stoweidlem48 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem48 46476

Description: This lemma is used to prove that 𝑥 built as in Lemma 2 of [BrosowskiDeutsh] p. 91, is such that x < ε on 𝐴. Here 𝑋 is used to represent 𝑥 in the paper, 𝐸 is used to represent ε in the paper, and 𝐷 is used to represent 𝐴 in the paper (because 𝐴 is always used to represent the subalgebra). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

**Hypotheses**
Ref	Expression
stoweidlem48.1	⊢ Ⅎ𝑖𝜑
stoweidlem48.2	⊢ Ⅎ𝑡𝜑
stoweidlem48.3	⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
stoweidlem48.4	⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
stoweidlem48.5	⊢ 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)
stoweidlem48.6	⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
stoweidlem48.7	⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀))
stoweidlem48.8	⊢ (𝜑 → 𝑀 ∈ ℕ)
stoweidlem48.9	⊢ (𝜑 → 𝑊:(1...𝑀)⟶𝑉)
stoweidlem48.10	⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)
stoweidlem48.11	⊢ (𝜑 → 𝐷 ⊆ ∪ ran 𝑊)
stoweidlem48.12	⊢ (𝜑 → 𝐷 ⊆ 𝑇)
stoweidlem48.13	⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊‘𝑖)((𝑈‘𝑖)‘𝑡) < 𝐸)
stoweidlem48.14	⊢ (𝜑 → 𝑇 ∈ V)
stoweidlem48.15	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem48.16	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem48.17	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)

**Assertion**
Ref	Expression
stoweidlem48	⊢ (𝜑 → ∀𝑡 ∈ 𝐷 (𝑋‘𝑡) < 𝐸)

Distinct variable groups: 𝑓,𝑔,ℎ,𝑡,𝐴 𝑓,𝑖,𝑇,ℎ,𝑡 𝑓,𝐹,𝑔 𝑓,𝑀,𝑔 𝑈,𝑓,𝑔,ℎ,𝑡 𝑓,𝑌,𝑔 𝜑,𝑓,𝑔 𝑇,𝑔 𝐷,𝑖 𝑖,𝐸 𝑖,𝑀 𝑈,𝑖 𝑖,𝑊 Allowed substitution hints: 𝜑(𝑡,ℎ,𝑖) 𝐴(𝑖) 𝐷(𝑡,𝑓,𝑔,ℎ) 𝑃(𝑡,𝑓,𝑔,ℎ,𝑖) 𝐸(𝑡,𝑓,𝑔,ℎ) 𝐹(𝑡,ℎ,𝑖) 𝑀(𝑡,ℎ) 𝑉(𝑡,𝑓,𝑔,ℎ,𝑖) 𝑊(𝑡,𝑓,𝑔,ℎ) 𝑋(𝑡,𝑓,𝑔,ℎ,𝑖) 𝑌(𝑡,ℎ,𝑖) 𝑍(𝑡,𝑓,𝑔,ℎ,𝑖)

Proof of Theorem stoweidlem48 Dummy variables 𝑗 𝑘 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		stoweidlem48.2	. 2 ⊢ Ⅎ𝑡𝜑
2		stoweidlem48.12	. . . . . 6 ⊢ (𝜑 → 𝐷 ⊆ 𝑇)
3	2	sselda 3921	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → 𝑡 ∈ 𝑇)
4		stoweidlem48.1	. . . . . 6 ⊢ Ⅎ𝑖𝜑
5		stoweidlem48.3	. . . . . . 7 ⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
6		nfra1 3261	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)
7		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑡𝐴
8	6, 7	nfrabw 3426	. . . . . . 7 ⊢ Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
9	5, 8	nfcxfr 2896	. . . . . 6 ⊢ Ⅎ𝑡𝑌
10		stoweidlem48.4	. . . . . 6 ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
11		stoweidlem48.5	. . . . . 6 ⊢ 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)
12		stoweidlem48.6	. . . . . 6 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
13		stoweidlem48.7	. . . . . 6 ⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀))
14		stoweidlem48.14	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ V)
15		stoweidlem48.8	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
16		stoweidlem48.10	. . . . . 6 ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)
17	5	eleq2i 2828	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝑌 ↔ 𝑓 ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)})
18		fveq1 6839	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑓 → (ℎ‘𝑡) = (𝑓‘𝑡))
19	18	breq2d 5097	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑓 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝑓‘𝑡)))
20	18	breq1d 5095	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑓 → ((ℎ‘𝑡) ≤ 1 ↔ (𝑓‘𝑡) ≤ 1))
21	19, 20	anbi12d 633	. . . . . . . . . . 11 ⊢ (ℎ = 𝑓 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1)))
22	21	ralbidv 3160	. . . . . . . . . 10 ⊢ (ℎ = 𝑓 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1)))
23	22	elrab 3634	. . . . . . . . 9 ⊢ (𝑓 ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} ↔ (𝑓 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1)))
