Theorem ovolshftlem1 25845

Description: Lemma for ovolshft 25847. (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 25802	. . . . . . . . . . . . . 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 592	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))))
4	3	simp1d 1160	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
5	3	simp2d 1161	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
6		ovolshft.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℝ)
7	6	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℝ)
8	3	simp3d 1162	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)))
9	4, 5, 7, 8	leadd1dd 11943	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) + 𝐶) ≤ ((2nd ‘(𝐹‘𝑛)) + 𝐶))
10		df-br 5107	. . . . . . . . . . 11 ⊢ (((1st ‘(𝐹‘𝑛)) + 𝐶) ≤ ((2nd ‘(𝐹‘𝑛)) + 𝐶) ↔ ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩ ∈ ≤ )
11	9, 10	sylib 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩ ∈ ≤ )
12	4, 7	readdcld 11351	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) + 𝐶) ∈ ℝ)
13	5, 7	readdcld 11351	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐹‘𝑛)) + 𝐶) ∈ ℝ)
14	12, 13	opelxpd 5694	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩ ∈ (ℝ × ℝ))
15	11, 14	elind 4100	. . . . . . . . 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 7125	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
18		eqid 2763	. . . . . . . . 9 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
19	18	ovolfsf 25807	. . . . . . . 8 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
20		ffn 6720	. . . . . . . 8 ⊢ (((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞) → ((abs ∘ − ) ∘ 𝐺) Fn ℕ)
21	17, 19, 20	3syl 19	. . . . . . 7 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐺) Fn ℕ)
22		eqid 2763	. . . . . . . . 9 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
23	22	ovolfsf 25807	. . . . . . . 8 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
24		ffn 6720	. . . . . . . 8 ⊢ (((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞) → ((abs ∘ − ) ∘ 𝐹) Fn ℕ)
25	1, 23, 24	3syl 19	. . . . . . 7 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐹) Fn ℕ)
26		opex 5436	. . . . . . . . . . . . . 14 ⊢ ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩ ∈ V
27	16	fvmpt2 7016	. . . . . . . . . . . . . 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 704	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝐺‘𝑛) = ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩)
29	28	fveq2d 6900	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2nd ‘(𝐺‘𝑛)) = (2nd ‘⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩))
30		ovex 7463	. . . . . . . . . . . . 13 ⊢ ((1st ‘(𝐹‘𝑛)) + 𝐶) ∈ V
31		ovex 7463	. . . . . . . . . . . . 13 ⊢ ((2nd ‘(𝐹‘𝑛)) + 𝐶) ∈ V
32	30, 31	op2nd 8020	. . . . . . . . . . . 12 ⊢ (2nd ‘⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩) = ((2nd ‘(𝐹‘𝑛)) + 𝐶)
33	29, 32	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (2nd ‘(𝐺‘𝑛)) = ((2nd ‘(𝐹‘𝑛)) + 𝐶))
34	28	fveq2d 6900	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (1st ‘(𝐺‘𝑛)) = (1st ‘⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩))
35	30, 31	op1st 8019	. . . . . . . . . . . 12 ⊢ (1st ‘⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩) = ((1st ‘(𝐹‘𝑛)) + 𝐶)
36	34, 35	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (1st ‘(𝐺‘𝑛)) = ((1st ‘(𝐹‘𝑛)) + 𝐶))
37	33, 36	oveq12d 7448	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))) = (((2nd ‘(𝐹‘𝑛)) + 𝐶) − ((1st ‘(𝐹‘𝑛)) + 𝐶)))
38	37	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))) = (((2nd ‘(𝐹‘𝑛)) + 𝐶) − ((1st ‘(𝐹‘𝑛)) + 𝐶)))
39	5	recnd 11350	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℂ)
40	4	recnd 11350	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℂ)
41	7	recnd 11350	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℂ)
42	39, 40, 41	pnpcan2d 11720	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((2nd ‘(𝐹‘𝑛)) + 𝐶) − ((1st ‘(𝐹‘𝑛)) + 𝐶)) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
43	38, 42	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
44	18	ovolfsval 25806	. . . . . . . . 9 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))))
45	17, 44	sylan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))))
46	22	ovolfsval 25806	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
47	1, 46	sylan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
48	43, 45, 47	3eqtr4d 2808	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))
49	21, 25, 48	eqfnfvd 7043	. . . . . 6 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐹))
50	49	seqeq3d 14165	. . . . 5 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐹)))
51		ovolshft.5	. . . . 5 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
52	50, 51	eqtr4di 2816	. . . 4 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) = 𝑆)
53	52	rneqd 5925	. . 3 ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) = ran 𝑆)
54	53	supeq1d 9451	. 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 2850	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴}))
57		oveq1 7437	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 − 𝐶) = (𝑦 − 𝐶))
58	57	eleq1d 2849	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥 − 𝐶) ∈ 𝐴 ↔ (𝑦 − 𝐶) ∈ 𝐴))
59	58	elrab 3648	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴} ↔ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴))
60	56, 59	bitrdi 290	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)))
61	60	biimpa 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴))
62		breq2 5110	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 − 𝐶) → ((1st ‘(𝐹‘𝑛)) < 𝑥 ↔ (1st ‘(𝐹‘𝑛)) < (𝑦 − 𝐶)))
63		breq1 5109	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 − 𝐶) → (𝑥 < (2nd ‘(𝐹‘𝑛)) ↔ (𝑦 − 𝐶) < (2nd ‘(𝐹‘𝑛))))
64	62, 63	anbi12d 644	. . . . . . . . 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 3190	. . . . . . . 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 25803	. . . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) ↔ ∀𝑥 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛)))))
70	66, 69	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))))
71	70	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))))
72		simprr 785	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) → (𝑦 − 𝐶) ∈ 𝐴)
73	65, 71, 72	rspcdva 3581	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < (𝑦 − 𝐶) ∧ (𝑦 − 𝐶) < (2nd ‘(𝐹‘𝑛))))
74	36	adantl 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺‘𝑛)) = ((1st ‘(𝐹‘𝑛)) + 𝐶))
75	74	breq1d 5116	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺‘𝑛)) < 𝑦 ↔ ((1st ‘(𝐹‘𝑛)) + 𝐶) < 𝑦))
76	4	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
77	6	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℝ)
78		simplrl 789	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑦 ∈ ℝ)
79	76, 77, 78	ltaddsubd 11929	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹‘𝑛)) + 𝐶) < 𝑦 ↔ (1st ‘(𝐹‘𝑛)) < (𝑦 − 𝐶)))
80	75, 79	bitrd 282	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺‘𝑛)) < 𝑦 ↔ (1st ‘(𝐹‘𝑛)) < (𝑦 − 𝐶)))
81	33	adantl 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺‘𝑛)) = ((2nd ‘(𝐹‘𝑛)) + 𝐶))
82	81	breq2d 5118	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑦 < (2nd ‘(𝐺‘𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹‘𝑛)) + 𝐶)))
83	5	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
84	78, 77, 83	ltsubaddd 11925	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝑦 − 𝐶) < (2nd ‘(𝐹‘𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹‘𝑛)) + 𝐶)))
85	82, 84	bitr4d 285	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑦 < (2nd ‘(𝐺‘𝑛)) ↔ (𝑦 − 𝐶) < (2nd ‘(𝐹‘𝑛))))
86	80, 85	anbi12d 644	. . . . . . . 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 3188	. . . . . . 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 260	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛))))
89	61, 88	syldan 603	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛))))
90	89	ralrimiva 3158	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛))))
91		ssrab2 4008	. . . . . 6 ⊢ {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴} ⊆ ℝ
92	55, 91	eqsstrdi 3957	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
93		ovolfioo 25803	. . . . 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 596	. . . 4 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) ↔ ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛)))))
95	90, 94	mpbird 260	. . 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 2763	. . . 4 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
98	96, 97	elovolmr 25812	. . 3 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐵 ⊆ ∪ ran ((,) ∘ 𝐺)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ 𝑀)
99	17, 95, 98	syl2anc 596	. 2 ⊢ (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ 𝑀)
100	54, 99	eqeltrrd 2865	1 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ 𝑀)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ⟨cop 4838  ∪ cuni 4870   class class class wbr 5106   ↦ cmpt 5189   × cxp 5653  ran crn 5656   ∘ ccom 5659   Fn wfn 6542  ⟶wf 6543  ‘cfv 6547  (class class class)co 7430  1^st c1st 8009  2^nd c2nd 8010   ↑_m cmap 8861  supcsup 9445  ℝcr 11212  0cc0 11213  1c1 11214   + caddc 11216  +∞cpnf 11353  ℝ^*cxr 11355   < clt 11356   ≤ cle 11357   − cmin 11554  ℕcn 12348  (,)cioo 13489  [,)cico 13491  seqcseq 14157  abscabs 15416

