smfmullem2 - Mathbox for Glauco Siliprandi

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Theorem smfmullem2 43998

Description: The multiplication of two sigma-measurable functions is measurable: this is the step (i) of the proof of Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

**Hypotheses**
Ref	Expression
smfmullem2.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
smfmullem2.k	⊢ 𝐾 = {𝑞 ∈ (ℚ ↑_m (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝐴}
smfmullem2.u	⊢ (𝜑 → 𝑈 ∈ ℝ)
smfmullem2.v	⊢ (𝜑 → 𝑉 ∈ ℝ)
smfmullem2.l	⊢ (𝜑 → (𝑈 · 𝑉) < 𝐴)
smfmullem2.p	⊢ (𝜑 → 𝑃 ∈ ℚ)
smfmullem2.r	⊢ (𝜑 → 𝑅 ∈ ℚ)
smfmullem2.s	⊢ (𝜑 → 𝑆 ∈ ℚ)
smfmullem2.z	⊢ (𝜑 → 𝑍 ∈ ℚ)
smfmullem2.p2	⊢ (𝜑 → 𝑃 ∈ ((𝑈 − 𝑌)(,)𝑈))
smfmullem2.42	⊢ (𝜑 → 𝑅 ∈ (𝑈(,)(𝑈 + 𝑌)))
smfmullem2.w2	⊢ (𝜑 → 𝑆 ∈ ((𝑉 − 𝑌)(,)𝑉))
smfmullem2.z2	⊢ (𝜑 → 𝑍 ∈ (𝑉(,)(𝑉 + 𝑌)))
smfmullem2.x	⊢ 𝑋 = ((𝐴 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉))))
smfmullem2.y	⊢ 𝑌 = if(1 ≤ 𝑋, 1, 𝑋)

**Assertion**
Ref	Expression
smfmullem2	⊢ (𝜑 → ∃𝑞 ∈ 𝐾 (𝑈 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝑉 ∈ ((𝑞‘2)(,)(𝑞‘3))))

Distinct variable groups: 𝐴,𝑞 𝑃,𝑞,𝑢,𝑣 𝑅,𝑞,𝑢,𝑣 𝑆,𝑞,𝑢,𝑣 𝑈,𝑞 𝑉,𝑞 𝑍,𝑞,𝑢,𝑣 𝜑,𝑢,𝑣 Allowed substitution hints: 𝜑(𝑞) 𝐴(𝑣,𝑢) 𝑈(𝑣,𝑢) 𝐾(𝑣,𝑢,𝑞) 𝑉(𝑣,𝑢) 𝑋(𝑣,𝑢,𝑞) 𝑌(𝑣,𝑢,𝑞)

**Proof of Theorem smfmullem2**
Step	Hyp	Ref	Expression
1		smfmullem2.p	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℚ)
2		smfmullem2.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ ℚ)
3		smfmullem2.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℚ)
4		smfmullem2.z	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ ℚ)
5	1, 2, 3, 4	s4cld 14438	. . . . . . 7 ⊢ (𝜑 → ⟨“𝑃𝑅𝑆𝑍”⟩ ∈ Word ℚ)
6		s4len 14464	. . . . . . . 8 ⊢ (♯‘⟨“𝑃𝑅𝑆𝑍”⟩) = 4
7	6	a1i 11	. . . . . . 7 ⊢ (𝜑 → (♯‘⟨“𝑃𝑅𝑆𝑍”⟩) = 4)
8	5, 7	jca 515	. . . . . 6 ⊢ (𝜑 → (⟨“𝑃𝑅𝑆𝑍”⟩ ∈ Word ℚ ∧ (♯‘⟨“𝑃𝑅𝑆𝑍”⟩) = 4))
9		qex 12557	. . . . . . . 8 ⊢ ℚ ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℚ ∈ V)
11		4nn0 12109	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → 4 ∈ ℕ₀)
13		wrdmap 14101	. . . . . . 7 ⊢ ((ℚ ∈ V ∧ 4 ∈ ℕ₀) → ((⟨“𝑃𝑅𝑆𝑍”⟩ ∈ Word ℚ ∧ (♯‘⟨“𝑃𝑅𝑆𝑍”⟩) = 4) ↔ ⟨“𝑃𝑅𝑆𝑍”⟩ ∈ (ℚ ↑_m (0..^4))))
14	10, 12, 13	syl2anc 587	. . . . . 6 ⊢ (𝜑 → ((⟨“𝑃𝑅𝑆𝑍”⟩ ∈ Word ℚ ∧ (♯‘⟨“𝑃𝑅𝑆𝑍”⟩) = 4) ↔ ⟨“𝑃𝑅𝑆𝑍”⟩ ∈ (ℚ ↑_m (0..^4))))
15	8, 14	mpbid 235	. . . . 5 ⊢ (𝜑 → ⟨“𝑃𝑅𝑆𝑍”⟩ ∈ (ℚ ↑_m (0..^4)))
16		3z 12210	. . . . . . . . . 10 ⊢ 3 ∈ ℤ
17		fzval3 13311	. . . . . . . . . 10 ⊢ (3 ∈ ℤ → (0...3) = (0..^(3 + 1)))
18	16, 17	ax-mp 5	. . . . . . . . 9 ⊢ (0...3) = (0..^(3 + 1))
19		3p1e4 11975	. . . . . . . . . 10 ⊢ (3 + 1) = 4
20	19	oveq2i 7224	. . . . . . . . 9 ⊢ (0..^(3 + 1)) = (0..^4)
21	18, 20	eqtri 2765	. . . . . . . 8 ⊢ (0...3) = (0..^4)
22	21	eqcomi 2746	. . . . . . 7 ⊢ (0..^4) = (0...3)
23	22	a1i 11	. . . . . 6 ⊢ (𝜑 → (0..^4) = (0...3))
24	23	oveq2d 7229	. . . . 5 ⊢ (𝜑 → (ℚ ↑_m (0..^4)) = (ℚ ↑_m (0...3)))
