smfmullem2 - Mathbox for Glauco Siliprandi

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Theorem smfmullem2 42444

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 14087	. . . . . . 7 ⊢ (𝜑 → ⟨“𝑃𝑅𝑆𝑍”⟩ ∈ Word ℚ)
6		s4len 14113	. . . . . . . 8 ⊢ (♯‘⟨“𝑃𝑅𝑆𝑍”⟩) = 4
7	6	a1i 11	. . . . . . 7 ⊢ (𝜑 → (♯‘⟨“𝑃𝑅𝑆𝑍”⟩) = 4)
8	5, 7	jca 504	. . . . . 6 ⊢ (𝜑 → (⟨“𝑃𝑅𝑆𝑍”⟩ ∈ Word ℚ ∧ (♯‘⟨“𝑃𝑅𝑆𝑍”⟩) = 4))
9		qex 12168	. . . . . . . 8 ⊢ ℚ ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℚ ∈ V)
11		4nn0 11721	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → 4 ∈ ℕ₀)
13		wrdmap 13698	. . . . . . 7 ⊢ ((ℚ ∈ V ∧ 4 ∈ ℕ₀) → ((⟨“𝑃𝑅𝑆𝑍”⟩ ∈ Word ℚ ∧ (♯‘⟨“𝑃𝑅𝑆𝑍”⟩) = 4) ↔ ⟨“𝑃𝑅𝑆𝑍”⟩ ∈ (ℚ ↑_𝑚 (0..^4))))
14	10, 12, 13	syl2anc 576	. . . . . 6 ⊢ (𝜑 → ((⟨“𝑃𝑅𝑆𝑍”⟩ ∈ Word ℚ ∧ (♯‘⟨“𝑃𝑅𝑆𝑍”⟩) = 4) ↔ ⟨“𝑃𝑅𝑆𝑍”⟩ ∈ (ℚ ↑_𝑚 (0..^4))))
15	8, 14	mpbid 224	. . . . 5 ⊢ (𝜑 → ⟨“𝑃𝑅𝑆𝑍”⟩ ∈ (ℚ ↑_𝑚 (0..^4)))
16		3z 11821	. . . . . . . . . 10 ⊢ 3 ∈ ℤ
17		fzval3 12914	. . . . . . . . . 10 ⊢ (3 ∈ ℤ → (0...3) = (0..^(3 + 1)))
18	16, 17	ax-mp 5	. . . . . . . . 9 ⊢ (0...3) = (0..^(3 + 1))
19		3p1e4 11585	. . . . . . . . . 10 ⊢ (3 + 1) = 4
20	19	oveq2i 6981	. . . . . . . . 9 ⊢ (0..^(3 + 1)) = (0..^4)
21	18, 20	eqtri 2796	. . . . . . . 8 ⊢ (0...3) = (0..^4)
22	21	eqcomi 2781	. . . . . . 7 ⊢ (0..^4) = (0...3)
23	22	a1i 11	. . . . . 6 ⊢ (𝜑 → (0..^4) = (0...3))
24	23	oveq2d 6986	. . . . 5 ⊢ (𝜑 → (ℚ ↑_𝑚 (0..^4)) = (ℚ ↑_𝑚 (0...3)))
25	15, 24	eleqtrd 2862	. . . 4 ⊢ (𝜑 → ⟨“𝑃𝑅𝑆𝑍”⟩ ∈ (ℚ ↑_𝑚 (0...3)))
26		simpr 477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))) → 𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)))
27		s4fv0 14109	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℚ → (⟨“𝑃𝑅𝑆𝑍”⟩‘0) = 𝑃)
28	1, 27	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (⟨“𝑃𝑅𝑆𝑍”⟩‘0) = 𝑃)
29		s4fv1 14110	. . . . . . . . . 10 ⊢ (𝑅 ∈ ℚ → (⟨“𝑃𝑅𝑆𝑍”⟩‘1) = 𝑅)
30	2, 29	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (⟨“𝑃𝑅𝑆𝑍”⟩‘1) = 𝑅)
31	28, 30	oveq12d 6988	. . . . . . . 8 ⊢ (𝜑 → ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)) = (𝑃(,)𝑅))
32	31	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))) → ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)) = (𝑃(,)𝑅))
33	26, 32	eleqtrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))) → 𝑢 ∈ (𝑃(,)𝑅))
34		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))) → 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)))
35		s4fv2 14111	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ ℚ → (⟨“𝑃𝑅𝑆𝑍”⟩‘2) = 𝑆)
36	3, 35	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⟨“𝑃𝑅𝑆𝑍”⟩‘2) = 𝑆)
37		s4fv3 14112	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ ℚ → (⟨“𝑃𝑅𝑆𝑍”⟩‘3) = 𝑍)
38	4, 37	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⟨“𝑃𝑅𝑆𝑍”⟩‘3) = 𝑍)
39	36, 38	oveq12d 6988	. . . . . . . . . . . 12 ⊢ (𝜑 → ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)) = (𝑆(,)𝑍))
40	39	adantr 473	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))) → ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)) = (𝑆(,)𝑍))
41	34, 40	eleqtrd 2862	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))) → 𝑣 ∈ (𝑆(,)𝑍))
42		simpr 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑣 ∈ (𝑆(,)𝑍))
43	41, 42	syldan 582	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))) → 𝑣 ∈ (𝑆(,)𝑍))
44	43	adantlr 702	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))) → 𝑣 ∈ (𝑆(,)𝑍))
45		smfmullem2.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
46	45	ad2antrr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝐴 ∈ ℝ)
47		smfmullem2.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ ℝ)
