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Theorem uniioombllem2 25484

Description: Lemma for uniioombl 25490. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 11-Dec-2016.) (Revised by AV, 13-Sep-2020.)

**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))
uniioombllem2.h	⊢ 𝐻 = (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))
uniioombllem2.k	⊢ 𝐾 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩))

**Assertion**
Ref	Expression
uniioombllem2	⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → seq1( + , (vol* ∘ 𝐻)) ⇝ (vol*‘(((,)‘(𝐺‘𝐽)) ∩ 𝐴)))

Distinct variable groups: 𝑥,𝑧,𝐹 𝑥,𝐺,𝑧 𝑥,𝐾,𝑧 𝑥,𝐴,𝑧 𝑥,𝐶,𝑧 𝑥,𝐻,𝑧 𝑥,𝐽,𝑧 𝜑,𝑥,𝑧 𝑥,𝑇,𝑧 Allowed substitution hints: 𝑆(𝑥,𝑧) 𝐸(𝑥,𝑧)

Proof of Theorem uniioombllem2 Dummy variables 𝑛 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 12836	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		eqid 2729	. . 3 ⊢ seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) = seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))
3		1zzd 12564	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 1 ∈ ℤ)
4		eqidd 2730	. . 3 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))‘𝑛) = (((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))‘𝑛))
5		uniioombl.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
6		uniioombl.2	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))
7		uniioombl.3	. . . . . . . . . 10 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
8		uniioombl.a	. . . . . . . . . 10 ⊢ 𝐴 = ∪ ran ((,) ∘ 𝐹)
9		uniioombl.e	. . . . . . . . . 10 ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)
10		uniioombl.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
11		uniioombl.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
12		uniioombl.s	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐺))
13		uniioombl.t	. . . . . . . . . 10 ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
14		uniioombl.v	. . . . . . . . . 10 ⊢ (𝜑 → sup(ran 𝑇, ℝ^, < ) ≤ ((vol‘𝐸) + 𝐶))
15	5, 6, 7, 8, 9, 10, 11, 12, 13, 14	uniioombllem2a 25483	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ∈ ran (,))
16		uniioombllem2.h	. . . . . . . . . 10 ⊢ 𝐻 = (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))
17	16	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐻 = (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))
18		uniioombllem2.k	. . . . . . . . . . . 12 ⊢ 𝐾 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩))
19	18	ioorf 25474	. . . . . . . . . . 11 ⊢ 𝐾:ran (,)⟶( ≤ ∩ (ℝ^* × ℝ^*))
20	19	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐾:ran (,)⟶( ≤ ∩ (ℝ^* × ℝ^*)))
21	20	feqmptd 6929	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐾 = (𝑦 ∈ ran (,) ↦ (𝐾‘𝑦)))
22		fveq2 6858	. . . . . . . . 9 ⊢ (𝑦 = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) → (𝐾‘𝑦) = (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))
23	15, 17, 21, 22	fmptco 7101	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐾 ∘ 𝐻) = (𝑧 ∈ ℕ ↦ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))))
24		inss2 4201	. . . . . . . . . . 11 ⊢ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ⊆ ((,)‘(𝐺‘𝐽))
25		inss2 4201	. . . . . . . . . . . . . . . 16 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
26	11	ffvelcdmda 7056	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) ∈ ( ≤ ∩ (ℝ × ℝ)))
27	25, 26	sselid 3944	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) ∈ (ℝ × ℝ))
28		1st2nd2 8007	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘𝐽) ∈ (ℝ × ℝ) → (𝐺‘𝐽) = ⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩)
29	27, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) = ⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩)