25	15, 24	eleqtrd 2840	. . . 4 ⊢ (𝜑 → ⟨“𝑃𝑅𝑆𝑍”⟩ ∈ (ℚ ↑_m (0...3)))
26		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))) → 𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)))
27		s4fv0 14460	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℚ → (⟨“𝑃𝑅𝑆𝑍”⟩‘0) = 𝑃)
28	1, 27	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (⟨“𝑃𝑅𝑆𝑍”⟩‘0) = 𝑃)
29		s4fv1 14461	. . . . . . . . . 10 ⊢ (𝑅 ∈ ℚ → (⟨“𝑃𝑅𝑆𝑍”⟩‘1) = 𝑅)
30	2, 29	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (⟨“𝑃𝑅𝑆𝑍”⟩‘1) = 𝑅)
31	28, 30	oveq12d 7231	. . . . . . . 8 ⊢ (𝜑 → ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)) = (𝑃(,)𝑅))
32	31	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))) → ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)) = (𝑃(,)𝑅))
33	26, 32	eleqtrd 2840	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))) → 𝑢 ∈ (𝑃(,)𝑅))
34		simpr 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))) → 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)))
35		s4fv2 14462	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ ℚ → (⟨“𝑃𝑅𝑆𝑍”⟩‘2) = 𝑆)
36	3, 35	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⟨“𝑃𝑅𝑆𝑍”⟩‘2) = 𝑆)
37		s4fv3 14463	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ ℚ → (⟨“𝑃𝑅𝑆𝑍”⟩‘3) = 𝑍)
38	4, 37	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⟨“𝑃𝑅𝑆𝑍”⟩‘3) = 𝑍)
39	36, 38	oveq12d 7231	. . . . . . . . . . . 12 ⊢ (𝜑 → ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)) = (𝑆(,)𝑍))
40	39	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))) → ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)) = (𝑆(,)𝑍))
41	34, 40	eleqtrd 2840	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))) → 𝑣 ∈ (𝑆(,)𝑍))
42		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑣 ∈ (𝑆(,)𝑍))
43	41, 42	syldan 594	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))) → 𝑣 ∈ (𝑆(,)𝑍))
44	43	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))) → 𝑣 ∈ (𝑆(,)𝑍))
45		smfmullem2.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
46	45	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝐴 ∈ ℝ)
47		smfmullem2.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ ℝ)
48	47	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑈 ∈ ℝ)
49		smfmullem2.v	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ ℝ)
50	49	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑉 ∈ ℝ)
51		smfmullem2.l	. . . . . . . . . 10 ⊢ (𝜑 → (𝑈 · 𝑉) < 𝐴)
52	51	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → (𝑈 · 𝑉) < 𝐴)
53		smfmullem2.x	. . . . . . . . 9 ⊢ 𝑋 = ((𝐴 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉))))
54		smfmullem2.y	. . . . . . . . 9 ⊢ 𝑌 = if(1 ≤ 𝑋, 1, 𝑋)
55		smfmullem2.p2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ((𝑈 − 𝑌)(,)𝑈))
56	55	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑃 ∈ ((𝑈 − 𝑌)(,)𝑈))
57		smfmullem2.42	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (𝑈(,)(𝑈 + 𝑌)))