24	17, 23	sylbb 219	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑌 → (𝑓 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1)))
25	24	simpld 494	. . . . . . 7 ⊢ (𝑓 ∈ 𝑌 → 𝑓 ∈ 𝐴)
26		stoweidlem48.15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
27	25, 26	sylan2 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ)
28		eqid 2736	. . . . . . 7 ⊢ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))
29		stoweidlem48.16	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
30	1, 5, 28, 26, 29	stoweidlem16 46444	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌)
31	4, 9, 10, 11, 12, 13, 14, 15, 16, 27, 30	fmuldfeq 46013	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑋‘𝑡) = (𝑍‘𝑡))
32	3, 31	syldan 592	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (𝑋‘𝑡) = (𝑍‘𝑡))
33		elnnuz 12828	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ_≥‘1))
34	15, 33	sylib 218	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
35	34	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → 𝑀 ∈ (ℤ_≥‘1))
36		nfv 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖 𝑡 ∈ 𝑇
37	4, 36	nfan 1901	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑡 ∈ 𝑇)
38	16	ffvelcdmda 7036	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖) ∈ 𝑌)
39		fveq1 6839	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = (𝑈‘𝑖) → (ℎ‘𝑡) = ((𝑈‘𝑖)‘𝑡))
40	39	breq2d 5097	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = (𝑈‘𝑖) → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ ((𝑈‘𝑖)‘𝑡)))
41	39	breq1d 5095	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = (𝑈‘𝑖) → ((ℎ‘𝑡) ≤ 1 ↔ ((𝑈‘𝑖)‘𝑡) ≤ 1))
42	40, 41	anbi12d 633	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = (𝑈‘𝑖) → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ ((𝑈‘𝑖)‘𝑡) ∧ ((𝑈‘𝑖)‘𝑡) ≤ 1)))
43	42	ralbidv 3160	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = (𝑈‘𝑖) → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ ((𝑈‘𝑖)‘𝑡) ∧ ((𝑈‘𝑖)‘𝑡) ≤ 1)))
44	43, 5	elrab2 3637	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈‘𝑖) ∈ 𝑌 ↔ ((𝑈‘𝑖) ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ ((𝑈‘𝑖)‘𝑡) ∧ ((𝑈‘𝑖)‘𝑡) ≤ 1)))
45	38, 44	sylib 218	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈‘𝑖) ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ ((𝑈‘𝑖)‘𝑡) ∧ ((𝑈‘𝑖)‘𝑡) ≤ 1)))
46	45	simpld 494	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖) ∈ 𝐴)
47		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝜑)
48	47, 46	jca 511	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑈‘𝑖) ∈ 𝐴))
49		eleq1 2824	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑈‘𝑖) → (𝑓 ∈ 𝐴 ↔ (𝑈‘𝑖) ∈ 𝐴))
50	49	anbi2d 631	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑈‘𝑖) → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ (𝑈‘𝑖) ∈ 𝐴)))
51		feq1 6646	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑈‘𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑈‘𝑖):𝑇⟶ℝ))
52	50, 51	imbi12d 344	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑈‘𝑖) → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈‘𝑖) ∈ 𝐴) → (𝑈‘𝑖):𝑇⟶ℝ)))
53	52, 26	vtoclg 3499	. . . . . . . . . . . . . 14 ⊢ ((𝑈‘𝑖) ∈ 𝐴 → ((𝜑 ∧ (𝑈‘𝑖) ∈ 𝐴) → (𝑈‘𝑖):𝑇⟶ℝ))
54	46, 48, 53	sylc 65	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖):𝑇⟶ℝ)
55	54	adantlr 716	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖):𝑇⟶ℝ)
56		simplr 769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡 ∈ 𝑇)
57	55, 56	ffvelcdmd 7037	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈‘𝑖)‘𝑡) ∈ ℝ)
58		eqid 2736	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))
59	37, 57, 58	fmptdf 7069	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)):(1...𝑀)⟶ℝ)
60		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
61		ovex 7400	. . . . . . . . . . . . 13 ⊢ (1...𝑀) ∈ V