30	29	fveq2d 6862	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,)‘(𝐺‘𝐽)) = ((,)‘⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩))
31		df-ov 7390	. . . . . . . . . . . . 13 ⊢ ((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽))) = ((,)‘⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩)
32	30, 31	eqtr4di 2782	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,)‘(𝐺‘𝐽)) = ((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽))))
33		ioossre 13368	. . . . . . . . . . . 12 ⊢ ((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽))) ⊆ ℝ
34	32, 33	eqsstrdi 3991	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,)‘(𝐺‘𝐽)) ⊆ ℝ)
35	32	fveq2d 6862	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (vol‘((,)‘(𝐺‘𝐽))) = (vol‘((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽)))))
36		ovolfcl 25367	. . . . . . . . . . . . . . 15 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐽 ∈ ℕ) → ((1^st ‘(𝐺‘𝐽)) ∈ ℝ ∧ (2^nd ‘(𝐺‘𝐽)) ∈ ℝ ∧ (1^st ‘(𝐺‘𝐽)) ≤ (2^nd ‘(𝐺‘𝐽))))
37	11, 36	sylan 580	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((1^st ‘(𝐺‘𝐽)) ∈ ℝ ∧ (2^nd ‘(𝐺‘𝐽)) ∈ ℝ ∧ (1^st ‘(𝐺‘𝐽)) ≤ (2^nd ‘(𝐺‘𝐽))))
38		ovolioo 25469	. . . . . . . . . . . . . 14 ⊢ (((1^st ‘(𝐺‘𝐽)) ∈ ℝ ∧ (2^nd ‘(𝐺‘𝐽)) ∈ ℝ ∧ (1^st ‘(𝐺‘𝐽)) ≤ (2^nd ‘(𝐺‘𝐽))) → (vol*‘((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽)))) = ((2^nd ‘(𝐺‘𝐽)) − (1^st ‘(𝐺‘𝐽))))
39	37, 38	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (vol*‘((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽)))) = ((2^nd ‘(𝐺‘𝐽)) − (1^st ‘(𝐺‘𝐽))))
40	35, 39	eqtrd 2764	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (vol*‘((,)‘(𝐺‘𝐽))) = ((2^nd ‘(𝐺‘𝐽)) − (1^st ‘(𝐺‘𝐽))))
41	37	simp2d 1143	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (2^nd ‘(𝐺‘𝐽)) ∈ ℝ)
42	37	simp1d 1142	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (1^st ‘(𝐺‘𝐽)) ∈ ℝ)
43	41, 42	resubcld 11606	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((2^nd ‘(𝐺‘𝐽)) − (1^st ‘(𝐺‘𝐽))) ∈ ℝ)
44	40, 43	eqeltrd 2828	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (vol*‘((,)‘(𝐺‘𝐽))) ∈ ℝ)
45		ovolsscl 25387	. . . . . . . . . . 11 ⊢ (((((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ⊆ ((,)‘(𝐺‘𝐽)) ∧ ((,)‘(𝐺‘𝐽)) ⊆ ℝ ∧ (vol‘((,)‘(𝐺‘𝐽))) ∈ ℝ) → (vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) ∈ ℝ)
46	24, 34, 44, 45	mp3an2i 1468	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (vol*‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) ∈ ℝ)
47	46	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (vol*‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) ∈ ℝ)
48	18	ioorcl 25478	. . . . . . . . 9 ⊢ (((((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ∈ ran (,) ∧ (vol*‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) ∈ ℝ) → (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) ∈ ( ≤ ∩ (ℝ × ℝ)))
49	15, 47, 48	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) ∈ ( ≤ ∩ (ℝ × ℝ)))
50	23, 49	fmpt3d 7088	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐾 ∘ 𝐻):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
51		eqid 2729	. . . . . . . 8 ⊢ ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)) = ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))
52	51	ovolfsf 25372	. . . . . . 7 ⊢ ((𝐾 ∘ 𝐻):ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)):ℕ⟶(0[,)+∞))
53	50, 52	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)):ℕ⟶(0[,)+∞))
54	53	ffvelcdmda 7056	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))‘𝑛) ∈ (0[,)+∞))
55		elrege0 13415	. . . . 5 ⊢ ((((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))‘𝑛) ∈ (0[,)+∞) ↔ ((((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))‘𝑛)))
56	54, 55	sylib 218	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → ((((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))‘𝑛)))
