stoweidlem48 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem48 43589

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 3144	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)
7		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑡𝐴
8	6, 7	nfrabw 3318	. . . . . . 7 ⊢ Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
9	5, 8	nfcxfr 2905	. . . . . 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 2830	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝑌 ↔ 𝑓 ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)})
18		fveq1 6773	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑓 → (ℎ‘𝑡) = (𝑓‘𝑡))
19	18	breq2d 5086	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑓 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝑓‘𝑡)))
20	18	breq1d 5084	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑓 → ((ℎ‘𝑡) ≤ 1 ↔ (𝑓‘𝑡) ≤ 1))
21	19, 20	anbi12d 631	. . . . . . . . . . 11 ⊢ (ℎ = 𝑓 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1)))
22	21	ralbidv 3112	. . . . . . . . . 10 ⊢ (ℎ = 𝑓 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1)))
23	22	elrab 3624	. . . . . . . . 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 218	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑌 → (𝑓 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1)))
25	24	simpld 495	. . . . . . 7 ⊢ (𝑓 ∈ 𝑌 → 𝑓 ∈ 𝐴)
26		stoweidlem48.15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
27	25, 26	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ)
28		eqid 2738	. . . . . . 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 43557	. . . . . 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 43124	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑋‘𝑡) = (𝑍‘𝑡))
32	3, 31	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (𝑋‘𝑡) = (𝑍‘𝑡))
33		elnnuz 12622	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ_≥‘1))
34	15, 33	sylib 217	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
35	34	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → 𝑀 ∈ (ℤ_≥‘1))
36		nfv 1917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖 𝑡 ∈ 𝑇
37	4, 36	nfan 1902	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑡 ∈ 𝑇)
38	16	ffvelrnda 6961	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖) ∈ 𝑌)
39		fveq1 6773	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = (𝑈‘𝑖) → (ℎ‘𝑡) = ((𝑈‘𝑖)‘𝑡))
40	39	breq2d 5086	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = (𝑈‘𝑖) → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ ((𝑈‘𝑖)‘𝑡)))
41	39	breq1d 5084	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = (𝑈‘𝑖) → ((ℎ‘𝑡) ≤ 1 ↔ ((𝑈‘𝑖)‘𝑡) ≤ 1))
42	40, 41	anbi12d 631	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = (𝑈‘𝑖) → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ ((𝑈‘𝑖)‘𝑡) ∧ ((𝑈‘𝑖)‘𝑡) ≤ 1)))
43	42	ralbidv 3112	. . . . . . . . . . . . . . . . 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 3627	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈‘𝑖) ∈ 𝑌 ↔ ((𝑈‘𝑖) ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ ((𝑈‘𝑖)‘𝑡) ∧ ((𝑈‘𝑖)‘𝑡) ≤ 1)))
45	38, 44	sylib 217	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈‘𝑖) ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ ((𝑈‘𝑖)‘𝑡) ∧ ((𝑈‘𝑖)‘𝑡) ≤ 1)))
46	45	simpld 495	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖) ∈ 𝐴)
47		simpl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝜑)
48	47, 46	jca 512	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑈‘𝑖) ∈ 𝐴))
49		eleq1 2826	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑈‘𝑖) → (𝑓 ∈ 𝐴 ↔ (𝑈‘𝑖) ∈ 𝐴))
50	49	anbi2d 629	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑈‘𝑖) → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ (𝑈‘𝑖) ∈ 𝐴)))
