smfmullem2 - Mathbox for Glauco Siliprandi

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Theorem smfmullem2 44326

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 14586	. . . . . . 7 ⊢ (𝜑 → ⟨“𝑃𝑅𝑆𝑍”⟩ ∈ Word ℚ)
6		s4len 14612	. . . . . . . 8 ⊢ (♯‘⟨“𝑃𝑅𝑆𝑍”⟩) = 4
7	6	a1i 11	. . . . . . 7 ⊢ (𝜑 → (♯‘⟨“𝑃𝑅𝑆𝑍”⟩) = 4)
8	5, 7	jca 512	. . . . . 6 ⊢ (𝜑 → (⟨“𝑃𝑅𝑆𝑍”⟩ ∈ Word ℚ ∧ (♯‘⟨“𝑃𝑅𝑆𝑍”⟩) = 4))
9		qex 12701	. . . . . . . 8 ⊢ ℚ ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℚ ∈ V)
11		4nn0 12252	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → 4 ∈ ℕ₀)
13		wrdmap 14249	. . . . . . 7 ⊢ ((ℚ ∈ V ∧ 4 ∈ ℕ₀) → ((⟨“𝑃𝑅𝑆𝑍”⟩ ∈ Word ℚ ∧ (♯‘⟨“𝑃𝑅𝑆𝑍”⟩) = 4) ↔ ⟨“𝑃𝑅𝑆𝑍”⟩ ∈ (ℚ ↑_m (0..^4))))
14	10, 12, 13	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((⟨“𝑃𝑅𝑆𝑍”⟩ ∈ Word ℚ ∧ (♯‘⟨“𝑃𝑅𝑆𝑍”⟩) = 4) ↔ ⟨“𝑃𝑅𝑆𝑍”⟩ ∈ (ℚ ↑_m (0..^4))))
15	8, 14	mpbid 231	. . . . 5 ⊢ (𝜑 → ⟨“𝑃𝑅𝑆𝑍”⟩ ∈ (ℚ ↑_m (0..^4)))
16		3z 12353	. . . . . . . . . 10 ⊢ 3 ∈ ℤ
17		fzval3 13456	. . . . . . . . . 10 ⊢ (3 ∈ ℤ → (0...3) = (0..^(3 + 1)))
18	16, 17	ax-mp 5	. . . . . . . . 9 ⊢ (0...3) = (0..^(3 + 1))
19		3p1e4 12118	. . . . . . . . . 10 ⊢ (3 + 1) = 4
20	19	oveq2i 7286	. . . . . . . . 9 ⊢ (0..^(3 + 1)) = (0..^4)
21	18, 20	eqtri 2766	. . . . . . . 8 ⊢ (0...3) = (0..^4)
22	21	eqcomi 2747	. . . . . . 7 ⊢ (0..^4) = (0...3)
23	22	a1i 11	. . . . . 6 ⊢ (𝜑 → (0..^4) = (0...3))
24	23	oveq2d 7291	. . . . 5 ⊢ (𝜑 → (ℚ ↑_m (0..^4)) = (ℚ ↑_m (0...3)))
25	15, 24	eleqtrd 2841	. . . 4 ⊢ (𝜑 → ⟨“𝑃𝑅𝑆𝑍”⟩ ∈ (ℚ ↑_m (0...3)))
26		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))) → 𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)))
27		s4fv0 14608	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℚ → (⟨“𝑃𝑅𝑆𝑍”⟩‘0) = 𝑃)
28	1, 27	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (⟨“𝑃𝑅𝑆𝑍”⟩‘0) = 𝑃)
29		s4fv1 14609	. . . . . . . . . 10 ⊢ (𝑅 ∈ ℚ → (⟨“𝑃𝑅𝑆𝑍”⟩‘1) = 𝑅)
30	2, 29	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (⟨“𝑃𝑅𝑆𝑍”⟩‘1) = 𝑅)
31	28, 30	oveq12d 7293	. . . . . . . 8 ⊢ (𝜑 → ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)) = (𝑃(,)𝑅))
32	31	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))) → ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)) = (𝑃(,)𝑅))
33	26, 32	eleqtrd 2841	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))) → 𝑢 ∈ (𝑃(,)𝑅))
34		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))) → 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)))
35		s4fv2 14610	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ ℚ → (⟨“𝑃𝑅𝑆𝑍”⟩‘2) = 𝑆)
36	3, 35	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⟨“𝑃𝑅𝑆𝑍”⟩‘2) = 𝑆)
37		s4fv3 14611	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ ℚ → (⟨“𝑃𝑅𝑆𝑍”⟩‘3) = 𝑍)
38	4, 37	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⟨“𝑃𝑅𝑆𝑍”⟩‘3) = 𝑍)
39	36, 38	oveq12d 7293	. . . . . . . . . . . 12 ⊢ (𝜑 → ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)) = (𝑆(,)𝑍))
40	39	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))) → ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)) = (𝑆(,)𝑍))
41	34, 40	eleqtrd 2841	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))) → 𝑣 ∈ (𝑆(,)𝑍))
42		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑣 ∈ (𝑆(,)𝑍))
43	41, 42	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))) → 𝑣 ∈ (𝑆(,)𝑍))
44	43	adantlr 712	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))) → 𝑣 ∈ (𝑆(,)𝑍))
45		smfmullem2.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
46	45	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝐴 ∈ ℝ)
