2 · (2 · 𝐶)))
133	78	2timesd 12415	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐶) = (𝐶 + 𝐶))
134	133	oveq2d 7378	. . . . . . 7 ⊢ (𝜑 → (2 · (2 · 𝐶)) = (2 · (𝐶 + 𝐶)))
135	127	2timesd 12415	. . . . . . 7 ⊢ (𝜑 → (2 · (𝐶 + 𝐶)) = ((𝐶 + 𝐶) + (𝐶 + 𝐶)))
136	132, 134, 135	3eqtrd 2776	. . . . . 6 ⊢ (𝜑 → ((2 · 2) · 𝐶) = ((𝐶 + 𝐶) + (𝐶 + 𝐶)))
137	130, 136	eqtr3id 2786	. . . . 5 ⊢ (𝜑 → (4 · 𝐶) = ((𝐶 + 𝐶) + (𝐶 + 𝐶)))
138	137	oveq2d 7378	. . . 4 ⊢ (𝜑 → ((vol‘𝐸) + (4 · 𝐶)) = ((vol‘𝐸) + ((𝐶 + 𝐶) + (𝐶 + 𝐶))))
139	128, 138	eqtr4d 2775	. . 3 ⊢ (𝜑 → (((vol‘𝐸) + (𝐶 + 𝐶)) + (𝐶 + 𝐶)) = ((vol‘𝐸) + (4 · 𝐶)))
140	126, 139	breqtrd 5112	. 2 ⊢ (𝜑 → (((vol‘(𝐾 ∩ 𝐴)) + (vol‘(𝐾 ∖ 𝐴))) + (𝐶 + 𝐶)) ≤ ((vol*‘𝐸) + (4 · 𝐶)))
141	16, 46, 50, 54, 140	letrd 11298	1 ⊢ (𝜑 → ((vol‘(𝐸 ∩ 𝐴)) + (vol‘(𝐸 ∖ 𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∖ cdif 3887 ∩ cin 3889 ⊆ wss 3890 𝒫 cpw 4542 ⟨cop 4574 ∪ cuni 4851 ∪ ciun 4934 Disj wdisj 5053 class class class wbr 5086 × cxp 5624 dom cdm 5626 ran crn 5627 “ cima 5629 ∘ ccom 5630 Fun wfun 6488 ⟶wf 6490 ‘cfv 6494 (class class class)co 7362 1^st c1st 7935 2^nd c2nd 7936 Fincfn 8888 supcsup 9348 ℝcr 11032 0cc0 11033 1c1 11034 + caddc 11036 · cmul 11038 ℝ^*cxr 11173 < clt 11174 ≤ cle 11175 − cmin 11372 / cdiv 11802 ℕcn 12169 2c2 12231 4c4 12233 ℝ⁺crp 12937 (,)cioo 13293 [,]cicc 13296 ...cfz 13456 seqcseq 13958 abscabs 15191 Σcsu 15643 vol*covol 25443 volcvol 25444

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-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-inf2 9557 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111

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-rmo 3343 df-reu 3344 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-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-se 5580 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7626 df-om 7813 df-1st 7937 df-2nd 7938 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-2o 8401 df-er 8638 df-map 8770 df-pm 8771 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-fi 9319 df-sup 9350 df-inf 9351 df-oi 9420 df-dju 9820 df-card 9858 df-acn 9861 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-n0 12433 df-z 12520 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-ioo 13297 df-ico 13299 df-icc 13300 df-fz 13457 df-fzo 13604 df-fl 13746 df-seq 13959 df-exp 14019 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-clim 15445 df-rlim 15446 df-sum 15644 df-rest 17380 df-topgen 17401 df-psmet 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-top 22873 df-topon 22890 df-bases 22925 df-cmp 23366 df-ovol 25445 df-vol 25446

This theorem is referenced by: uniioombllem6 25569

W3C validator