57	56	simpld 494	. . 3 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))‘𝑛) ∈ ℝ)
58	56	simprd 495	. . 3 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → 0 ≤ (((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))‘𝑛))
59	23	fveq1d 6860	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((𝐾 ∘ 𝐻)‘𝑧) = ((𝑧 ∈ ℕ ↦ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))‘𝑧))
60		fvex 6871	. . . . . . . . . . . . . . . 16 ⊢ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) ∈ V
61		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℕ ↦ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))) = (𝑧 ∈ ℕ ↦ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))
62	61	fvmpt2 6979	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℕ ∧ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) ∈ V) → ((𝑧 ∈ ℕ ↦ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))‘𝑧) = (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))
63	60, 62	mpan2 691	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℕ → ((𝑧 ∈ ℕ ↦ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))‘𝑧) = (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))
64	59, 63	sylan9eq 2784	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((𝐾 ∘ 𝐻)‘𝑧) = (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))
65	64	fveq2d 6862	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘((𝐾 ∘ 𝐻)‘𝑧)) = ((,)‘(𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))))
66	18	ioorinv 25477	. . . . . . . . . . . . . 14 ⊢ ((((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ∈ ran (,) → ((,)‘(𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))
67	15, 66	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))
68	65, 67	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘((𝐾 ∘ 𝐻)‘𝑧)) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))
69	68	ralrimiva 3125	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∀𝑧 ∈ ℕ ((,)‘((𝐾 ∘ 𝐻)‘𝑧)) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))
70		2fveq3 6863	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → ((,)‘((𝐾 ∘ 𝐻)‘𝑧)) = ((,)‘((𝐾 ∘ 𝐻)‘𝑥)))
71		2fveq3 6863	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → ((,)‘(𝐹‘𝑧)) = ((,)‘(𝐹‘𝑥)))
72	71	ineq1d 4182	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) = (((,)‘(𝐹‘𝑥)) ∩ ((,)‘(𝐺‘𝐽))))
73	70, 72	eqeq12d 2745	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (((,)‘((𝐾 ∘ 𝐻)‘𝑧)) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ↔ ((,)‘((𝐾 ∘ 𝐻)‘𝑥)) = (((,)‘(𝐹‘𝑥)) ∩ ((,)‘(𝐺‘𝐽)))))
74	73	rspccva 3587	. . . . . . . . . . 11 ⊢ ((∀𝑧 ∈ ℕ ((,)‘((𝐾 ∘ 𝐻)‘𝑧)) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ∧ 𝑥 ∈ ℕ) → ((,)‘((𝐾 ∘ 𝐻)‘𝑥)) = (((,)‘(𝐹‘𝑥)) ∩ ((,)‘(𝐺‘𝐽))))
75	69, 74	sylan 580	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑥 ∈ ℕ) → ((,)‘((𝐾 ∘ 𝐻)‘𝑥)) = (((,)‘(𝐹‘𝑥)) ∩ ((,)‘(𝐺‘𝐽))))
76		inss1 4200	. . . . . . . . . 10 ⊢ (((,)‘(𝐹‘𝑥)) ∩ ((,)‘(𝐺‘𝐽))) ⊆ ((,)‘(𝐹‘𝑥))
77	75, 76	eqsstrdi 3991	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑥 ∈ ℕ) → ((,)‘((𝐾 ∘ 𝐻)‘𝑥)) ⊆ ((,)‘(𝐹‘𝑥)))
78	77	ralrimiva 3125	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∀𝑥 ∈ ℕ ((,)‘((𝐾 ∘ 𝐻)‘𝑥)) ⊆ ((,)‘(𝐹‘𝑥)))
79	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))
80		disjss2 5077	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℕ ((,)‘((𝐾 ∘ 𝐻)‘𝑥)) ⊆ ((,)‘(𝐹‘𝑥)) → (Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)) → Disj 𝑥 ∈ ℕ ((,)‘((𝐾 ∘ 𝐻)‘𝑥))))
81	78, 79, 80	sylc 65	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → Disj 𝑥 ∈ ℕ ((,)‘((𝐾 ∘ 𝐻)‘𝑥)))
82	50, 81, 2	uniioovol 25480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (vol‘∪ ran ((,) ∘ (𝐾 ∘ 𝐻))) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ^, < ))