58	57	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑅 ∈ (𝑈(,)(𝑈 + 𝑌)))
59		smfmullem2.w2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ ((𝑉 − 𝑌)(,)𝑉))
60	59	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑆 ∈ ((𝑉 − 𝑌)(,)𝑉))
61		smfmullem2.z2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ (𝑉(,)(𝑉 + 𝑌)))
62	61	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑍 ∈ (𝑉(,)(𝑉 + 𝑌)))
63		simplr 769	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑢 ∈ (𝑃(,)𝑅))
64		simpr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑣 ∈ (𝑆(,)𝑍))
65	46, 48, 50, 52, 53, 54, 56, 58, 60, 62, 63, 64	smfmullem1 43997	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → (𝑢 · 𝑣) < 𝐴)
66	44, 65	syldan 594	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))) → (𝑢 · 𝑣) < 𝐴)
67	66	ralrimiva 3105	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) → ∀𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))(𝑢 · 𝑣) < 𝐴)
68	33, 67	syldan 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))) → ∀𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))(𝑢 · 𝑣) < 𝐴)
69	68	ralrimiva 3105	. . . 4 ⊢ (𝜑 → ∀𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))∀𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))(𝑢 · 𝑣) < 𝐴)
70	25, 69	jca 515	. . 3 ⊢ (𝜑 → (⟨“𝑃𝑅𝑆𝑍”⟩ ∈ (ℚ ↑_m (0...3)) ∧ ∀𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))∀𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))(𝑢 · 𝑣) < 𝐴))
71		fveq1 6716	. . . . . . 7 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (𝑞‘0) = (⟨“𝑃𝑅𝑆𝑍”⟩‘0))
72		fveq1 6716	. . . . . . 7 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (𝑞‘1) = (⟨“𝑃𝑅𝑆𝑍”⟩‘1))
73	71, 72	oveq12d 7231	. . . . . 6 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → ((𝑞‘0)(,)(𝑞‘1)) = ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)))
74	73	raleqdv 3325	. . . . 5 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝐴 ↔ ∀𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝐴))
75		fveq1 6716	. . . . . . . 8 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (𝑞‘2) = (⟨“𝑃𝑅𝑆𝑍”⟩‘2))
76		fveq1 6716	. . . . . . . 8 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (𝑞‘3) = (⟨“𝑃𝑅𝑆𝑍”⟩‘3))
77	75, 76	oveq12d 7231	. . . . . . 7 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → ((𝑞‘2)(,)(𝑞‘3)) = ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)))
78	77	raleqdv 3325	. . . . . 6 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝐴 ↔ ∀𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))(𝑢 · 𝑣) < 𝐴))
79	78	ralbidv 3118	. . . . 5 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (∀𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝐴 ↔ ∀𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))∀𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))(𝑢 · 𝑣) < 𝐴))
