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Theorem uniioombllem2 25511

Description: Lemma for uniioombl 25517. (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 12775	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		eqid 2731	. . 3 ⊢ seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) = seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))
3		1zzd 12503	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 1 ∈ ℤ)
4		eqidd 2732	. . 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 25510	. . . . . . . . 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 25501	. . . . . . . . . . 11 ⊢ 𝐾:ran (,)⟶( ≤ ∩ (ℝ^* × ℝ^*))
20	19	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐾:ran (,)⟶( ≤ ∩ (ℝ^* × ℝ^*)))
21	20	feqmptd 6890	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐾 = (𝑦 ∈ ran (,) ↦ (𝐾‘𝑦)))
22		fveq2 6822	. . . . . . . . 9 ⊢ (𝑦 = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) → (𝐾‘𝑦) = (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))
23	15, 17, 21, 22	fmptco 7062	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐾 ∘ 𝐻) = (𝑧 ∈ ℕ ↦ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))))
24		inss2 4185	. . . . . . . . . . 11 ⊢ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ⊆ ((,)‘(𝐺‘𝐽))
25		inss2 4185	. . . . . . . . . . . . . . . 16 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
26	11	ffvelcdmda 7017	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) ∈ ( ≤ ∩ (ℝ × ℝ)))
27	25, 26	sselid 3927	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) ∈ (ℝ × ℝ))
28		1st2nd2 7960	. . . . . . . . . . . . . . 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 6826	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,)‘(𝐺‘𝐽)) = ((,)‘⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩))
31		df-ov 7349	. . . . . . . . . . . . 13 ⊢ ((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽))) = ((,)‘⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩)
32	30, 31	eqtr4di 2784	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,)‘(𝐺‘𝐽)) = ((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽))))
33		ioossre 13307	. . . . . . . . . . . 12 ⊢ ((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽))) ⊆ ℝ
34	32, 33	eqsstrdi 3974	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,)‘(𝐺‘𝐽)) ⊆ ℝ)
35	32	fveq2d 6826	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (vol‘((,)‘(𝐺‘𝐽))) = (vol‘((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽)))))
36		ovolfcl 25394	. . . . . . . . . . . . . . 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 25496	. . . . . . . . . . . . . 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 2766	. . . . . . . . . . . 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 11545	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((2^nd ‘(𝐺‘𝐽)) − (1^st ‘(𝐺‘𝐽))) ∈ ℝ)
44	40, 43	eqeltrd 2831	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (vol*‘((,)‘(𝐺‘𝐽))) ∈ ℝ)
45		ovolsscl 25414	. . . . . . . . . . 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 25505	. . . . . . . . 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 7049	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐾 ∘ 𝐻):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
51		eqid 2731	. . . . . . . 8 ⊢ ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)) = ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))
52	51	ovolfsf 25399	. . . . . . 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 7017	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))‘𝑛) ∈ (0[,)+∞))
55		elrege0 13354	. . . . 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 6824	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((𝐾 ∘ 𝐻)‘𝑧) = ((𝑧 ∈ ℕ ↦ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))‘𝑧))
60		fvex 6835	. . . . . . . . . . . . . . . 16 ⊢ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) ∈ V
61		eqid 2731	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℕ ↦ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))) = (𝑧 ∈ ℕ ↦ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))
62	61	fvmpt2 6940	. . . . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((𝐾 ∘ 𝐻)‘𝑧) = (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))
65	64	fveq2d 6826	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘((𝐾 ∘ 𝐻)‘𝑧)) = ((,)‘(𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))))
66	18	ioorinv 25504	. . . . . . . . . . . . . 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 2766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘((𝐾 ∘ 𝐻)‘𝑧)) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))