62		mptexg 7176	. . . . . . . . . . . . 13 ⊢ ((1...𝑀) ∈ V → (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) ∈ V)
63	61, 62	mp1i 13	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) ∈ V)
64	12	fvmpt2 6959	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝑇 ∧ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) ∈ V) → (𝐹‘𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
65	60, 63, 64	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
66	65	feq1d 6650	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡):(1...𝑀)⟶ℝ ↔ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)):(1...𝑀)⟶ℝ))
67	59, 66	mpbird 257	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡):(1...𝑀)⟶ℝ)
68	3, 67	syldan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (𝐹‘𝑡):(1...𝑀)⟶ℝ)
69	68	ffvelcdmda 7036	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑘) ∈ ℝ)
70		remulcl 11123	. . . . . . . 8 ⊢ ((𝑘 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑘 · 𝑗) ∈ ℝ)
71	70	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ (𝑘 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝑘 · 𝑗) ∈ ℝ)
72	35, 69, 71	seqcl 13984	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (seq1( · , (𝐹‘𝑡))‘𝑀) ∈ ℝ)
73	13	fvmpt2 6959	. . . . . 6 ⊢ ((𝑡 ∈ 𝑇 ∧ (seq1( · , (𝐹‘𝑡))‘𝑀) ∈ ℝ) → (𝑍‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀))
74	3, 72, 73	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (𝑍‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀))
75		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑖𝑇
76		nfmpt1 5184	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))
77	75, 76	nfmpt 5183	. . . . . . . 8 ⊢ Ⅎ𝑖(𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
78	12, 77	nfcxfr 2896	. . . . . . 7 ⊢ Ⅎ𝑖𝐹
79		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑖𝑡
80	78, 79	nffv 6850	. . . . . 6 ⊢ Ⅎ𝑖(𝐹‘𝑡)
81		nfv 1916	. . . . . . 7 ⊢ Ⅎ𝑖 𝑡 ∈ 𝐷
82	4, 81	nfan 1901	. . . . . 6 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑡 ∈ 𝐷)
83		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑗seq1( · , (𝐹‘𝑡))
84		eqid 2736	. . . . . 6 ⊢ seq1( · , (𝐹‘𝑡)) = seq1( · , (𝐹‘𝑡))
85	15	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → 𝑀 ∈ ℕ)
86		simpll 767	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 𝜑)
87		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ (1...𝑀))
88	3	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡 ∈ 𝑇)
89	45	simprd 495	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ 𝑇 (0 ≤ ((𝑈‘𝑖)‘𝑡) ∧ ((𝑈‘𝑖)‘𝑡) ≤ 1))
90	89	r19.21bi 3229	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝑇) → (0 ≤ ((𝑈‘𝑖)‘𝑡) ∧ ((𝑈‘𝑖)‘𝑡) ≤ 1))
91	90	simpld 494	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝑈‘𝑖)‘𝑡))
92	86, 87, 88, 91	syl21anc 838	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 0 ≤ ((𝑈‘𝑖)‘𝑡))
93	65	fveq1d 6842	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘𝑖))
94	86, 88, 93	syl2anc 585	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘𝑖))
95	86, 88, 87, 57	syl21anc 838	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈‘𝑖)‘𝑡) ∈ ℝ)
96	58	fvmpt2 6959	. . . . . . . . 9 ⊢ ((𝑖 ∈ (1...𝑀) ∧ ((𝑈‘𝑖)‘𝑡) ∈ ℝ) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘𝑖) = ((𝑈‘𝑖)‘𝑡))
97	87, 95, 96	syl2anc 585	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘𝑖) = ((𝑈‘𝑖)‘𝑡))
98	94, 97	eqtrd 2771	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) = ((𝑈‘𝑖)‘𝑡))
99	92, 98	breqtrrd 5113	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 0 ≤ ((𝐹‘𝑡)‘𝑖))
100	90	simprd 495	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝑇) → ((𝑈‘𝑖)‘𝑡) ≤ 1)
101	86, 87, 88, 100	syl21anc 838	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈‘𝑖)‘𝑡) ≤ 1)
102	98, 101	eqbrtrd 5107	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) ≤ 1)
