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Theorem stoweidlem39 42201
Description: This lemma is used to prove that there exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91: assuming that 𝑟 is a finite subset of 𝑊, 𝑥 indexes a finite set of functions in the subalgebra (of the Stone Weierstrass theorem), such that for all i ranging in the finite indexing set, 0 ≤ xi ≤ 1, xi < ε / m on V(ti), and xi > 1 - ε / m on 𝐵. Here 𝐷 is used to represent A in the paper's Lemma 2 (because 𝐴 is used for the subalgebra), 𝑀 is used to represent m in the paper, 𝐸 is used to represent ε, and vi is used to represent V(ti). 𝑊 is just a local definition, used to shorten statements. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem39.1 𝜑
stoweidlem39.2 𝑡𝜑
stoweidlem39.3 𝑤𝜑
stoweidlem39.4 𝑈 = (𝑇𝐵)
stoweidlem39.5 𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
stoweidlem39.6 𝑊 = {𝑤𝐽 ∣ ∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡))}
stoweidlem39.7 (𝜑𝑟 ∈ (𝒫 𝑊 ∩ Fin))
stoweidlem39.8 (𝜑𝐷 𝑟)
stoweidlem39.9 (𝜑𝐷 ≠ ∅)
stoweidlem39.10 (𝜑𝐸 ∈ ℝ+)
stoweidlem39.11 (𝜑𝐵𝑇)
stoweidlem39.12 (𝜑𝑊 ∈ V)
stoweidlem39.13 (𝜑𝐴 ∈ V)
Assertion
Ref Expression
stoweidlem39 (𝜑 → ∃𝑚 ∈ ℕ ∃𝑣(𝑣:(1...𝑚)⟶𝑊𝐷 ran 𝑣 ∧ ∃𝑥(𝑥:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣𝑖)((𝑥𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑚)) < ((𝑥𝑖)‘𝑡)))))
Distinct variable groups:   𝑒,,𝑚,𝑡,𝑤   𝐴,𝑒,,𝑡,𝑤   𝑒,𝐸,,𝑡,𝑤   𝑇,𝑒,,𝑤   𝑈,𝑒,,𝑤   ,𝑖,𝑟,𝑣,𝑥,𝑚,𝑡,𝑤   𝐴,𝑖,𝑥   𝑖,𝐸,𝑥   𝑇,𝑖,𝑥   𝑈,𝑖,𝑥   𝜑,𝑖,𝑚,𝑣   𝑤,𝑌,𝑥   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑤,𝑡,𝑒,,𝑟)   𝐴(𝑣,𝑚,𝑟)   𝐵(𝑤,𝑣,𝑡,𝑒,,𝑖,𝑚,𝑟)   𝐷(𝑥,𝑤,𝑣,𝑡,𝑒,,𝑖,𝑚,𝑟)   𝑇(𝑣,𝑡,𝑚,𝑟)   𝑈(𝑣,𝑡,𝑚,𝑟)   𝐸(𝑣,𝑚,𝑟)   𝐽(𝑥,𝑤,𝑣,𝑡,𝑒,,𝑖,𝑚,𝑟)   𝑊(𝑥,𝑤,𝑣,𝑡,𝑒,,𝑖,𝑚,𝑟)   𝑌(𝑣,𝑡,𝑒,,𝑖,𝑚,𝑟)

Proof of Theorem stoweidlem39
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 stoweidlem39.8 . . . . . . 7 (𝜑𝐷 𝑟)
2 stoweidlem39.9 . . . . . . 7 (𝜑𝐷 ≠ ∅)
31, 2jca 512 . . . . . 6 (𝜑 → (𝐷 𝑟𝐷 ≠ ∅))
4 ssn0 4351 . . . . . 6 ((𝐷 𝑟𝐷 ≠ ∅) → 𝑟 ≠ ∅)
5 unieq 4838 . . . . . . . 8 (𝑟 = ∅ → 𝑟 = ∅)
6 uni0 4857 . . . . . . . 8 ∅ = ∅
75, 6syl6eq 2869 . . . . . . 7 (𝑟 = ∅ → 𝑟 = ∅)
87necon3i 3045 . . . . . 6 ( 𝑟 ≠ ∅ → 𝑟 ≠ ∅)
93, 4, 83syl 18 . . . . 5 (𝜑𝑟 ≠ ∅)
109neneqd 3018 . . . 4 (𝜑 → ¬ 𝑟 = ∅)
11 stoweidlem39.7 . . . . . 6 (𝜑𝑟 ∈ (𝒫 𝑊 ∩ Fin))