83	67	mpteq2dva 5200	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝑧 ∈ ℕ ↦ ((,)‘(𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))) = (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))
84		rexpssxrxp 11219	. . . . . . . . . . . . . 14 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
85	25, 84	sstri 3956	. . . . . . . . . . . . 13 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
86	85, 49	sselid 3944	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) ∈ (ℝ^* × ℝ^*))
87		ioof 13408	. . . . . . . . . . . . . 14 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
88	87	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ)
89	88	feqmptd 6929	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (,) = (𝑦 ∈ (ℝ^* × ℝ^*) ↦ ((,)‘𝑦)))
90		fveq2 6858	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) → ((,)‘𝑦) = ((,)‘(𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))))
91	86, 23, 89, 90	fmptco 7101	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,) ∘ (𝐾 ∘ 𝐻)) = (𝑧 ∈ ℕ ↦ ((,)‘(𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))))
92	83, 91, 17	3eqtr4d 2774	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,) ∘ (𝐾 ∘ 𝐻)) = 𝐻)
93	92	rneqd 5902	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ran ((,) ∘ (𝐾 ∘ 𝐻)) = ran 𝐻)
94	93	unieqd 4884	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∪ ran ((,) ∘ (𝐾 ∘ 𝐻)) = ∪ ran 𝐻)
95		fvex 6871	. . . . . . . . . . . . . 14 ⊢ ((,)‘(𝐹‘𝑧)) ∈ V
96	95	inex1 5272	. . . . . . . . . . . . 13 ⊢ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ∈ V
97	16	fvmpt2 6979	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℕ ∧ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ∈ V) → (𝐻‘𝑧) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))
98	96, 97	mpan2 691	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℕ → (𝐻‘𝑧) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))
99		incom 4172	. . . . . . . . . . . 12 ⊢ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) = (((,)‘(𝐺‘𝐽)) ∩ ((,)‘(𝐹‘𝑧)))
100	98, 99	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℕ → (𝐻‘𝑧) = (((,)‘(𝐺‘𝐽)) ∩ ((,)‘(𝐹‘𝑧))))
101	100	iuneq2i 4977	. . . . . . . . . 10 ⊢ ∪ 𝑧 ∈ ℕ (𝐻‘𝑧) = ∪ 𝑧 ∈ ℕ (((,)‘(𝐺‘𝐽)) ∩ ((,)‘(𝐹‘𝑧)))
102		iunin2 5035	. . . . . . . . . 10 ⊢ ∪ 𝑧 ∈ ℕ (((,)‘(𝐺‘𝐽)) ∩ ((,)‘(𝐹‘𝑧))) = (((,)‘(𝐺‘𝐽)) ∩ ∪ 𝑧 ∈ ℕ ((,)‘(𝐹‘𝑧)))
103	101, 102	eqtri 2752	. . . . . . . . 9 ⊢ ∪ 𝑧 ∈ ℕ (𝐻‘𝑧) = (((,)‘(𝐺‘𝐽)) ∩ ∪ 𝑧 ∈ ℕ ((,)‘(𝐹‘𝑧)))
104	15, 16	fmptd 7086	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐻:ℕ⟶ran (,))
105	104	ffnd 6689	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐻 Fn ℕ)
106		fniunfv 7221	. . . . . . . . . 10 ⊢ (𝐻 Fn ℕ → ∪ 𝑧 ∈ ℕ (𝐻‘𝑧) = ∪ ran 𝐻)
107	105, 106	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∪ 𝑧 ∈ ℕ (𝐻‘𝑧) = ∪ ran 𝐻)
108	103, 107	eqtr3id 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (((,)‘(𝐺‘𝐽)) ∩ ∪ 𝑧 ∈ ℕ ((,)‘(𝐹‘𝑧))) = ∪ ran 𝐻)
109	5	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
110		fvco3 6960	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑧 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑧) = ((,)‘(𝐹‘𝑧)))
111	109, 110	sylan 580	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑧) = ((,)‘(𝐹‘𝑧)))
112	111	iuneq2dv 4980	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∪ 𝑧 ∈ ℕ (((,) ∘ 𝐹)‘𝑧) = ∪ 𝑧 ∈ ℕ ((,)‘(𝐹‘𝑧)))
113		ffn 6688	. . . . . . . . . . . . . 14 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
114	87, 113	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (,) Fn (ℝ^* × ℝ^*)
115		fss 6704	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^)) → 𝐹:ℕ⟶(ℝ^ × ℝ^*))
116	109, 85, 115	sylancl 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐹:ℕ⟶(ℝ^* × ℝ^*))