103		stoweidlem48.17	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
104	103	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → 𝐸 ∈ ℝ⁺)
105		stoweidlem48.11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ⊆ ∪ ran 𝑊)
106	105	sselda 3921	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → 𝑡 ∈ ∪ ran 𝑊)
107		eluni 4853	. . . . . . . . . 10 ⊢ (𝑡 ∈ ∪ ran 𝑊 ↔ ∃𝑤(𝑡 ∈ 𝑤 ∧ 𝑤 ∈ ran 𝑊))
108	106, 107	sylib 218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → ∃𝑤(𝑡 ∈ 𝑤 ∧ 𝑤 ∈ ran 𝑊))
109		stoweidlem48.9	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑊:(1...𝑀)⟶𝑉)
110		ffn 6668	. . . . . . . . . . . . . . . 16 ⊢ (𝑊:(1...𝑀)⟶𝑉 → 𝑊 Fn (1...𝑀))
111		fvelrnb 6900	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 Fn (1...𝑀) → (𝑤 ∈ ran 𝑊 ↔ ∃𝑗 ∈ (1...𝑀)(𝑊‘𝑗) = 𝑤))
112	109, 110, 111	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑤 ∈ ran 𝑊 ↔ ∃𝑗 ∈ (1...𝑀)(𝑊‘𝑗) = 𝑤))
113	112	biimpa 476	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ ran 𝑊) → ∃𝑗 ∈ (1...𝑀)(𝑊‘𝑗) = 𝑤)
114	113	adantrl 717	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑤 ∧ 𝑤 ∈ ran 𝑊)) → ∃𝑗 ∈ (1...𝑀)(𝑊‘𝑗) = 𝑤)
115		simplr 769	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑤) ∧ (𝑊‘𝑗) = 𝑤) → 𝑡 ∈ 𝑤)
116		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑤) ∧ (𝑊‘𝑗) = 𝑤) → (𝑊‘𝑗) = 𝑤)
117	115, 116	eleqtrrd 2839	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑤) ∧ (𝑊‘𝑗) = 𝑤) → 𝑡 ∈ (𝑊‘𝑗))
118	117	ex 412	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑤) → ((𝑊‘𝑗) = 𝑤 → 𝑡 ∈ (𝑊‘𝑗)))
119	118	reximdv 3152	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑤) → (∃𝑗 ∈ (1...𝑀)(𝑊‘𝑗) = 𝑤 → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊‘𝑗)))
120	119	adantrr 718	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑤 ∧ 𝑤 ∈ ran 𝑊)) → (∃𝑗 ∈ (1...𝑀)(𝑊‘𝑗) = 𝑤 → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊‘𝑗)))
121	114, 120	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑤 ∧ 𝑤 ∈ ran 𝑊)) → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊‘𝑗))
122	121	ex 412	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑡 ∈ 𝑤 ∧ 𝑤 ∈ ran 𝑊) → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊‘𝑗)))
123	122	exlimdv 1935	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑤(𝑡 ∈ 𝑤 ∧ 𝑤 ∈ ran 𝑊) → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊‘𝑗)))
124	123	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (∃𝑤(𝑡 ∈ 𝑤 ∧ 𝑤 ∈ ran 𝑊) → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊‘𝑗)))
125	108, 124	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊‘𝑗))
126		simplll 775	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑗)) → 𝜑)
127		simplr 769	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑗)) → 𝑗 ∈ (1...𝑀))
128		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑗)) → 𝑡 ∈ (𝑊‘𝑗))
129		nfv 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖 𝑗 ∈ (1...𝑀)
130		nfv 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖 𝑡 ∈ (𝑊‘𝑗)
131	4, 129, 130	nf3an 1903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊‘𝑗))
132		nfv 1916	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖((𝑈‘𝑗)‘𝑡) < 𝐸
133	131, 132	nfim 1898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊‘𝑗)) → ((𝑈‘𝑗)‘𝑡) < 𝐸)
134		eleq1 2824	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ (1...𝑀) ↔ 𝑗 ∈ (1...𝑀)))
135		fveq2 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑗 → (𝑊‘𝑖) = (𝑊‘𝑗))
136	135	eleq2d 2822	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (𝑡 ∈ (𝑊‘𝑖) ↔ 𝑡 ∈ (𝑊‘𝑗)))
137	134, 136	3anbi23d 1442	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊‘𝑖)) ↔ (𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊‘𝑗))))
138		fveq2 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑗 → (𝑈‘𝑖) = (𝑈‘𝑗))
139	138	fveq1d 6842	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → ((𝑈‘𝑖)‘𝑡) = ((𝑈‘𝑗)‘𝑡))