47		smfmullem2.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ ℝ)
48	47	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑈 ∈ ℝ)
49		smfmullem2.v	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ ℝ)
50	49	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑉 ∈ ℝ)
51		smfmullem2.l	. . . . . . . . . 10 ⊢ (𝜑 → (𝑈 · 𝑉) < 𝐴)
52	51	ad2antrr 723	. . . . . . . . 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 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑃 ∈ ((𝑈 − 𝑌)(,)𝑈))
57		smfmullem2.42	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (𝑈(,)(𝑈 + 𝑌)))
58	57	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑅 ∈ (𝑈(,)(𝑈 + 𝑌)))
59		smfmullem2.w2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ ((𝑉 − 𝑌)(,)𝑉))
60	59	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑆 ∈ ((𝑉 − 𝑌)(,)𝑉))
61		smfmullem2.z2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ (𝑉(,)(𝑉 + 𝑌)))
62	61	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑍 ∈ (𝑉(,)(𝑉 + 𝑌)))
63		simplr 766	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑢 ∈ (𝑃(,)𝑅))
64		simpr 485	. . . . . . . . 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 44325	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → (𝑢 · 𝑣) < 𝐴)
66	44, 65	syldan 591	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))) → (𝑢 · 𝑣) < 𝐴)
67	66	ralrimiva 3103	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) → ∀𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))(𝑢 · 𝑣) < 𝐴)
68	33, 67	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))) → ∀𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))(𝑢 · 𝑣) < 𝐴)
69	68	ralrimiva 3103	. . . 4 ⊢ (𝜑 → ∀𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))∀𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))(𝑢 · 𝑣) < 𝐴)
70	25, 69	jca 512	. . 3 ⊢ (𝜑 → (⟨“𝑃𝑅𝑆𝑍”⟩ ∈ (ℚ ↑_m (0...3)) ∧ ∀𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))∀𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))(𝑢 · 𝑣) < 𝐴))
71		fveq1 6773	. . . . . . 7 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (𝑞‘0) = (⟨“𝑃𝑅𝑆𝑍”⟩‘0))
72		fveq1 6773	. . . . . . 7 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (𝑞‘1) = (⟨“𝑃𝑅𝑆𝑍”⟩‘1))
73	71, 72	oveq12d 7293	. . . . . 6 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → ((𝑞‘0)(,)(𝑞‘1)) = ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)))
74	73	raleqdv 3348	. . . . 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 6773	. . . . . . . 8 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (𝑞‘2) = (⟨“𝑃𝑅𝑆𝑍”⟩‘2))
76		fveq1 6773	. . . . . . . 8 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (𝑞‘3) = (⟨“𝑃𝑅𝑆𝑍”⟩‘3))
77	75, 76	oveq12d 7293	. . . . . . 7 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → ((𝑞‘2)(,)(𝑞‘3)) = ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)))
78	77	raleqdv 3348	. . . . . 6 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝐴 ↔ ∀𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))(𝑢 · 𝑣) < 𝐴))
79	78	ralbidv 3112	. . . . 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 278	. . . 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 3627	. . 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 233	. 2 ⊢ (𝜑 → ⟨“𝑃𝑅𝑆𝑍”⟩ ∈ 𝐾)
84		qssre 12699	. . . . . . 7 ⊢ ℚ ⊆ ℝ
85	84, 1	sselid 3919	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ)
86	85	rexrd 11025	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℝ^*)
87	84, 2	sselid 3919	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℝ)