51		feq1 6581	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑈‘𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑈‘𝑖):𝑇⟶ℝ))
52	50, 51	imbi12d 345	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑈‘𝑖) → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈‘𝑖) ∈ 𝐴) → (𝑈‘𝑖):𝑇⟶ℝ)))
53	52, 26	vtoclg 3505	. . . . . . . . . . . . . 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 712	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖):𝑇⟶ℝ)
56		simplr 766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡 ∈ 𝑇)
57	55, 56	ffvelrnd 6962	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈‘𝑖)‘𝑡) ∈ ℝ)
58		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))
59	37, 57, 58	fmptdf 6991	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)):(1...𝑀)⟶ℝ)
60		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
61		ovex 7308	. . . . . . . . . . . . 13 ⊢ (1...𝑀) ∈ V
62		mptexg 7097	. . . . . . . . . . . . 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 6886	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝑇 ∧ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) ∈ V) → (𝐹‘𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
65	60, 63, 64	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
66	65	feq1d 6585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡):(1...𝑀)⟶ℝ ↔ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)):(1...𝑀)⟶ℝ))
67	59, 66	mpbird 256	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡):(1...𝑀)⟶ℝ)
68	3, 67	syldan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (𝐹‘𝑡):(1...𝑀)⟶ℝ)
69	68	ffvelrnda 6961	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑘) ∈ ℝ)
70		remulcl 10956	. . . . . . . 8 ⊢ ((𝑘 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑘 · 𝑗) ∈ ℝ)
71	70	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ (𝑘 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝑘 · 𝑗) ∈ ℝ)
72	35, 69, 71	seqcl 13743	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (seq1( · , (𝐹‘𝑡))‘𝑀) ∈ ℝ)
73	13	fvmpt2 6886	. . . . . 6 ⊢ ((𝑡 ∈ 𝑇 ∧ (seq1( · , (𝐹‘𝑡))‘𝑀) ∈ ℝ) → (𝑍‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀))
74	3, 72, 73	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (𝑍‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀))
75		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑖𝑇
76		nfmpt1 5182	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))
77	75, 76	nfmpt 5181	. . . . . . . 8 ⊢ Ⅎ𝑖(𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
78	12, 77	nfcxfr 2905	. . . . . . 7 ⊢ Ⅎ𝑖𝐹
79		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑖𝑡
80	78, 79	nffv 6784	. . . . . 6 ⊢ Ⅎ𝑖(𝐹‘𝑡)
81		nfv 1917	. . . . . . 7 ⊢ Ⅎ𝑖 𝑡 ∈ 𝐷
82	4, 81	nfan 1902	. . . . . 6 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑡 ∈ 𝐷)
83		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑗seq1( · , (𝐹‘𝑡))
84		eqid 2738	. . . . . 6 ⊢ seq1( · , (𝐹‘𝑡)) = seq1( · , (𝐹‘𝑡))
85	15	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → 𝑀 ∈ ℕ)
86		simpll 764	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 𝜑)
87		simpr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ (1...𝑀))
88	3	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡 ∈ 𝑇)
89	45	simprd 496	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ 𝑇 (0 ≤ ((𝑈‘𝑖)‘𝑡) ∧ ((𝑈‘𝑖)‘𝑡) ≤ 1))
90	89	r19.21bi 3134	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝑇) → (0 ≤ ((𝑈‘𝑖)‘𝑡) ∧ ((𝑈‘𝑖)‘𝑡) ≤ 1))
91	90	simpld 495	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝑈‘𝑖)‘𝑡))
92	86, 87, 88, 91	syl21anc 835	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 0 ≤ ((𝑈‘𝑖)‘𝑡))