140	139	breq1d 5095	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (((𝑈‘𝑖)‘𝑡) < 𝐸 ↔ ((𝑈‘𝑗)‘𝑡) < 𝐸))
141	137, 140	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊‘𝑖)) → ((𝑈‘𝑖)‘𝑡) < 𝐸) ↔ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊‘𝑗)) → ((𝑈‘𝑗)‘𝑡) < 𝐸)))
142		stoweidlem48.13	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊‘𝑖)((𝑈‘𝑖)‘𝑡) < 𝐸)
143	142	r19.21bi 3229	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → ((𝑈‘𝑖)‘𝑡) < 𝐸)
144	143	3impa 1110	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊‘𝑖)) → ((𝑈‘𝑖)‘𝑡) < 𝐸)
145	133, 141, 144	chvarfv 2248	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊‘𝑗)) → ((𝑈‘𝑗)‘𝑡) < 𝐸)
146	126, 127, 128, 145	syl3anc 1374	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑗)) → ((𝑈‘𝑗)‘𝑡) < 𝐸)
147	146	ex 412	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀)) → (𝑡 ∈ (𝑊‘𝑗) → ((𝑈‘𝑗)‘𝑡) < 𝐸))
148	147	reximdva 3150	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊‘𝑗) → ∃𝑗 ∈ (1...𝑀)((𝑈‘𝑗)‘𝑡) < 𝐸))
149	125, 148	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → ∃𝑗 ∈ (1...𝑀)((𝑈‘𝑗)‘𝑡) < 𝐸)
150	82, 129	nfan 1901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀))
151		nfcv 2898	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖𝑗
152	80, 151	nffv 6850	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖((𝐹‘𝑡)‘𝑗)
153	152	nfeq1 2914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖((𝐹‘𝑡)‘𝑗) = ((𝑈‘𝑗)‘𝑡)
154	150, 153	nfim 1898	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑗) = ((𝑈‘𝑗)‘𝑡))
155	134	anbi2d 631	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) ↔ ((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀))))
156		fveq2 6840	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → ((𝐹‘𝑡)‘𝑖) = ((𝐹‘𝑡)‘𝑗))
157	156, 139	eqeq12d 2752	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (((𝐹‘𝑡)‘𝑖) = ((𝑈‘𝑖)‘𝑡) ↔ ((𝐹‘𝑡)‘𝑗) = ((𝑈‘𝑗)‘𝑡)))
158	155, 157	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → ((((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) = ((𝑈‘𝑖)‘𝑡)) ↔ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑗) = ((𝑈‘𝑗)‘𝑡))))
159	154, 158, 98	chvarfv 2248	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑗) = ((𝑈‘𝑗)‘𝑡))
160	159	breq1d 5095	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀)) → (((𝐹‘𝑡)‘𝑗) < 𝐸 ↔ ((𝑈‘𝑗)‘𝑡) < 𝐸))
161	160	biimprd 248	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀)) → (((𝑈‘𝑗)‘𝑡) < 𝐸 → ((𝐹‘𝑡)‘𝑗) < 𝐸))
162	161	reximdva 3150	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (∃𝑗 ∈ (1...𝑀)((𝑈‘𝑗)‘𝑡) < 𝐸 → ∃𝑗 ∈ (1...𝑀)((𝐹‘𝑡)‘𝑗) < 𝐸))
163	149, 162	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → ∃𝑗 ∈ (1...𝑀)((𝐹‘𝑡)‘𝑗) < 𝐸)
164	80, 82, 83, 84, 85, 68, 99, 102, 104, 163	fmul01lt1 46016	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (seq1( · , (𝐹‘𝑡))‘𝑀) < 𝐸)
165	74, 164	eqbrtrd 5107	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (𝑍‘𝑡) < 𝐸)
166	32, 165	eqbrtrd 5107	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (𝑋‘𝑡) < 𝐸)
167	166	ex 412	. 2 ⊢ (𝜑 → (𝑡 ∈ 𝐷 → (𝑋‘𝑡) < 𝐸))
168	1, 167	ralrimi 3235	1 ⊢ (𝜑 → ∀𝑡 ∈ 𝐷 (𝑋‘𝑡) < 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∃wex 1781 Ⅎwnf 1785 ∈ wcel 2114 ∀wral 3051 ∃wrex 3061 {crab 3389 Vcvv 3429 ⊆ wss 3889 ∪ cuni 4850 class class class wbr 5085 ↦ cmpt 5166 ran crn 5632 Fn wfn 6493 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ∈ cmpo 7369 ℝcr 11037 0cc0 11038 1c1 11039 · cmul 11043 < clt 11179 ≤ cle 11180 ℕcn 12174 ℤ_≥cuz 12788 ℝ⁺crp 12942 ...cfz 13461 seqcseq 13963

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-n0 12438 df-z 12525 df-uz 12789 df-rp 12943 df-fz 13462 df-fzo 13609 df-seq 13964

This theorem is referenced by: stoweidlem51 46479

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