12 elinel2 4170 . . . . . 6 (𝑟 ∈ (𝒫 𝑊 ∩ Fin) → 𝑟 ∈ Fin)
1311, 12syl 17 . . . . 5 (𝜑𝑟 ∈ Fin)
14 fz1f1o 15055 . . . . 5 (𝑟 ∈ Fin → (𝑟 = ∅ ∨ ((♯‘𝑟) ∈ ℕ ∧ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto𝑟)))
15 pm2.53 845 . . . . 5 ((𝑟 = ∅ ∨ ((♯‘𝑟) ∈ ℕ ∧ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto𝑟)) → (¬ 𝑟 = ∅ → ((♯‘𝑟) ∈ ℕ ∧ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto𝑟)))
1613, 14, 153syl 18 . . . 4 (𝜑 → (¬ 𝑟 = ∅ → ((♯‘𝑟) ∈ ℕ ∧ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto𝑟)))
1710, 16mpd 15 . . 3 (𝜑 → ((♯‘𝑟) ∈ ℕ ∧ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto𝑟))
18 oveq2 7153 . . . . . 6 (𝑚 = (♯‘𝑟) → (1...𝑚) = (1...(♯‘𝑟)))
1918f1oeq2d 6604 . . . . 5 (𝑚 = (♯‘𝑟) → (𝑣:(1...𝑚)–1-1-onto𝑟𝑣:(1...(♯‘𝑟))–1-1-onto𝑟))
2019exbidv 1913 . . . 4 (𝑚 = (♯‘𝑟) → (∃𝑣 𝑣:(1...𝑚)–1-1-onto𝑟 ↔ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto𝑟))
2120rspcev 3620 . . 3 (((♯‘𝑟) ∈ ℕ ∧ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto𝑟) → ∃𝑚 ∈ ℕ ∃𝑣 𝑣:(1...𝑚)–1-1-onto𝑟)
2217, 21syl 17 . 2 (𝜑 → ∃𝑚 ∈ ℕ ∃𝑣 𝑣:(1...𝑚)–1-1-onto𝑟)
23 f1of 6608 . . . . . . . 8 (𝑣:(1...𝑚)–1-1-onto𝑟𝑣:(1...𝑚)⟶𝑟)
2423adantl 482 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝑣:(1...𝑚)⟶𝑟)
25 simpll 763 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝜑)
26 elinel1 4169 . . . . . . . . 9 (𝑟 ∈ (𝒫 𝑊 ∩ Fin) → 𝑟 ∈ 𝒫 𝑊)
2726elpwid 4549 . . . . . . . 8 (𝑟 ∈ (𝒫 𝑊 ∩ Fin) → 𝑟𝑊)
2825, 11, 273syl 18 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝑟𝑊)
2924, 28fssd 6521 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝑣:(1...𝑚)⟶𝑊)
301ad2antrr 722 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝐷 𝑟)
31 dff1o2 6613 . . . . . . . . . 10 (𝑣:(1...𝑚)–1-1-onto𝑟 ↔ (𝑣 Fn (1...𝑚) ∧ Fun 𝑣 ∧ ran 𝑣 = 𝑟))
3231simp3bi 1139 . . . . . . . . 9 (𝑣:(1...𝑚)–1-1-onto𝑟 → ran 𝑣 = 𝑟)
3332unieqd 4840 . . . . . . . 8 (𝑣:(1...𝑚)–1-1-onto𝑟 ran 𝑣 = 𝑟)
3433adantl 482 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → ran 𝑣 = 𝑟)
3530, 34sseqtrrd 4005 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝐷 ran 𝑣)
36 stoweidlem39.1 . . . . . . . . 9 𝜑
37 nfv 1906 . . . . . . . . 9 𝑚 ∈ ℕ
3836, 37nfan 1891 . . . . . . . 8 (𝜑𝑚 ∈ ℕ)
39 nfv 1906 . . . . . . . 8 𝑣:(1...𝑚)–1-1-onto𝑟
4038, 39nfan 1891 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟)