80	74, 79	bitrd 282	. . . 4 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝐴 ↔ ∀𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))∀𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))(𝑢 · 𝑣) < 𝐴))
81		smfmullem2.k	. . . 4 ⊢ 𝐾 = {𝑞 ∈ (ℚ ↑_m (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝐴}
82	80, 81	elrab2 3605	. . 3 ⊢ (⟨“𝑃𝑅𝑆𝑍”⟩ ∈ 𝐾 ↔ (⟨“𝑃𝑅𝑆𝑍”⟩ ∈ (ℚ ↑_m (0...3)) ∧ ∀𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))∀𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))(𝑢 · 𝑣) < 𝐴))
83	70, 82	sylibr 237	. 2 ⊢ (𝜑 → ⟨“𝑃𝑅𝑆𝑍”⟩ ∈ 𝐾)
84		qssre 12555	. . . . . . 7 ⊢ ℚ ⊆ ℝ
85	84, 1	sseldi 3899	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ)
86	85	rexrd 10883	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℝ^*)
87	84, 2	sseldi 3899	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℝ)
88	87	rexrd 10883	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
89	54	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 = if(1 ≤ 𝑋, 1, 𝑋))
90		1rp 12590	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ⁺
91	90	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℝ⁺)
92	53	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 = ((𝐴 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉)))))
93	47, 49	remulcld 10863	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 · 𝑉) ∈ ℝ)
94		difrp 12624	. . . . . . . . . . . . . . 15 ⊢ (((𝑈 · 𝑉) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑈 · 𝑉) < 𝐴 ↔ (𝐴 − (𝑈 · 𝑉)) ∈ ℝ⁺))
95	93, 45, 94	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑈 · 𝑉) < 𝐴 ↔ (𝐴 − (𝑈 · 𝑉)) ∈ ℝ⁺))
96	51, 95	mpbid 235	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 − (𝑈 · 𝑉)) ∈ ℝ⁺)
97		1red 10834	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℝ)
98	47	recnd 10861	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑈 ∈ ℂ)
99	98	abscld 15000	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (abs‘𝑈) ∈ ℝ)
100	49	recnd 10861	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑉 ∈ ℂ)
101	100	abscld 15000	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (abs‘𝑉) ∈ ℝ)
102	99, 101	readdcld 10862	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((abs‘𝑈) + (abs‘𝑉)) ∈ ℝ)
103	97, 102	readdcld 10862	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 + ((abs‘𝑈) + (abs‘𝑉))) ∈ ℝ)
104		0re 10835	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
105	104	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℝ)
106	91	rpgt0d 12631	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 1)
107	98	absge0d 15008	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 ≤ (abs‘𝑈))
108	100	absge0d 15008	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 0 ≤ (abs‘𝑉))
109	99, 101	addge01d 11420	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0 ≤ (abs‘𝑉) ↔ (abs‘𝑈) ≤ ((abs‘𝑈) + (abs‘𝑉))))
110	108, 109	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (abs‘𝑈) ≤ ((abs‘𝑈) + (abs‘𝑉)))
