Theorem ovolshftlem1 24673

Description: Lemma for ovolshft 24675. (Contributed by Mario Carneiro, 22-Mar-2014.)

Hypotheses

Ref	Expression
ovolshft.1	⊢ (𝜑 → 𝐴 ⊆ ℝ)
ovolshft.2	⊢ (𝜑 → 𝐶 ∈ ℝ)
ovolshft.3	⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴})
ovolshft.4	⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}
ovolshft.5	⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
ovolshft.6	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩)
ovolshft.7	⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
ovolshft.8	⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹))

Assertion

Ref	Expression
ovolshftlem1	⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ 𝑀)

Distinct variable groups:   𝑓,𝑛,𝑥,𝑦,𝐴   𝐶,𝑓,𝑛,𝑥,𝑦   𝑛,𝐹,𝑥   𝑓,𝐺,𝑛,𝑦   𝐵,𝑓,𝑛,𝑦   𝜑,𝑓,𝑛,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝑆(𝑥,𝑦,𝑓,𝑛)   𝐹(𝑦,𝑓)   𝐺(𝑥)   𝑀(𝑥,𝑦,𝑓,𝑛)

Proof of Theorem ovolshftlem1

Step	Hyp	Ref	Expression
1		ovolshft.7	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2		ovolfcl 24630	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))))
3	1, 2	sylan 580	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))))
4	3	simp1d 1141	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
5	3	simp2d 1142	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
6		ovolshft.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℝ)
7	6	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℝ)
8	3	simp3d 1143	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)))
9	4, 5, 7, 8	leadd1dd 11589	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) + 𝐶) ≤ ((2nd ‘(𝐹‘𝑛)) + 𝐶))
10		df-br 5075	. . . . . . . . . . 11 ⊢ (((1st ‘(𝐹‘𝑛)) + 𝐶) ≤ ((2nd ‘(𝐹‘𝑛)) + 𝐶) ↔ ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩ ∈ ≤ )
11	9, 10	sylib 217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩ ∈ ≤ )
12	4, 7	readdcld 11004	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) + 𝐶) ∈ ℝ)
13	5, 7	readdcld 11004	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐹‘𝑛)) + 𝐶) ∈ ℝ)
14	12, 13	opelxpd 5627	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩ ∈ (ℝ × ℝ))
15	11, 14	elind 4128	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
16		ovolshft.6	. . . . . . . . 9 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩)
17	15, 16	fmptd 6988	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
18		eqid 2738	. . . . . . . . 9 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
19	18	ovolfsf 24635	. . . . . . . 8 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
20		ffn 6600	. . . . . . . 8 ⊢ (((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞) → ((abs ∘ − ) ∘ 𝐺) Fn ℕ)
21	17, 19, 20	3syl 18	. . . . . . 7 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐺) Fn ℕ)
22		eqid 2738	. . . . . . . . 9 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
23	22	ovolfsf 24635	. . . . . . . 8 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
24		ffn 6600	. . . . . . . 8 ⊢ (((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞) → ((abs ∘ − ) ∘ 𝐹) Fn ℕ)
25	1, 23, 24	3syl 18	. . . . . . 7 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐹) Fn ℕ)
26		opex 5379	. . . . . . . . . . . . . 14 ⊢ ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩ ∈ V
27	16	fvmpt2 6886	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩ ∈ V) → (𝐺‘𝑛) = ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩)
28	26, 27	mpan2 688	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝐺‘𝑛) = ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩)
29	28	fveq2d 6778	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2nd ‘(𝐺‘𝑛)) = (2nd ‘⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩))
30		ovex 7308	. . . . . . . . . . . . 13 ⊢ ((1st ‘(𝐹‘𝑛)) + 𝐶) ∈ V
31		ovex 7308	. . . . . . . . . . . . 13 ⊢ ((2nd ‘(𝐹‘𝑛)) + 𝐶) ∈ V
32	30, 31	op2nd 7840	. . . . . . . . . . . 12 ⊢ (2nd ‘⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩) = ((2nd ‘(𝐹‘𝑛)) + 𝐶)
33	29, 32	eqtrdi 2794	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (2nd ‘(𝐺‘𝑛)) = ((2nd ‘(𝐹‘𝑛)) + 𝐶))
34	28	fveq2d 6778	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (1st ‘(𝐺‘𝑛)) = (1st ‘⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩))
35	30, 31	op1st 7839	. . . . . . . . . . . 12 ⊢ (1st ‘⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩) = ((1st ‘(𝐹‘𝑛)) + 𝐶)
36	34, 35	eqtrdi 2794	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (1st ‘(𝐺‘𝑛)) = ((1st ‘(𝐹‘𝑛)) + 𝐶))
37	33, 36	oveq12d 7293	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))) = (((2nd ‘(𝐹‘𝑛)) + 𝐶) − ((1st ‘(𝐹‘𝑛)) + 𝐶)))
38	37	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))) = (((2nd ‘(𝐹‘𝑛)) + 𝐶) − ((1st ‘(𝐹‘𝑛)) + 𝐶)))
39	5	recnd 11003	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℂ)
40	4	recnd 11003	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℂ)
41	7	recnd 11003	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℂ)
42	39, 40, 41	pnpcan2d 11370	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((2nd ‘(𝐹‘𝑛)) + 𝐶) − ((1st ‘(𝐹‘𝑛)) + 𝐶)) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
43	38, 42	eqtrd 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
44	18	ovolfsval 24634	. . . . . . . . 9 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))))
45	17, 44	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))))
46	22	ovolfsval 24634	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
47	1, 46	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
48	43, 45, 47	3eqtr4d 2788	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))
49	21, 25, 48	eqfnfvd 6912	. . . . . 6 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐹))
50	49	seqeq3d 13729	. . . . 5 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐹)))
51		ovolshft.5	. . . . 5 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