93	65	fveq1d 6776	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘𝑖))
94	86, 88, 93	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘𝑖))
95	86, 88, 87, 57	syl21anc 835	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈‘𝑖)‘𝑡) ∈ ℝ)
96	58	fvmpt2 6886	. . . . . . . . 9 ⊢ ((𝑖 ∈ (1...𝑀) ∧ ((𝑈‘𝑖)‘𝑡) ∈ ℝ) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘𝑖) = ((𝑈‘𝑖)‘𝑡))
97	87, 95, 96	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))‘𝑖) = ((𝑈‘𝑖)‘𝑡))
98	94, 97	eqtrd 2778	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) = ((𝑈‘𝑖)‘𝑡))
99	92, 98	breqtrrd 5102	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 0 ≤ ((𝐹‘𝑡)‘𝑖))
100	90	simprd 496	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝑇) → ((𝑈‘𝑖)‘𝑡) ≤ 1)
101	86, 87, 88, 100	syl21anc 835	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈‘𝑖)‘𝑡) ≤ 1)
102	98, 101	eqbrtrd 5096	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) ≤ 1)
103		stoweidlem48.17	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
104	103	adantr 481	. . . . . 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 4842	. . . . . . . . . 10 ⊢ (𝑡 ∈ ∪ ran 𝑊 ↔ ∃𝑤(𝑡 ∈ 𝑤 ∧ 𝑤 ∈ ran 𝑊))
108	106, 107	sylib 217	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → ∃𝑤(𝑡 ∈ 𝑤 ∧ 𝑤 ∈ ran 𝑊))
109		stoweidlem48.9	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑊:(1...𝑀)⟶𝑉)
110		ffn 6600	. . . . . . . . . . . . . . . 16 ⊢ (𝑊:(1...𝑀)⟶𝑉 → 𝑊 Fn (1...𝑀))
111		fvelrnb 6830	. . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ ran 𝑊) → ∃𝑗 ∈ (1...𝑀)(𝑊‘𝑗) = 𝑤)
114	113	adantrl 713	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑤 ∧ 𝑤 ∈ ran 𝑊)) → ∃𝑗 ∈ (1...𝑀)(𝑊‘𝑗) = 𝑤)
115		simplr 766	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑤) ∧ (𝑊‘𝑗) = 𝑤) → 𝑡 ∈ 𝑤)
116		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑤) ∧ (𝑊‘𝑗) = 𝑤) → (𝑊‘𝑗) = 𝑤)
117	115, 116	eleqtrrd 2842	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑤) ∧ (𝑊‘𝑗) = 𝑤) → 𝑡 ∈ (𝑊‘𝑗))
118	117	ex 413	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑤) → ((𝑊‘𝑗) = 𝑤 → 𝑡 ∈ (𝑊‘𝑗)))
119	118	reximdv 3202	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑤) → (∃𝑗 ∈ (1...𝑀)(𝑊‘𝑗) = 𝑤 → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊‘𝑗)))
120	119	adantrr 714	. . . . . . . . . . . . 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 413	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑡 ∈ 𝑤 ∧ 𝑤 ∈ ran 𝑊) → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊‘𝑗)))
123	122	exlimdv 1936	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑤(𝑡 ∈ 𝑤 ∧ 𝑤 ∈ ran 𝑊) → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊‘𝑗)))
124	123	adantr 481	. . . . . . . . 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 772	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑗)) → 𝜑)
127		simplr 766	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑗)) → 𝑗 ∈ (1...𝑀))
128		simpr 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑗)) → 𝑡 ∈ (𝑊‘𝑗))
129		nfv 1917	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖 𝑗 ∈ (1...𝑀)
130		nfv 1917	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖 𝑡 ∈ (𝑊‘𝑗)
131	4, 129, 130	nf3an 1904	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊‘𝑗))
132		nfv 1917	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖((𝑈‘𝑗)‘𝑡) < 𝐸
133	131, 132	nfim 1899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊‘𝑗)) → ((𝑈‘𝑗)‘𝑡) < 𝐸)
134		eleq1 2826	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ (1...𝑀) ↔ 𝑗 ∈ (1...𝑀)))
135		fveq2 6774	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑗 → (𝑊‘𝑖) = (𝑊‘𝑗))