69	68	ralrimiva 3124	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∀𝑧 ∈ ℕ ((,)‘((𝐾 ∘ 𝐻)‘𝑧)) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))
70		2fveq3 6827	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → ((,)‘((𝐾 ∘ 𝐻)‘𝑧)) = ((,)‘((𝐾 ∘ 𝐻)‘𝑥)))
71		2fveq3 6827	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → ((,)‘(𝐹‘𝑧)) = ((,)‘(𝐹‘𝑥)))
72	71	ineq1d 4166	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) = (((,)‘(𝐹‘𝑥)) ∩ ((,)‘(𝐺‘𝐽))))
73	70, 72	eqeq12d 2747	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (((,)‘((𝐾 ∘ 𝐻)‘𝑧)) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ↔ ((,)‘((𝐾 ∘ 𝐻)‘𝑥)) = (((,)‘(𝐹‘𝑥)) ∩ ((,)‘(𝐺‘𝐽)))))
74	73	rspccva 3571	. . . . . . . . . . 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 4184	. . . . . . . . . 10 ⊢ (((,)‘(𝐹‘𝑥)) ∩ ((,)‘(𝐺‘𝐽))) ⊆ ((,)‘(𝐹‘𝑥))
77	75, 76	eqsstrdi 3974	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑥 ∈ ℕ) → ((,)‘((𝐾 ∘ 𝐻)‘𝑥)) ⊆ ((,)‘(𝐹‘𝑥)))
78	77	ralrimiva 3124	. . . . . . . 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 5059	. . . . . . . 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 25507	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (vol‘∪ ran ((,) ∘ (𝐾 ∘ 𝐻))) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ^, < ))
83	67	mpteq2dva 5182	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝑧 ∈ ℕ ↦ ((,)‘(𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))) = (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))
84		rexpssxrxp 11157	. . . . . . . . . . . . . 14 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
85	25, 84	sstri 3939	. . . . . . . . . . . . 13 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
86	85, 49	sselid 3927	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) ∈ (ℝ^* × ℝ^*))
87		ioof 13347	. . . . . . . . . . . . . 14 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
88	87	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ)
89	88	feqmptd 6890	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (,) = (𝑦 ∈ (ℝ^* × ℝ^*) ↦ ((,)‘𝑦)))
90		fveq2 6822	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) → ((,)‘𝑦) = ((,)‘(𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))))
91	86, 23, 89, 90	fmptco 7062	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,) ∘ (𝐾 ∘ 𝐻)) = (𝑧 ∈ ℕ ↦ ((,)‘(𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))))
92	83, 91, 17	3eqtr4d 2776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,) ∘ (𝐾 ∘ 𝐻)) = 𝐻)
93	92	rneqd 5877	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ran ((,) ∘ (𝐾 ∘ 𝐻)) = ran 𝐻)
94	93	unieqd 4869	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∪ ran ((,) ∘ (𝐾 ∘ 𝐻)) = ∪ ran 𝐻)
95		fvex 6835	. . . . . . . . . . . . . 14 ⊢ ((,)‘(𝐹‘𝑧)) ∈ V
96	95	inex1 5253	. . . . . . . . . . . . 13 ⊢ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ∈ V
97	16	fvmpt2 6940	. . . . . . . . . . . . 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 4156	. . . . . . . . . . . 12 ⊢ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) = (((,)‘(𝐺‘𝐽)) ∩ ((,)‘(𝐹‘𝑧)))
100	98, 99	eqtrdi 2782	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℕ → (𝐻‘𝑧) = (((,)‘(𝐺‘𝐽)) ∩ ((,)‘(𝐹‘𝑧))))
101	100	iuneq2i 4961	. . . . . . . . . 10 ⊢ ∪ 𝑧 ∈ ℕ (𝐻‘𝑧) = ∪ 𝑧 ∈ ℕ (((,)‘(𝐺‘𝐽)) ∩ ((,)‘(𝐹‘𝑧)))
102		iunin2 5017	. . . . . . . . . 10 ⊢ ∪ 𝑧 ∈ ℕ (((,)‘(𝐺‘𝐽)) ∩ ((,)‘(𝐹‘𝑧))) = (((,)‘(𝐺‘𝐽)) ∩ ∪ 𝑧 ∈ ℕ ((,)‘(𝐹‘𝑧)))
103	101, 102	eqtri 2754	. . . . . . . . 9 ⊢ ∪ 𝑧 ∈ ℕ (𝐻‘𝑧) = (((,)‘(𝐺‘𝐽)) ∩ ∪ 𝑧 ∈ ℕ ((,)‘(𝐹‘𝑧)))
104	15, 16	fmptd 7047	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐻:ℕ⟶ran (,))
105	104	ffnd 6652	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐻 Fn ℕ)
106		fniunfv 7181	. . . . . . . . . 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 2780	. . . . . . . 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 6921	. . . . . . . . . . . 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 4964	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∪ 𝑧 ∈ ℕ (((,) ∘ 𝐹)‘𝑧) = ∪ 𝑧 ∈ ℕ ((,)‘(𝐹‘𝑧)))
113		ffn 6651	. . . . . . . . . . . . . 14 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
114	87, 113	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (,) Fn (ℝ^* × ℝ^*)
115		fss 6667	. . . . . . . . . . . . . 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 6688	. . . . . . . . . . . . 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 7181	. . . . . . . . . . . 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 2784	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∪ 𝑧 ∈ ℕ (((,) ∘ 𝐹)‘𝑧) = 𝐴)
122	112, 121	eqtr3d 2768	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∪ 𝑧 ∈ ℕ ((,)‘(𝐹‘𝑧)) = 𝐴)
123	122	ineq2d 4167	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (((,)‘(𝐺‘𝐽)) ∩ ∪ 𝑧 ∈ ℕ ((,)‘(𝐹‘𝑧))) = (((,)‘(𝐺‘𝐽)) ∩ 𝐴))