88	87	rexrd 11025	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
89	54	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 = if(1 ≤ 𝑋, 1, 𝑋))
90		1rp 12734	. . . . . . . . . . . 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 11005	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 · 𝑉) ∈ ℝ)
94		difrp 12768	. . . . . . . . . . . . . . 15 ⊢ (((𝑈 · 𝑉) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑈 · 𝑉) < 𝐴 ↔ (𝐴 − (𝑈 · 𝑉)) ∈ ℝ⁺))
95	93, 45, 94	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑈 · 𝑉) < 𝐴 ↔ (𝐴 − (𝑈 · 𝑉)) ∈ ℝ⁺))
96	51, 95	mpbid 231	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 − (𝑈 · 𝑉)) ∈ ℝ⁺)
97		1red 10976	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℝ)
98	47	recnd 11003	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑈 ∈ ℂ)
99	98	abscld 15148	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (abs‘𝑈) ∈ ℝ)
100	49	recnd 11003	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑉 ∈ ℂ)
101	100	abscld 15148	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (abs‘𝑉) ∈ ℝ)
102	99, 101	readdcld 11004	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((abs‘𝑈) + (abs‘𝑉)) ∈ ℝ)
103	97, 102	readdcld 11004	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 + ((abs‘𝑈) + (abs‘𝑉))) ∈ ℝ)
104		0re 10977	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
105	104	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℝ)
106	91	rpgt0d 12775	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 1)
107	98	absge0d 15156	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 ≤ (abs‘𝑈))
108	100	absge0d 15156	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 0 ≤ (abs‘𝑉))
109	99, 101	addge01d 11563	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0 ≤ (abs‘𝑉) ↔ (abs‘𝑈) ≤ ((abs‘𝑈) + (abs‘𝑉))))
110	108, 109	mpbid 231	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (abs‘𝑈) ≤ ((abs‘𝑈) + (abs‘𝑉)))
111	105, 99, 102, 107, 110	letrd 11132	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ≤ ((abs‘𝑈) + (abs‘𝑉)))
112	97, 102	addge01d 11563	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0 ≤ ((abs‘𝑈) + (abs‘𝑉)) ↔ 1 ≤ (1 + ((abs‘𝑈) + (abs‘𝑉)))))
113	111, 112	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ (1 + ((abs‘𝑈) + (abs‘𝑉))))
114	105, 97, 103, 106, 113	ltletrd 11135	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < (1 + ((abs‘𝑈) + (abs‘𝑉))))
115	103, 114	elrpd 12769	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1 + ((abs‘𝑈) + (abs‘𝑉))) ∈ ℝ⁺)
116	96, 115	rpdivcld 12789	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉)))) ∈ ℝ⁺)
117	92, 116	eqeltrd 2839	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
118	91, 117	ifcld 4505	. . . . . . . . . 10 ⊢ (𝜑 → if(1 ≤ 𝑋, 1, 𝑋) ∈ ℝ⁺)
119	89, 118	eqeltrd 2839	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
120	119	rpred 12772	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ℝ)
121	47, 120	resubcld 11403	. . . . . . 7 ⊢ (𝜑 → (𝑈 − 𝑌) ∈ ℝ)
122	121	rexrd 11025	. . . . . 6 ⊢ (𝜑 → (𝑈 − 𝑌) ∈ ℝ^*)
123	47	rexrd 11025	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ^*)
124		iooltub 43048	. . . . . 6 ⊢ (((𝑈 − 𝑌) ∈ ℝ^* ∧ 𝑈 ∈ ℝ^* ∧ 𝑃 ∈ ((𝑈 − 𝑌)(,)𝑈)) → 𝑃 < 𝑈)
125	122, 123, 55, 124	syl3anc 1370	. . . . 5 ⊢ (𝜑 → 𝑃 < 𝑈)
126	47, 120	readdcld 11004	. . . . . . 7 ⊢ (𝜑 → (𝑈 + 𝑌) ∈ ℝ)
127	126	rexrd 11025	. . . . . 6 ⊢ (𝜑 → (𝑈 + 𝑌) ∈ ℝ^*)
128		ioogtlb 43033	. . . . . 6 ⊢ ((𝑈 ∈ ℝ^* ∧ (𝑈 + 𝑌) ∈ ℝ^* ∧ 𝑅 ∈ (𝑈(,)(𝑈 + 𝑌))) → 𝑈 < 𝑅)
129	123, 127, 57, 128	syl3anc 1370	. . . . 5 ⊢ (𝜑 → 𝑈 < 𝑅)