117		fnfco 6725	. . . . . . . . . . . . 13 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ 𝐹:ℕ⟶(ℝ^ × ℝ^*)) → ((,) ∘ 𝐹) Fn ℕ)
118	114, 116, 117	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,) ∘ 𝐹) Fn ℕ)
119		fniunfv 7221	. . . . . . . . . . . 12 ⊢ (((,) ∘ 𝐹) Fn ℕ → ∪ 𝑧 ∈ ℕ (((,) ∘ 𝐹)‘𝑧) = ∪ ran ((,) ∘ 𝐹))
120	118, 119	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∪ 𝑧 ∈ ℕ (((,) ∘ 𝐹)‘𝑧) = ∪ ran ((,) ∘ 𝐹))
121	120, 8	eqtr4di 2782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∪ 𝑧 ∈ ℕ (((,) ∘ 𝐹)‘𝑧) = 𝐴)
122	112, 121	eqtr3d 2766	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∪ 𝑧 ∈ ℕ ((,)‘(𝐹‘𝑧)) = 𝐴)
123	122	ineq2d 4183	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (((,)‘(𝐺‘𝐽)) ∩ ∪ 𝑧 ∈ ℕ ((,)‘(𝐹‘𝑧))) = (((,)‘(𝐺‘𝐽)) ∩ 𝐴))
124	94, 108, 123	3eqtr2d 2770	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∪ ran ((,) ∘ (𝐾 ∘ 𝐻)) = (((,)‘(𝐺‘𝐽)) ∩ 𝐴))
125	124	fveq2d 6862	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (vol‘∪ ran ((,) ∘ (𝐾 ∘ 𝐻))) = (vol‘(((,)‘(𝐺‘𝐽)) ∩ 𝐴)))
126	82, 125	eqtr3d 2766	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ^, < ) = (vol‘(((,)‘(𝐺‘𝐽)) ∩ 𝐴)))
127		inss1 4200	. . . . . 6 ⊢ (((,)‘(𝐺‘𝐽)) ∩ 𝐴) ⊆ ((,)‘(𝐺‘𝐽))
128		ovolsscl 25387	. . . . . 6 ⊢ (((((,)‘(𝐺‘𝐽)) ∩ 𝐴) ⊆ ((,)‘(𝐺‘𝐽)) ∧ ((,)‘(𝐺‘𝐽)) ⊆ ℝ ∧ (vol‘((,)‘(𝐺‘𝐽))) ∈ ℝ) → (vol‘(((,)‘(𝐺‘𝐽)) ∩ 𝐴)) ∈ ℝ)
129	127, 34, 44, 128	mp3an2i 1468	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (vol*‘(((,)‘(𝐺‘𝐽)) ∩ 𝐴)) ∈ ℝ)
130	126, 129	eqeltrd 2828	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ^*, < ) ∈ ℝ)
131	51, 2	ovolsf 25373	. . . . . . . . 9 ⊢ ((𝐾 ∘ 𝐻):ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))):ℕ⟶(0[,)+∞))
132	50, 131	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))):ℕ⟶(0[,)+∞))
133	132	frnd 6696	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ⊆ (0[,)+∞))
134		icossxr 13393	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ^*
135	133, 134	sstrdi 3959	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ⊆ ℝ^*)
136	132	ffnd 6689	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) Fn ℕ)
137		fnfvelrn 7052	. . . . . . 7 ⊢ ((seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) Fn ℕ ∧ 𝑦 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))))
138	136, 137	sylan 580	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑦 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))))
139		supxrub 13284	. . . . . 6 ⊢ ((ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ⊆ ℝ^* ∧ (seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))) → (seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ^*, < ))
140	135, 138, 139	syl2an2r 685	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑦 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ^*, < ))
141	140	ralrimiva 3125	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∀𝑦 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ^*, < ))
142		brralrspcev 5167	. . . 4 ⊢ ((sup(ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ^, < ) ∈ ℝ ∧ ∀𝑦 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ^, < )) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ 𝑥)
143	130, 141, 142	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ 𝑥)
144	1, 2, 3, 4, 57, 58, 143	isumsup2 15812	. 2 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ⇝ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ, < ))
145	51	ovolfs2 25472	. . . . 5 ⊢ ((𝐾 ∘ 𝐻):ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)) = ((vol* ∘ (,)) ∘ (𝐾 ∘ 𝐻)))
146	50, 145	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)) = ((vol* ∘ (,)) ∘ (𝐾 ∘ 𝐻)))
147		coass 6238	. . . . 5 ⊢ ((vol* ∘ (,)) ∘ (𝐾 ∘ 𝐻)) = (vol* ∘ ((,) ∘ (𝐾 ∘ 𝐻)))
148	92	coeq2d 5826	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (vol* ∘ ((,) ∘ (𝐾 ∘ 𝐻))) = (vol* ∘ 𝐻))