136	135	eleq2d 2824	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (𝑡 ∈ (𝑊‘𝑖) ↔ 𝑡 ∈ (𝑊‘𝑗)))
137	134, 136	3anbi23d 1438	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊‘𝑖)) ↔ (𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊‘𝑗))))
138		fveq2 6774	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑗 → (𝑈‘𝑖) = (𝑈‘𝑗))
139	138	fveq1d 6776	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → ((𝑈‘𝑖)‘𝑡) = ((𝑈‘𝑗)‘𝑡))
140	139	breq1d 5084	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (((𝑈‘𝑖)‘𝑡) < 𝐸 ↔ ((𝑈‘𝑗)‘𝑡) < 𝐸))
141	137, 140	imbi12d 345	. . . . . . . . . . . 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 3134	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → ((𝑈‘𝑖)‘𝑡) < 𝐸)
144	143	3impa 1109	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊‘𝑖)) → ((𝑈‘𝑖)‘𝑡) < 𝐸)
145	133, 141, 144	chvarfv 2233	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊‘𝑗)) → ((𝑈‘𝑗)‘𝑡) < 𝐸)
146	126, 127, 128, 145	syl3anc 1370	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑗)) → ((𝑈‘𝑗)‘𝑡) < 𝐸)
147	146	ex 413	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀)) → (𝑡 ∈ (𝑊‘𝑗) → ((𝑈‘𝑗)‘𝑡) < 𝐸))
148	147	reximdva 3203	. . . . . . . 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 1902	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀))
151		nfcv 2907	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖𝑗
152	80, 151	nffv 6784	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖((𝐹‘𝑡)‘𝑗)
153	152	nfeq1 2922	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖((𝐹‘𝑡)‘𝑗) = ((𝑈‘𝑗)‘𝑡)
154	150, 153	nfim 1899	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑗) = ((𝑈‘𝑗)‘𝑡))
155	134	anbi2d 629	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑖 ∈ (1...𝑀)) ↔ ((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀))))
156		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → ((𝐹‘𝑡)‘𝑖) = ((𝐹‘𝑡)‘𝑗))
157	156, 139	eqeq12d 2754	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (((𝐹‘𝑡)‘𝑖) = ((𝑈‘𝑖)‘𝑡) ↔ ((𝐹‘𝑡)‘𝑗) = ((𝑈‘𝑗)‘𝑡)))
158	155, 157	imbi12d 345	. . . . . . . . . . 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 2233	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑗) = ((𝑈‘𝑗)‘𝑡))
160	159	breq1d 5084	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀)) → (((𝐹‘𝑡)‘𝑗) < 𝐸 ↔ ((𝑈‘𝑗)‘𝑡) < 𝐸))
161	160	biimprd 247	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑗 ∈ (1...𝑀)) → (((𝑈‘𝑗)‘𝑡) < 𝐸 → ((𝐹‘𝑡)‘𝑗) < 𝐸))
162	161	reximdva 3203	. . . . . . 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 43127	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (seq1( · , (𝐹‘𝑡))‘𝑀) < 𝐸)
165	74, 164	eqbrtrd 5096	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (𝑍‘𝑡) < 𝐸)
166	32, 165	eqbrtrd 5096	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (𝑋‘𝑡) < 𝐸)
167	166	ex 413	. 2 ⊢ (𝜑 → (𝑡 ∈ 𝐷 → (𝑋‘𝑡) < 𝐸))
168	1, 167	ralrimi 3141	1 ⊢ (𝜑 → ∀𝑡 ∈ 𝐷 (𝑋‘𝑡) < 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∃wex 1782 Ⅎwnf 1786 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 {crab 3068 Vcvv 3432 ⊆ wss 3887 ∪ cuni 4839 class class class wbr 5074 ↦ cmpt 5157 ran crn 5590 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 ℝcr 10870 0cc0 10871 1c1 10872 · cmul 10876 < clt 11009 ≤ cle 11010 ℕcn 11973 ℤ_≥cuz 12582 ℝ⁺crp 12730 ...cfz 13239 seqcseq 13721

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

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-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-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-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722

This theorem is referenced by: stoweidlem51 43592

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