Theorem ovolshftlem1 24406

Description: Lemma for ovolshft 24408. (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 24363	. . . . . . . . . . . . . 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 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))))
4	3	simp1d 1144	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
5	3	simp2d 1145	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
6		ovolshft.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℝ)
7	6	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℝ)
8	3	simp3d 1146	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)))
9	4, 5, 7, 8	leadd1dd 11446	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) + 𝐶) ≤ ((2nd ‘(𝐹‘𝑛)) + 𝐶))
10		df-br 5054	. . . . . . . . . . 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 10862	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) + 𝐶) ∈ ℝ)
13	5, 7	readdcld 10862	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐹‘𝑛)) + 𝐶) ∈ ℝ)
14	12, 13	opelxpd 5589	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩ ∈ (ℝ × ℝ))
15	11, 14	elind 4108	. . . . . . . . 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 6931	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
18		eqid 2737	. . . . . . . . 9 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
19	18	ovolfsf 24368	. . . . . . . 8 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
20		ffn 6545	. . . . . . . 8 ⊢ (((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞) → ((abs ∘ − ) ∘ 𝐺) Fn ℕ)
21	17, 19, 20	3syl 18	. . . . . . 7 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐺) Fn ℕ)
22		eqid 2737	. . . . . . . . 9 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
23	22	ovolfsf 24368	. . . . . . . 8 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
24		ffn 6545	. . . . . . . 8 ⊢ (((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞) → ((abs ∘ − ) ∘ 𝐹) Fn ℕ)
25	1, 23, 24	3syl 18	. . . . . . 7 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐹) Fn ℕ)
26		opex 5348	. . . . . . . . . . . . . 14 ⊢ ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩ ∈ V
27	16	fvmpt2 6829	. . . . . . . . . . . . . 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 691	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝐺‘𝑛) = ⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩)
29	28	fveq2d 6721	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2nd ‘(𝐺‘𝑛)) = (2nd ‘⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩))
30		ovex 7246	. . . . . . . . . . . . 13 ⊢ ((1st ‘(𝐹‘𝑛)) + 𝐶) ∈ V
31		ovex 7246	. . . . . . . . . . . . 13 ⊢ ((2nd ‘(𝐹‘𝑛)) + 𝐶) ∈ V
32	30, 31	op2nd 7770	. . . . . . . . . . . 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 6721	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (1st ‘(𝐺‘𝑛)) = (1st ‘⟨((1st ‘(𝐹‘𝑛)) + 𝐶), ((2nd ‘(𝐹‘𝑛)) + 𝐶)⟩))
35	30, 31	op1st 7769	. . . . . . . . . . . 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 7231	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))) = (((2nd ‘(𝐹‘𝑛)) + 𝐶) − ((1st ‘(𝐹‘𝑛)) + 𝐶)))
38	37	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))) = (((2nd ‘(𝐹‘𝑛)) + 𝐶) − ((1st ‘(𝐹‘𝑛)) + 𝐶)))
39	5	recnd 10861	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℂ)
40	4	recnd 10861	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℂ)
41	7	recnd 10861	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℂ)
42	39, 40, 41	pnpcan2d 11227	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((2nd ‘(𝐹‘𝑛)) + 𝐶) − ((1st ‘(𝐹‘𝑛)) + 𝐶)) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
43	38, 42	eqtrd 2777	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
44	18	ovolfsval 24367	. . . . . . . . 9 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))))
45	17, 44	sylan 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))))
46	22	ovolfsval 24367	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
47	1, 46	sylan 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
48	43, 45, 47	3eqtr4d 2787	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))
49	21, 25, 48	eqfnfvd 6855	. . . . . 6 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐹))
50	49	seqeq3d 13582	. . . . 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 5807	. . 3 ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) = ran 𝑆)
54	53	supeq1d 9062	. 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 2823	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴}))
57		oveq1 7220	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 − 𝐶) = (𝑦 − 𝐶))
58	57	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥 − 𝐶) ∈ 𝐴 ↔ (𝑦 − 𝐶) ∈ 𝐴))
59	58	elrab 3602	. . . . . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴))
62		breq2 5057	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 − 𝐶) → ((1st ‘(𝐹‘𝑛)) < 𝑥 ↔ (1st ‘(𝐹‘𝑛)) < (𝑦 − 𝐶)))
63		breq1 5056	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 − 𝐶) → (𝑥 < (2nd ‘(𝐹‘𝑛)) ↔ (𝑦 − 𝐶) < (2nd ‘(𝐹‘𝑛))))
64	62, 63	anbi12d 634	. . . . . . . . 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 3216	. . . . . . . 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 24364	. . . . . . . . . . 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 587	. . . . . . . . . 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 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))))
72		simprr 773	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) → (𝑦 − 𝐶) ∈ 𝐴)
73	65, 71, 72	rspcdva 3539	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < (𝑦 − 𝐶) ∧ (𝑦 − 𝐶) < (2nd ‘(𝐹‘𝑛))))
74	36	adantl 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺‘𝑛)) = ((1st ‘(𝐹‘𝑛)) + 𝐶))
75	74	breq1d 5063	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺‘𝑛)) < 𝑦 ↔ ((1st ‘(𝐹‘𝑛)) + 𝐶) < 𝑦))
76	4	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
77	6	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℝ)
78		simplrl 777	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑦 ∈ ℝ)
79	76, 77, 78	ltaddsubd 11432	. . . . . . . . . 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 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺‘𝑛)) = ((2nd ‘(𝐹‘𝑛)) + 𝐶))
82	81	breq2d 5065	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑦 < (2nd ‘(𝐺‘𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹‘𝑛)) + 𝐶)))
83	5	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝑦 − 𝐶) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
84	78, 77, 83	ltsubaddd 11428	. . . . . . . . . 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 634	. . . . . . . 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 3215	. . . . . . 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 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛))))
90	89	ralrimiva 3105	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛))))
91		ssrab2 3993	. . . . . 6 ⊢ {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴} ⊆ ℝ
92	55, 91	eqsstrdi 3955	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
93		ovolfioo 24364	. . . . 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 587	. . . 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 2737	. . . 4 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
98	96, 97	elovolmr 24373	. . 3 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐵 ⊆ ∪ ran ((,) ∘ 𝐺)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ 𝑀)
99	17, 95, 98	syl2anc 587	. 2 ⊢ (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ 𝑀)
100	54, 99	eqeltrrd 2839	1 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ 𝑀)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3061  ∃wrex 3062  {crab 3065  Vcvv 3408   ∩ cin 3865   ⊆ wss 3866  ⟨cop 4547  ∪ cuni 4819   class class class wbr 5053   ↦ cmpt 5135   × cxp 5549  ran crn 5552   ∘ ccom 5555   Fn wfn 6375  ⟶wf 6376  ‘cfv 6380  (class class class)co 7213  1^st c1st 7759  2^nd c2nd 7760   ↑_m cmap 8508  supcsup 9056  ℝcr 10728  0cc0 10729  1c1 10730   + caddc 10732  +∞cpnf 10864  ℝ^*cxr 10866   < clt 10867   ≤ cle 10868   − cmin 11062  ℕcn 11830  (,)cioo 12935  [,)cico 12937  seqcseq 13574  abscabs 14797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-sup 9058  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-n0 12091  df-z 12177  df-uz 12439  df-rp 12587  df-ioo 12939  df-ico 12941  df-fz 13096  df-seq 13575  df-exp 13636  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799

This theorem is referenced by:  ovolshftlem2  24407