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-cnex 11269  ax-resscn 11270  ax-1cn 11271  ax-icn 11272  ax-addcl 11273  ax-addrcl 11274  ax-mulcl 11275  ax-mulrcl 11276  ax-mulcom 11277  ax-addass 11278  ax-mulass 11279  ax-distr 11280  ax-i2m1 11281  ax-1ne0 11282  ax-1rid 11283  ax-rnegex 11284  ax-rrecex 11285  ax-cnre 11286  ax-pre-lttri 11287  ax-pre-lttrn 11288  ax-pre-ltadd 11289  ax-pre-mulgt0 11290  ax-pre-sup 11291

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7888  df-1st 8011  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-er 8731  df-map 8863  df-en 8988  df-dom 8989  df-sdom 8990  df-sup 9447  df-pnf 11358  df-mnf 11359  df-xr 11360  df-ltxr 11361  df-le 11362  df-sub 11556  df-neg 11557  df-div 11987  df-nn 12349  df-2 12418  df-3 12419  df-n0 12620  df-z 12707  df-uz 12979  df-rp 13134  df-ioo 13493  df-ico 13495  df-fz 13653  df-seq 14158  df-exp 14218  df-cj 15281  df-re 15282  df-im 15283  df-sqrt 15417  df-abs 15418

This theorem is used by:  ovolshftlem2  25846