149	147, 148	eqtrid 2776	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((vol* ∘ (,)) ∘ (𝐾 ∘ 𝐻)) = (vol* ∘ 𝐻))
150	146, 149	eqtrd 2764	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)) = (vol* ∘ 𝐻))
151	150	seqeq3d 13974	. 2 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) = seq1( + , (vol* ∘ 𝐻)))
152		rge0ssre 13417	. . . . 5 ⊢ (0[,)+∞) ⊆ ℝ
153	133, 152	sstrdi 3959	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ⊆ ℝ)
154		1nn 12197	. . . . . . 7 ⊢ 1 ∈ ℕ
155	132	fdmd 6698	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → dom seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) = ℕ)
156	154, 155	eleqtrrid 2835	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 1 ∈ dom seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))))
157	156	ne0d 4305	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → dom seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ≠ ∅)
158		dm0rn0 5888	. . . . . 6 ⊢ (dom seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) = ∅ ↔ ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) = ∅)
159	158	necon3bii 2977	. . . . 5 ⊢ (dom seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ≠ ∅ ↔ ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ≠ ∅)
160	157, 159	sylib 218	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ≠ ∅)
161		breq1 5110	. . . . . . . 8 ⊢ (𝑧 = (seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) → (𝑧 ≤ 𝑥 ↔ (seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ 𝑥))
162	161	ralrn 7060	. . . . . . 7 ⊢ (seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) Fn ℕ → (∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))𝑧 ≤ 𝑥 ↔ ∀𝑦 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ 𝑥))
163	136, 162	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))𝑧 ≤ 𝑥 ↔ ∀𝑦 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ 𝑥))
164	163	rexbidv 3157	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ 𝑥))
165	143, 164	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))𝑧 ≤ 𝑥)
166		supxrre 13287	. . . 4 ⊢ ((ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ⊆ ℝ ∧ ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))𝑧 ≤ 𝑥) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ^*, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ, < ))
167	153, 160, 165, 166	syl3anc 1373	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ^*, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ, < ))
168	167, 126	eqtr3d 2766	. 2 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ, < ) = (vol*‘(((,)‘(𝐺‘𝐽)) ∩ 𝐴)))
169	144, 151, 168	3brtr3d 5138	1 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → seq1( + , (vol* ∘ 𝐻)) ⇝ (vol*‘(((,)‘(𝐺‘𝐽)) ∩ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 Vcvv 3447 ∩ cin 3913 ⊆ wss 3914 ∅c0 4296 ifcif 4488 𝒫 cpw 4563 ⟨cop 4595 ∪ cuni 4871 ∪ ciun 4955 Disj wdisj 5074 class class class wbr 5107 ↦ cmpt 5188 × cxp 5636 dom cdm 5638 ran crn 5639 ∘ ccom 5642 Fn wfn 6506 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 1^st c1st 7966 2^nd c2nd 7967 supcsup 9391 infcinf 9392 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 +∞cpnf 11205 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 − cmin 11405 ℕcn 12186 ℝ⁺crp 12951 (,)cioo 13306 [,)cico 13308 seqcseq 13966 abscabs 15200 ⇝ cli 15450 vol*covol 25363

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-disj 5075 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-rlim 15455 df-sum 15653 df-rest 17385 df-topgen 17406 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-top 22781 df-topon 22798 df-bases 22833 df-cmp 23274 df-ovol 25365 df-vol 25366

This theorem is referenced by: uniioombllem6 25489

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