130	86, 88, 47, 125, 129	eliood 43036	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (𝑃(,)𝑅))
131	31	eqcomd 2744	. . . 4 ⊢ (𝜑 → (𝑃(,)𝑅) = ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)))
132	130, 131	eleqtrd 2841	. . 3 ⊢ (𝜑 → 𝑈 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)))
133	84, 3	sselid 3919	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℝ)
134	133	rexrd 11025	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ ℝ^*)
135	84, 4	sselid 3919	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ ℝ)
136	135	rexrd 11025	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ ℝ^*)
137	49, 120	resubcld 11403	. . . . . . 7 ⊢ (𝜑 → (𝑉 − 𝑌) ∈ ℝ)
138	137	rexrd 11025	. . . . . 6 ⊢ (𝜑 → (𝑉 − 𝑌) ∈ ℝ^*)
139	49	rexrd 11025	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ ℝ^*)
140		iooltub 43048	. . . . . 6 ⊢ (((𝑉 − 𝑌) ∈ ℝ^* ∧ 𝑉 ∈ ℝ^* ∧ 𝑆 ∈ ((𝑉 − 𝑌)(,)𝑉)) → 𝑆 < 𝑉)
141	138, 139, 59, 140	syl3anc 1370	. . . . 5 ⊢ (𝜑 → 𝑆 < 𝑉)
142	49, 120	readdcld 11004	. . . . . . 7 ⊢ (𝜑 → (𝑉 + 𝑌) ∈ ℝ)
143	142	rexrd 11025	. . . . . 6 ⊢ (𝜑 → (𝑉 + 𝑌) ∈ ℝ^*)
144		ioogtlb 43033	. . . . . 6 ⊢ ((𝑉 ∈ ℝ^* ∧ (𝑉 + 𝑌) ∈ ℝ^* ∧ 𝑍 ∈ (𝑉(,)(𝑉 + 𝑌))) → 𝑉 < 𝑍)
145	139, 143, 61, 144	syl3anc 1370	. . . . 5 ⊢ (𝜑 → 𝑉 < 𝑍)
146	134, 136, 49, 141, 145	eliood 43036	. . . 4 ⊢ (𝜑 → 𝑉 ∈ (𝑆(,)𝑍))
147	39	eqcomd 2744	. . . 4 ⊢ (𝜑 → (𝑆(,)𝑍) = ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)))
148	146, 147	eleqtrd 2841	. . 3 ⊢ (𝜑 → 𝑉 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)))
149	132, 148	jca 512	. 2 ⊢ (𝜑 → (𝑈 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)) ∧ 𝑉 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))))
150		nfv 1917	. . 3 ⊢ Ⅎ𝑞(𝑈 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)) ∧ 𝑉 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)))
151		nfcv 2907	. . 3 ⊢ Ⅎ𝑞⟨“𝑃𝑅𝑆𝑍”⟩
152		nfrab1 3317	. . . 4 ⊢ Ⅎ𝑞{𝑞 ∈ (ℚ ↑_m (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝐴}
153	81, 152	nfcxfr 2905	. . 3 ⊢ Ⅎ𝑞𝐾
154	73	eleq2d 2824	. . . 4 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (𝑈 ∈ ((𝑞‘0)(,)(𝑞‘1)) ↔ 𝑈 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))))
155	77	eleq2d 2824	. . . 4 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (𝑉 ∈ ((𝑞‘2)(,)(𝑞‘3)) ↔ 𝑉 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))))
156	154, 155	anbi12d 631	. . 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 42620	. 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 584	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 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 {crab 3068 Vcvv 3432 ifcif 4459 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 − cmin 11205 / cdiv 11632 2c2 12028 3c3 12029 4c4 12030 ℕ₀cn0 12233 ℤcz 12319 ℚcq 12688 ℝ⁺crp 12730 (,)cioo 13079 ...cfz 13239 ..^cfzo 13382 ♯chash 14044 Word cword 14217 ⟨“cs4 14556 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-rep 5209 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-int 4880 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-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-card 9697 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-4 12038 df-n0 12234 df-z 12320 df-uz 12583 df-q 12689 df-rp 12731 df-ioo 13083 df-icc 13086 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-word 14218 df-concat 14274 df-s1 14301 df-s2 14561 df-s3 14562 df-s4 14563 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947

This theorem is referenced by: smfmullem3 44327

W3C validator