111	105, 99, 102, 107, 110	letrd 10989	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ≤ ((abs‘𝑈) + (abs‘𝑉)))
112	97, 102	addge01d 11420	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0 ≤ ((abs‘𝑈) + (abs‘𝑉)) ↔ 1 ≤ (1 + ((abs‘𝑈) + (abs‘𝑉)))))
113	111, 112	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ (1 + ((abs‘𝑈) + (abs‘𝑉))))
114	105, 97, 103, 106, 113	ltletrd 10992	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < (1 + ((abs‘𝑈) + (abs‘𝑉))))
115	103, 114	elrpd 12625	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1 + ((abs‘𝑈) + (abs‘𝑉))) ∈ ℝ⁺)
116	96, 115	rpdivcld 12645	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉)))) ∈ ℝ⁺)
117	92, 116	eqeltrd 2838	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
118	91, 117	ifcld 4485	. . . . . . . . . 10 ⊢ (𝜑 → if(1 ≤ 𝑋, 1, 𝑋) ∈ ℝ⁺)
119	89, 118	eqeltrd 2838	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
120	119	rpred 12628	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ℝ)
121	47, 120	resubcld 11260	. . . . . . 7 ⊢ (𝜑 → (𝑈 − 𝑌) ∈ ℝ)
122	121	rexrd 10883	. . . . . 6 ⊢ (𝜑 → (𝑈 − 𝑌) ∈ ℝ^*)
123	47	rexrd 10883	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ^*)
124		iooltub 42723	. . . . . 6 ⊢ (((𝑈 − 𝑌) ∈ ℝ^* ∧ 𝑈 ∈ ℝ^* ∧ 𝑃 ∈ ((𝑈 − 𝑌)(,)𝑈)) → 𝑃 < 𝑈)
125	122, 123, 55, 124	syl3anc 1373	. . . . 5 ⊢ (𝜑 → 𝑃 < 𝑈)
126	47, 120	readdcld 10862	. . . . . . 7 ⊢ (𝜑 → (𝑈 + 𝑌) ∈ ℝ)
127	126	rexrd 10883	. . . . . 6 ⊢ (𝜑 → (𝑈 + 𝑌) ∈ ℝ^*)
128		ioogtlb 42708	. . . . . 6 ⊢ ((𝑈 ∈ ℝ^* ∧ (𝑈 + 𝑌) ∈ ℝ^* ∧ 𝑅 ∈ (𝑈(,)(𝑈 + 𝑌))) → 𝑈 < 𝑅)
129	123, 127, 57, 128	syl3anc 1373	. . . . 5 ⊢ (𝜑 → 𝑈 < 𝑅)
130	86, 88, 47, 125, 129	eliood 42711	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (𝑃(,)𝑅))
131	31	eqcomd 2743	. . . 4 ⊢ (𝜑 → (𝑃(,)𝑅) = ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)))
132	130, 131	eleqtrd 2840	. . 3 ⊢ (𝜑 → 𝑈 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)))
133	84, 3	sseldi 3899	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℝ)
134	133	rexrd 10883	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ ℝ^*)
135	84, 4	sseldi 3899	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ ℝ)
136	135	rexrd 10883	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ ℝ^*)
137	49, 120	resubcld 11260	. . . . . . 7 ⊢ (𝜑 → (𝑉 − 𝑌) ∈ ℝ)
138	137	rexrd 10883	. . . . . 6 ⊢ (𝜑 → (𝑉 − 𝑌) ∈ ℝ^*)
139	49	rexrd 10883	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ ℝ^*)
140		iooltub 42723	. . . . . 6 ⊢ (((𝑉 − 𝑌) ∈ ℝ^* ∧ 𝑉 ∈ ℝ^* ∧ 𝑆 ∈ ((𝑉 − 𝑌)(,)𝑉)) → 𝑆 < 𝑉)
141	138, 139, 59, 140	syl3anc 1373	. . . . 5 ⊢ (𝜑 → 𝑆 < 𝑉)
142	49, 120	readdcld 10862	. . . . . . 7 ⊢ (𝜑 → (𝑉 + 𝑌) ∈ ℝ)
143	142	rexrd 10883	. . . . . 6 ⊢ (𝜑 → (𝑉 + 𝑌) ∈ ℝ^*)
144		ioogtlb 42708	. . . . . 6 ⊢ ((𝑉 ∈ ℝ^* ∧ (𝑉 + 𝑌) ∈ ℝ^* ∧ 𝑍 ∈ (𝑉(,)(𝑉 + 𝑌))) → 𝑉 < 𝑍)
145	139, 143, 61, 144	syl3anc 1373	. . . . 5 ⊢ (𝜑 → 𝑉 < 𝑍)