52	50, 51	eqtr4di 2796	. . . 4 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) = 𝑆)
53	52	rneqd 5847	. . 3 ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) = ran 𝑆)
54	53	supeq1d 9205	. 2 ⊢ (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) = sup(ran 𝑆, ℝ*, < ))
55		ovolshft.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴})
56	55	eleq2d 2824	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴}))
57		oveq1 7282	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 − 𝐶) = (𝑦 − 𝐶))
58	57	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥 − 𝐶) ∈ 𝐴 ↔ (𝑦 − 𝐶) ∈ 𝐴))
59	58	elrab 3624	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴} ↔ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴))
60	56, 59	bitrdi 287	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)))
61	60	biimpa 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴))
62		breq2 5078	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 − 𝐶) → ((1st ‘(𝐹‘𝑛)) < 𝑥 ↔ (1st ‘(𝐹‘𝑛)) < (𝑦 − 𝐶)))
63		breq1 5077	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 − 𝐶) → (𝑥 < (2nd ‘(𝐹‘𝑛)) ↔ (𝑦 − 𝐶) < (2nd ‘(𝐹‘𝑛))))
64	62, 63	anbi12d 631	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 − 𝐶) → (((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) ↔ ((1st ‘(𝐹‘𝑛)) < (𝑦 − 𝐶) ∧ (𝑦 − 𝐶) < (2nd ‘(𝐹‘𝑛)))))
65	64	rexbidv 3226	. . . . . . . 8 ⊢ (𝑥 = (𝑦 − 𝐶) → (∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < (𝑦 − 𝐶) ∧ (𝑦 − 𝐶) < (2nd ‘(𝐹‘𝑛)))))
66		ovolshft.8	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹))
67		ovolshft.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
68		ovolfioo 24631	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) ↔ ∀𝑥 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛)))))
69	67, 1, 68	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) ↔ ∀𝑥 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛)))))
70	66, 69	mpbid 231	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))))
71	70	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))))
72		simprr 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) → (𝑦 − 𝐶) ∈ 𝐴)
73	65, 71, 72	rspcdva 3562	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < (𝑦 − 𝐶) ∧ (𝑦 − 𝐶) < (2nd ‘(𝐹‘𝑛))))
74	36	adantl 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺‘𝑛)) = ((1st ‘(𝐹‘𝑛)) + 𝐶))
75	74	breq1d 5084	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺‘𝑛)) < 𝑦 ↔ ((1st ‘(𝐹‘𝑛)) + 𝐶) < 𝑦))
76	4	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
77	6	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℝ)
78		simplrl 774	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑦 ∈ ℝ)
79	76, 77, 78	ltaddsubd 11575	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹‘𝑛)) + 𝐶) < 𝑦 ↔ (1st ‘(𝐹‘𝑛)) < (𝑦 − 𝐶)))
80	75, 79	bitrd 278	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺‘𝑛)) < 𝑦 ↔ (1st ‘(𝐹‘𝑛)) < (𝑦 − 𝐶)))
81	33	adantl 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺‘𝑛)) = ((2nd ‘(𝐹‘𝑛)) + 𝐶))
82	81	breq2d 5086	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑦 < (2nd ‘(𝐺‘𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹‘𝑛)) + 𝐶)))
83	5	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
84	78, 77, 83	ltsubaddd 11571	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝑦 − 𝐶) < (2nd ‘(𝐹‘𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹‘𝑛)) + 𝐶)))
85	82, 84	bitr4d 281	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑦 < (2nd ‘(𝐺‘𝑛)) ↔ (𝑦 − 𝐶) < (2nd ‘(𝐹‘𝑛))))
86	80, 85	anbi12d 631	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛))) ↔ ((1st ‘(𝐹‘𝑛)) < (𝑦 − 𝐶) ∧ (𝑦 − 𝐶) < (2nd ‘(𝐹‘𝑛)))))
87	86	rexbidva 3225	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) → (∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛))) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < (𝑦 − 𝐶) ∧ (𝑦 − 𝐶) < (2nd ‘(𝐹‘𝑛)))))
88	73, 87	mpbird 256	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛))))
89	61, 88	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛))))
90	89	ralrimiva 3103	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛))))
91		ssrab2 4013	. . . . . 6 ⊢ {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴} ⊆ ℝ
92	55, 91	eqsstrdi 3975	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
93		ovolfioo 24631	. . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) ↔ ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛)))))
94	92, 17, 93	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) ↔ ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛)))))
95	90, 94	mpbird 256	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐺))
96		ovolshft.4	. . . 4 ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}
97		eqid 2738	. . . 4 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
98	96, 97	elovolmr 24640	. . 3 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐵 ⊆ ∪ ran ((,) ∘ 𝐺)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ 𝑀)
99	17, 95, 98	syl2anc 584	. 2 ⊢ (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ 𝑀)
100	54, 99	eqeltrrd 2840	1 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ 𝑀)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  {crab 3068  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ⟨cop 4567  ∪ cuni 4839   class class class wbr 5074   ↦ cmpt 5157   × cxp 5587  ran crn 5590   ∘ ccom 5593   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830   ↑_m cmap 8615  supcsup 9199  ℝcr 10870  0cc0 10871  1c1 10872   + caddc 10874  +∞cpnf 11006  ℝ^*cxr 11008   < clt 11009   ≤ cle 11010   − cmin 11205  ℕcn 11973  (,)cioo 13079  [,)cico 13081  seqcseq 13721  abscabs 14945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-sup 9201  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-ioo 13083  df-ico 13085  df-fz 13240  df-seq 13722  df-exp 13783  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947

This theorem is referenced by:  ovolshftlem2  24674