48	47	ad2antrr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑈 ∈ ℝ)
49		smfmullem2.v	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ ℝ)
50	49	ad2antrr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑉 ∈ ℝ)
51		smfmullem2.l	. . . . . . . . . 10 ⊢ (𝜑 → (𝑈 · 𝑉) < 𝐴)
52	51	ad2antrr 713	. . . . . . . . 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 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑃 ∈ ((𝑈 − 𝑌)(,)𝑈))
57		smfmullem2.42	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (𝑈(,)(𝑈 + 𝑌)))
58	57	ad2antrr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑅 ∈ (𝑈(,)(𝑈 + 𝑌)))
59		smfmullem2.w2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ ((𝑉 − 𝑌)(,)𝑉))
60	59	ad2antrr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑆 ∈ ((𝑉 − 𝑌)(,)𝑉))
61		smfmullem2.z2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ (𝑉(,)(𝑉 + 𝑌)))
62	61	ad2antrr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑍 ∈ (𝑉(,)(𝑉 + 𝑌)))
63		simplr 756	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → 𝑢 ∈ (𝑃(,)𝑅))
64		simpr 477	. . . . . . . . 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 42443	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ (𝑆(,)𝑍)) → (𝑢 · 𝑣) < 𝐴)
66	44, 65	syldan 582	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) ∧ 𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))) → (𝑢 · 𝑣) < 𝐴)
67	66	ralrimiva 3126	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑃(,)𝑅)) → ∀𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))(𝑢 · 𝑣) < 𝐴)
68	33, 67	syldan 582	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))) → ∀𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))(𝑢 · 𝑣) < 𝐴)
69	68	ralrimiva 3126	. . . 4 ⊢ (𝜑 → ∀𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))∀𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))(𝑢 · 𝑣) < 𝐴)
70	25, 69	jca 504	. . 3 ⊢ (𝜑 → (⟨“𝑃𝑅𝑆𝑍”⟩ ∈ (ℚ ↑_𝑚 (0...3)) ∧ ∀𝑢 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))∀𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))(𝑢 · 𝑣) < 𝐴))
71		fveq1 6492	. . . . . . 7 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (𝑞‘0) = (⟨“𝑃𝑅𝑆𝑍”⟩‘0))
72		fveq1 6492	. . . . . . 7 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (𝑞‘1) = (⟨“𝑃𝑅𝑆𝑍”⟩‘1))
73	71, 72	oveq12d 6988	. . . . . 6 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → ((𝑞‘0)(,)(𝑞‘1)) = ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)))
74	73	raleqdv 3349	. . . . 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 6492	. . . . . . . 8 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (𝑞‘2) = (⟨“𝑃𝑅𝑆𝑍”⟩‘2))
76		fveq1 6492	. . . . . . . 8 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (𝑞‘3) = (⟨“𝑃𝑅𝑆𝑍”⟩‘3))
77	75, 76	oveq12d 6988	. . . . . . 7 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → ((𝑞‘2)(,)(𝑞‘3)) = ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)))
78	77	raleqdv 3349	. . . . . 6 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝐴 ↔ ∀𝑣 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))(𝑢 · 𝑣) < 𝐴))
79	78	ralbidv 3141	. . . . 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 271	. . . 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 3593	. . 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 226	. 2 ⊢ (𝜑 → ⟨“𝑃𝑅𝑆𝑍”⟩ ∈ 𝐾)
84		qssre 12166	. . . . . . 7 ⊢ ℚ ⊆ ℝ
85	84, 1	sseldi 3852	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℝ)
86	85	rexrd 10482	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℝ^*)
87	84, 2	sseldi 3852	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℝ)