146	134, 136, 49, 141, 145	eliood 42711	. . . 4 ⊢ (𝜑 → 𝑉 ∈ (𝑆(,)𝑍))
147	39	eqcomd 2743	. . . 4 ⊢ (𝜑 → (𝑆(,)𝑍) = ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)))
148	146, 147	eleqtrd 2840	. . 3 ⊢ (𝜑 → 𝑉 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)))
149	132, 148	jca 515	. 2 ⊢ (𝜑 → (𝑈 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)) ∧ 𝑉 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))))
150		nfv 1922	. . 3 ⊢ Ⅎ𝑞(𝑈 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)) ∧ 𝑉 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)))
151		nfcv 2904	. . 3 ⊢ Ⅎ𝑞⟨“𝑃𝑅𝑆𝑍”⟩
152		nfrab1 3296	. . . 4 ⊢ Ⅎ𝑞{𝑞 ∈ (ℚ ↑_m (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝐴}
153	81, 152	nfcxfr 2902	. . 3 ⊢ Ⅎ𝑞𝐾
154	73	eleq2d 2823	. . . 4 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (𝑈 ∈ ((𝑞‘0)(,)(𝑞‘1)) ↔ 𝑈 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))))
155	77	eleq2d 2823	. . . 4 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (𝑉 ∈ ((𝑞‘2)(,)(𝑞‘3)) ↔ 𝑉 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))))
156	154, 155	anbi12d 634	. . 3 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → ((𝑈 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝑉 ∈ ((𝑞‘2)(,)(𝑞‘3))) ↔ (𝑈 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)) ∧ 𝑉 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)))))
157	150, 151, 153, 156	rspcef 42293	. 2 ⊢ ((⟨“𝑃𝑅𝑆𝑍”⟩ ∈ 𝐾 ∧ (𝑈 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)) ∧ 𝑉 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)))) → ∃𝑞 ∈ 𝐾 (𝑈 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝑉 ∈ ((𝑞‘2)(,)(𝑞‘3))))
158	83, 149, 157	syl2anc 587	1 ⊢ (𝜑 → ∃𝑞 ∈ 𝐾 (𝑈 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝑉 ∈ ((𝑞‘2)(,)(𝑞‘3))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ∃wrex 3062 {crab 3065 Vcvv 3408 ifcif 4439 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 ↑_m cmap 8508 ℝcr 10728 0cc0 10729 1c1 10730 + caddc 10732 · cmul 10734 ℝ^*cxr 10866 < clt 10867 ≤ cle 10868 − cmin 11062 / cdiv 11489 2c2 11885 3c3 11886 4c4 11887 ℕ₀cn0 12090 ℤcz 12176 ℚcq 12544 ℝ⁺crp 12586 (,)cioo 12935 ...cfz 13095 ..^cfzo 13238 ♯chash 13896 Word cword 14069 ⟨“cs4 14408 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-rep 5179 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-int 4860 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-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-card 9555 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-4 11895 df-n0 12091 df-z 12177 df-uz 12439 df-q 12545 df-rp 12587 df-ioo 12939 df-icc 12942 df-fz 13096 df-fzo 13239 df-seq 13575 df-exp 13636 df-hash 13897 df-word 14070 df-concat 14126 df-s1 14153 df-s2 14413 df-s3 14414 df-s4 14415 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799

This theorem is referenced by: smfmullem3 43999

W3C validator