124	94, 108, 123	3eqtr2d 2772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∪ ran ((,) ∘ (𝐾 ∘ 𝐻)) = (((,)‘(𝐺‘𝐽)) ∩ 𝐴))
125	124	fveq2d 6826	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (vol‘∪ ran ((,) ∘ (𝐾 ∘ 𝐻))) = (vol‘(((,)‘(𝐺‘𝐽)) ∩ 𝐴)))
126	82, 125	eqtr3d 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ^, < ) = (vol‘(((,)‘(𝐺‘𝐽)) ∩ 𝐴)))
127		inss1 4184	. . . . . 6 ⊢ (((,)‘(𝐺‘𝐽)) ∩ 𝐴) ⊆ ((,)‘(𝐺‘𝐽))
128		ovolsscl 25414	. . . . . 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 2831	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ^*, < ) ∈ ℝ)
131	51, 2	ovolsf 25400	. . . . . . . . 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 6659	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ⊆ (0[,)+∞))
134		icossxr 13332	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ^*
135	133, 134	sstrdi 3942	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ⊆ ℝ^*)
136	132	ffnd 6652	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) Fn ℕ)
137		fnfvelrn 7013	. . . . . . 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 13223	. . . . . 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 3124	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∀𝑦 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ^*, < ))
142		brralrspcev 5149	. . . 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 15753	. 2 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ⇝ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ, < ))
145	51	ovolfs2 25499	. . . . 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 6213	. . . . 5 ⊢ ((vol* ∘ (,)) ∘ (𝐾 ∘ 𝐻)) = (vol* ∘ ((,) ∘ (𝐾 ∘ 𝐻)))
148	92	coeq2d 5801	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (vol* ∘ ((,) ∘ (𝐾 ∘ 𝐻))) = (vol* ∘ 𝐻))
149	147, 148	eqtrid 2778	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((vol* ∘ (,)) ∘ (𝐾 ∘ 𝐻)) = (vol* ∘ 𝐻))
150	146, 149	eqtrd 2766	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)) = (vol* ∘ 𝐻))
151	150	seqeq3d 13916	. 2 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) = seq1( + , (vol* ∘ 𝐻)))
152		rge0ssre 13356	. . . . 5 ⊢ (0[,)+∞) ⊆ ℝ
153	133, 152	sstrdi 3942	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ⊆ ℝ)
154		1nn 12136	. . . . . . 7 ⊢ 1 ∈ ℕ
155	132	fdmd 6661	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → dom seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) = ℕ)
156	154, 155	eleqtrrid 2838	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 1 ∈ dom seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))))
157	156	ne0d 4289	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → dom seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ≠ ∅)
158		dm0rn0 5863	. . . . . 6 ⊢ (dom seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) = ∅ ↔ ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) = ∅)
159	158	necon3bii 2980	. . . . 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 5092	. . . . . . . 8 ⊢ (𝑧 = (seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) → (𝑧 ≤ 𝑥 ↔ (seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ 𝑥))
162	161	ralrn 7021	. . . . . . 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 3156	. . . . 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 13226	. . . 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 2768	. 2 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ, < ) = (vol*‘(((,)‘(𝐺‘𝐽)) ∩ 𝐴)))
169	144, 151, 168	3brtr3d 5120	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 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 Vcvv 3436 ∩ cin 3896 ⊆ wss 3897 ∅c0 4280 ifcif 4472 𝒫 cpw 4547 ⟨cop 4579 ∪ cuni 4856 ∪ ciun 4939 Disj wdisj 5056 class class class wbr 5089 ↦ cmpt 5170 × cxp 5612 dom cdm 5614 ran crn 5615 ∘ ccom 5618 Fn wfn 6476 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 1^st c1st 7919 2^nd c2nd 7920 supcsup 9324 infcinf 9325 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 +∞cpnf 11143 ℝ^*cxr 11145 < clt 11146 ≤ cle 11147 − cmin 11344 ℕcn 12125 ℝ⁺crp 12890 (,)cioo 13245 [,)cico 13247 seqcseq 13908 abscabs 15141 ⇝ cli 15391 vol*covol 25390

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-disj 5057 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-dju 9794 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-n0 12382 df-z 12469 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-rlim 15396 df-sum 15594 df-rest 17326 df-topgen 17347 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-top 22809 df-topon 22826 df-bases 22861 df-cmp 23302 df-ovol 25392 df-vol 25393

This theorem is referenced by: uniioombllem6 25516

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