Theorem ovolshftlem1 25360

Description: Lemma for ovolshft 25362. (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 25317	. . . . . . . . . . . . . 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 579	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))))
4	3	simp1d 1139	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
5	3	simp2d 1140	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
6		ovolshft.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℝ)
7	6	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℝ)
8	3	simp3d 1141	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)))
9	4, 5, 7, 8	leadd1dd 11825	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) + 𝐶) ≤ ((2nd ‘(𝐹‘𝑛)) + 𝐶))
10		df-br 5139	. . . . . . . . . . 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 11240	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) + 𝐶) ∈ ℝ)
13	5, 7	readdcld 11240	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐹‘𝑛)) + 𝐶) ∈ ℝ)
14	12, 13	opelxpd 5705	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩ ∈ (ℝ × ℝ))
15	11, 14	elind 4186	. . . . . . . . 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 7105	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
18		eqid 2724	. . . . . . . . 9 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
19	18	ovolfsf 25322	. . . . . . . 8 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
20		ffn 6707	. . . . . . . 8 ⊢ (((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞) → ((abs ∘ − ) ∘ 𝐺) Fn ℕ)
21	17, 19, 20	3syl 18	. . . . . . 7 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐺) Fn ℕ)
22		eqid 2724	. . . . . . . . 9 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
23	22	ovolfsf 25322	. . . . . . . 8 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
24		ffn 6707	. . . . . . . 8 ⊢ (((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞) → ((abs ∘ − ) ∘ 𝐹) Fn ℕ)
25	1, 23, 24	3syl 18	. . . . . . 7 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐹) Fn ℕ)
26		opex 5454	. . . . . . . . . . . . . 14 ⊢ ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩ ∈ V
27	16	fvmpt2 6999	. . . . . . . . . . . . . 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 6885	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2nd ‘(𝐺‘𝑛)) = (2nd ‘⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩))
30		ovex 7434	. . . . . . . . . . . . 13 ⊢ ((1st ‘(𝐹‘𝑛)) + 𝐶) ∈ V
31		ovex 7434	. . . . . . . . . . . . 13 ⊢ ((2nd ‘(𝐹‘𝑛)) + 𝐶) ∈ V
32	30, 31	op2nd 7977	. . . . . . . . . . . 12 ⊢ (2nd ‘⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩) = ((2nd ‘(𝐹‘𝑛)) + 𝐶)
33	29, 32	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (2nd ‘(𝐺‘𝑛)) = ((2nd ‘(𝐹‘𝑛)) + 𝐶))
34	28	fveq2d 6885	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (1st ‘(𝐺‘𝑛)) = (1st ‘⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩))
35	30, 31	op1st 7976	. . . . . . . . . . . 12 ⊢ (1st ‘⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩) = ((1st ‘(𝐹‘𝑛)) + 𝐶)
36	34, 35	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (1st ‘(𝐺‘𝑛)) = ((1st ‘(𝐹‘𝑛)) + 𝐶))
37	33, 36	oveq12d 7419	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))) = (((2nd ‘(𝐹‘𝑛)) + 𝐶) − ((1st ‘(𝐹‘𝑛)) + 𝐶)))
38	37	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))) = (((2nd ‘(𝐹‘𝑛)) + 𝐶) − ((1st ‘(𝐹‘𝑛)) + 𝐶)))
39	5	recnd 11239	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℂ)
40	4	recnd 11239	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℂ)
41	7	recnd 11239	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℂ)
42	39, 40, 41	pnpcan2d 11606	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((2nd ‘(𝐹‘𝑛)) + 𝐶) − ((1st ‘(𝐹‘𝑛)) + 𝐶)) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
43	38, 42	eqtrd 2764	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
44	18	ovolfsval 25321	. . . . . . . . 9 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))))
45	17, 44	sylan 579	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))))
46	22	ovolfsval 25321	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
47	1, 46	sylan 579	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
48	43, 45, 47	3eqtr4d 2774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))
49	21, 25, 48	eqfnfvd 7025	. . . . . 6 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐹))
50	49	seqeq3d 13971	. . . . 5 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐹)))