88	87	rexrd 10482	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
89	54	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 = if(1 ≤ 𝑋, 1, 𝑋))
90		1rp 12201	. . . . . . . . . . . 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 10462	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 · 𝑉) ∈ ℝ)
94		difrp 12237	. . . . . . . . . . . . . . 15 ⊢ (((𝑈 · 𝑉) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑈 · 𝑉) < 𝐴 ↔ (𝐴 − (𝑈 · 𝑉)) ∈ ℝ⁺))
95	93, 45, 94	syl2anc 576	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑈 · 𝑉) < 𝐴 ↔ (𝐴 − (𝑈 · 𝑉)) ∈ ℝ⁺))
96	51, 95	mpbid 224	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 − (𝑈 · 𝑉)) ∈ ℝ⁺)
97		1red 10432	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℝ)
98	47	recnd 10460	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑈 ∈ ℂ)
99	98	abscld 14647	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (abs‘𝑈) ∈ ℝ)
100	49	recnd 10460	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑉 ∈ ℂ)
101	100	abscld 14647	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (abs‘𝑉) ∈ ℝ)
102	99, 101	readdcld 10461	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((abs‘𝑈) + (abs‘𝑉)) ∈ ℝ)
103	97, 102	readdcld 10461	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 + ((abs‘𝑈) + (abs‘𝑉))) ∈ ℝ)
104		0re 10433	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
105	104	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℝ)
106	91	rpgt0d 12244	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 1)
107	98	absge0d 14655	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 ≤ (abs‘𝑈))
108	100	absge0d 14655	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 0 ≤ (abs‘𝑉))
109	99, 101	addge01d 11021	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0 ≤ (abs‘𝑉) ↔ (abs‘𝑈) ≤ ((abs‘𝑈) + (abs‘𝑉))))
110	108, 109	mpbid 224	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (abs‘𝑈) ≤ ((abs‘𝑈) + (abs‘𝑉)))
111	105, 99, 102, 107, 110	letrd 10589	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ≤ ((abs‘𝑈) + (abs‘𝑉)))
112	97, 102	addge01d 11021	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0 ≤ ((abs‘𝑈) + (abs‘𝑉)) ↔ 1 ≤ (1 + ((abs‘𝑈) + (abs‘𝑉)))))
113	111, 112	mpbid 224	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ (1 + ((abs‘𝑈) + (abs‘𝑉))))
114	105, 97, 103, 106, 113	ltletrd 10592	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < (1 + ((abs‘𝑈) + (abs‘𝑉))))
115	103, 114	elrpd 12238	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1 + ((abs‘𝑈) + (abs‘𝑉))) ∈ ℝ⁺)
116	96, 115	rpdivcld 12258	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉)))) ∈ ℝ⁺)
117	92, 116	eqeltrd 2860	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
118	91, 117	ifcld 4389	. . . . . . . . . 10 ⊢ (𝜑 → if(1 ≤ 𝑋, 1, 𝑋) ∈ ℝ⁺)
119	89, 118	eqeltrd 2860	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
120	119	rpred 12241	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ℝ)
121	47, 120	resubcld 10861	. . . . . . 7 ⊢ (𝜑 → (𝑈 − 𝑌) ∈ ℝ)
122	121	rexrd 10482	. . . . . 6 ⊢ (𝜑 → (𝑈 − 𝑌) ∈ ℝ^*)
123	47	rexrd 10482	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ^*)
124		iooltub 41163	. . . . . 6 ⊢ (((𝑈 − 𝑌) ∈ ℝ^* ∧ 𝑈 ∈ ℝ^* ∧ 𝑃 ∈ ((𝑈 − 𝑌)(,)𝑈)) → 𝑃 < 𝑈)
125	122, 123, 55, 124	syl3anc 1351	. . . . 5 ⊢ (𝜑 → 𝑃 < 𝑈)
126	47, 120	readdcld 10461	. . . . . . 7 ⊢ (𝜑 → (𝑈 + 𝑌) ∈ ℝ)
127	126	rexrd 10482	. . . . . 6 ⊢ (𝜑 → (𝑈 + 𝑌) ∈ ℝ^*)
128		ioogtlb 41147	. . . . . 6 ⊢ ((𝑈 ∈ ℝ^* ∧ (𝑈 + 𝑌) ∈ ℝ^* ∧ 𝑅 ∈ (𝑈(,)(𝑈 + 𝑌))) → 𝑈 < 𝑅)
129	123, 127, 57, 128	syl3anc 1351	. . . . 5 ⊢ (𝜑 → 𝑈 < 𝑅)