51		ovolshft.5	. . . . 5 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
52	50, 51	eqtr4di 2782	. . . 4 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) = 𝑆)
53	52	rneqd 5927	. . 3 ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) = ran 𝑆)
54	53	supeq1d 9437	. 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 2811	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴}))
57		oveq1 7408	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 − 𝐶) = (𝑦 − 𝐶))
58	57	eleq1d 2810	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥 − 𝐶) ∈ 𝐴 ↔ (𝑦 − 𝐶) ∈ 𝐴))
59	58	elrab 3675	. . . . . . . 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 476	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴))
62		breq2 5142	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 − 𝐶) → ((1st ‘(𝐹‘𝑛)) < 𝑥 ↔ (1st ‘(𝐹‘𝑛)) < (𝑦 − 𝐶)))
63		breq1 5141	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 − 𝐶) → (𝑥 < (2nd ‘(𝐹‘𝑛)) ↔ (𝑦 − 𝐶) < (2nd ‘(𝐹‘𝑛))))
64	62, 63	anbi12d 630	. . . . . . . . 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 3170	. . . . . . . 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 25318	. . . . . . . . . . 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 583	. . . . . . . . . 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 480	. . . . . . . 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 3605	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < (𝑦 − 𝐶) ∧ (𝑦 − 𝐶) < (2nd ‘(𝐹‘𝑛))))
74	36	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺‘𝑛)) = ((1st ‘(𝐹‘𝑛)) + 𝐶))
75	74	breq1d 5148	. . . . . . . . . 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 11811	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹‘𝑛)) + 𝐶) < 𝑦 ↔ (1st ‘(𝐹‘𝑛)) < (𝑦 − 𝐶)))
80	75, 79	bitrd 279	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺‘𝑛)) < 𝑦 ↔ (1st ‘(𝐹‘𝑛)) < (𝑦 − 𝐶)))
81	33	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺‘𝑛)) = ((2nd ‘(𝐹‘𝑛)) + 𝐶))
82	81	breq2d 5150	. . . . . . . . . 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 11807	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝑦 − 𝐶) < (2nd ‘(𝐹‘𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹‘𝑛)) + 𝐶)))
85	82, 84	bitr4d 282	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑦 < (2nd ‘(𝐺‘𝑛)) ↔ (𝑦 − 𝐶) < (2nd ‘(𝐹‘𝑛))))
86	80, 85	anbi12d 630	. . . . . . . 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 3168	. . . . . . 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 257	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛))))
89	61, 88	syldan 590	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛))))
90	89	ralrimiva 3138	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛))))
91		ssrab2 4069	. . . . . 6 ⊢ {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴} ⊆ ℝ
92	55, 91	eqsstrdi 4028	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
93		ovolfioo 25318	. . . . 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 583	. . . 4 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) ↔ ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛)))))
95	90, 94	mpbird 257	. . 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 2724	. . . 4 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
98	96, 97	elovolmr 25327	. . 3 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐵 ⊆ ∪ ran ((,) ∘ 𝐺)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ 𝑀)
99	17, 95, 98	syl2anc 583	. 2 ⊢ (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ 𝑀)
100	54, 99	eqeltrrd 2826	1 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ 𝑀)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3053  ∃wrex 3062  {crab 3424  Vcvv 3466   ∩ cin 3939   ⊆ wss 3940  ⟨cop 4626  ∪ cuni 4899   class class class wbr 5138   ↦ cmpt 5221   × cxp 5664  ran crn 5667   ∘ ccom 5670   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7401  1^st c1st 7966  2^nd c2nd 7967   ↑_m cmap 8816  supcsup 9431  ℝcr 11105  0cc0 11106  1c1 11107   + caddc 11109  +∞cpnf 11242  ℝ^*cxr 11244   < clt 11245   ≤ cle 11246   − cmin 11441  ℕcn 12209  (,)cioo 13321  [,)cico 13323  seqcseq 13963  abscabs 15178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-sup 9433  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-n0 12470  df-z 12556  df-uz 12820  df-rp 12972  df-ioo 13325  df-ico 13327  df-fz 13482  df-seq 13964  df-exp 14025  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180

This theorem is referenced by:  ovolshftlem2  25361