130	86, 88, 47, 125, 129	eliood 41150	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (𝑃(,)𝑅))
131	31	eqcomd 2778	. . . 4 ⊢ (𝜑 → (𝑃(,)𝑅) = ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)))
132	130, 131	eleqtrd 2862	. . 3 ⊢ (𝜑 → 𝑈 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)))
133	84, 3	sseldi 3852	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℝ)
134	133	rexrd 10482	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ ℝ^*)
135	84, 4	sseldi 3852	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ ℝ)
136	135	rexrd 10482	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ ℝ^*)
137	49, 120	resubcld 10861	. . . . . . 7 ⊢ (𝜑 → (𝑉 − 𝑌) ∈ ℝ)
138	137	rexrd 10482	. . . . . 6 ⊢ (𝜑 → (𝑉 − 𝑌) ∈ ℝ^*)
139	49	rexrd 10482	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ ℝ^*)
140		iooltub 41163	. . . . . 6 ⊢ (((𝑉 − 𝑌) ∈ ℝ^* ∧ 𝑉 ∈ ℝ^* ∧ 𝑆 ∈ ((𝑉 − 𝑌)(,)𝑉)) → 𝑆 < 𝑉)
141	138, 139, 59, 140	syl3anc 1351	. . . . 5 ⊢ (𝜑 → 𝑆 < 𝑉)
142	49, 120	readdcld 10461	. . . . . . 7 ⊢ (𝜑 → (𝑉 + 𝑌) ∈ ℝ)
143	142	rexrd 10482	. . . . . 6 ⊢ (𝜑 → (𝑉 + 𝑌) ∈ ℝ^*)
144		ioogtlb 41147	. . . . . 6 ⊢ ((𝑉 ∈ ℝ^* ∧ (𝑉 + 𝑌) ∈ ℝ^* ∧ 𝑍 ∈ (𝑉(,)(𝑉 + 𝑌))) → 𝑉 < 𝑍)
145	139, 143, 61, 144	syl3anc 1351	. . . . 5 ⊢ (𝜑 → 𝑉 < 𝑍)
146	134, 136, 49, 141, 145	eliood 41150	. . . 4 ⊢ (𝜑 → 𝑉 ∈ (𝑆(,)𝑍))
147	39	eqcomd 2778	. . . 4 ⊢ (𝜑 → (𝑆(,)𝑍) = ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)))
148	146, 147	eleqtrd 2862	. . 3 ⊢ (𝜑 → 𝑉 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)))
149	132, 148	jca 504	. 2 ⊢ (𝜑 → (𝑈 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)) ∧ 𝑉 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))))
150		nfv 1873	. . 3 ⊢ Ⅎ𝑞(𝑈 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1)) ∧ 𝑉 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3)))
151		nfcv 2926	. . 3 ⊢ Ⅎ𝑞⟨“𝑃𝑅𝑆𝑍”⟩
152		nfrab1 3318	. . . 4 ⊢ Ⅎ𝑞{𝑞 ∈ (ℚ ↑_𝑚 (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝐴}
153	81, 152	nfcxfr 2924	. . 3 ⊢ Ⅎ𝑞𝐾
154	73	eleq2d 2845	. . . 4 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (𝑈 ∈ ((𝑞‘0)(,)(𝑞‘1)) ↔ 𝑈 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘0)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘1))))
155	77	eleq2d 2845	. . . 4 ⊢ (𝑞 = ⟨“𝑃𝑅𝑆𝑍”⟩ → (𝑉 ∈ ((𝑞‘2)(,)(𝑞‘3)) ↔ 𝑉 ∈ ((⟨“𝑃𝑅𝑆𝑍”⟩‘2)(,)(⟨“𝑃𝑅𝑆𝑍”⟩‘3))))
156	154, 155	anbi12d 621	. . 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 40698	. 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 576	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 198 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ∀wral 3082 ∃wrex 3083 {crab 3086 Vcvv 3409 ifcif 4344 class class class wbr 4923 ‘cfv 6182 (class class class)co 6970 ↑_𝑚 cmap 8198 ℝcr 10326 0cc0 10327 1c1 10328 + caddc 10330 · cmul 10332 ℝ^*cxr 10465 < clt 10466 ≤ cle 10467 − cmin 10662 / cdiv 11090 2c2 11488 3c3 11489 4c4 11490 ℕ₀cn0 11700 ℤcz 11786 ℚcq 12155 ℝ⁺crp 12197 (,)cioo 12547 ...cfz 12701 ..^cfzo 12842 ♯chash 13498 Word cword 13662 ⟨“cs4 14057 abscabs 14444

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-oadd 7901 df-er 8081 df-map 8200 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-sup 8693 df-card 9154 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-n0 11701 df-z 11787 df-uz 12052 df-q 12156 df-rp 12198 df-ioo 12551 df-icc 12554 df-fz 12702 df-fzo 12843 df-seq 13178 df-exp 13238 df-hash 13499 df-word 13663 df-concat 13724 df-s1 13749 df-s2 14062 df-s3 14063 df-s4 14064 df-cj 14309 df-re 14310 df-im 14311 df-sqrt 14445 df-abs 14446

This theorem is referenced by: smfmullem3 42445

W3C validator