41 stoweidlem39.2 . . . . . . . . 9 𝑡𝜑
42 nfv 1906 . . . . . . . . 9 𝑡 𝑚 ∈ ℕ
4341, 42nfan 1891 . . . . . . . 8 𝑡(𝜑𝑚 ∈ ℕ)
44 nfv 1906 . . . . . . . 8 𝑡 𝑣:(1...𝑚)–1-1-onto𝑟
4543, 44nfan 1891 . . . . . . 7 𝑡((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟)
46 stoweidlem39.3 . . . . . . . . 9 𝑤𝜑
47 nfv 1906 . . . . . . . . 9 𝑤 𝑚 ∈ ℕ
4846, 47nfan 1891 . . . . . . . 8 𝑤(𝜑𝑚 ∈ ℕ)
49 nfv 1906 . . . . . . . 8 𝑤 𝑣:(1...𝑚)–1-1-onto𝑟
5048, 49nfan 1891 . . . . . . 7 𝑤((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟)
51 stoweidlem39.5 . . . . . . 7 𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
52 stoweidlem39.6 . . . . . . 7 𝑊 = {𝑤𝐽 ∣ ∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡))}
53 eqid 2818 . . . . . . 7 (𝑤𝑟 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑚)) < (𝑡))}) = (𝑤𝑟 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑚)) < (𝑡))})
54 simplr 765 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝑚 ∈ ℕ)
55 simpr 485 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝑣:(1...𝑚)–1-1-onto𝑟)
56 stoweidlem39.10 . . . . . . . 8 (𝜑𝐸 ∈ ℝ+)
5756ad2antrr 722 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝐸 ∈ ℝ+)
58 stoweidlem39.11 . . . . . . . . . . . 12 (𝜑𝐵𝑇)
5958sselda 3964 . . . . . . . . . . 11 ((𝜑𝑏𝐵) → 𝑏𝑇)
60 notnot 144 . . . . . . . . . . . . . . 15 (𝑏𝐵 → ¬ ¬ 𝑏𝐵)
6160intnand 489 . . . . . . . . . . . . . 14 (𝑏𝐵 → ¬ (𝑏𝑇 ∧ ¬ 𝑏𝐵))
6261adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑏𝐵) → ¬ (𝑏𝑇 ∧ ¬ 𝑏𝐵))
63 eldif 3943 . . . . . . . . . . . . 13 (𝑏 ∈ (𝑇𝐵) ↔ (𝑏𝑇 ∧ ¬ 𝑏𝐵))
6462, 63sylnibr 330 . . . . . . . . . . . 12 ((𝜑𝑏𝐵) → ¬ 𝑏 ∈ (𝑇𝐵))
65 stoweidlem39.4 . . . . . . . . . . . . 13 𝑈 = (𝑇𝐵)
6665eleq2i 2901 . . . . . . . . . . . 12 (𝑏𝑈𝑏 ∈ (𝑇𝐵))
6764, 66sylnibr 330 . . . . . . . . . . 11 ((𝜑𝑏𝐵) → ¬ 𝑏𝑈)
6859, 67eldifd 3944 . . . . . . . . . 10 ((𝜑𝑏𝐵) → 𝑏 ∈ (𝑇𝑈))
6968ralrimiva 3179 . . . . . . . . 9 (𝜑 → ∀𝑏𝐵 𝑏 ∈ (𝑇𝑈))
70 dfss3 3953 . . . . . . . . 9 (𝐵 ⊆ (𝑇𝑈) ↔ ∀𝑏𝐵 𝑏 ∈ (𝑇𝑈))
7169, 70sylibr 235 . . . . . . . 8 (𝜑𝐵 ⊆ (𝑇𝑈))
7271ad2antrr 722 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝐵 ⊆ (𝑇𝑈))
73 stoweidlem39.12 . . . . . . . 8 (𝜑𝑊 ∈ V)
7473ad2antrr 722 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝑊 ∈ V)
75 stoweidlem39.13 . . . . . . . 8 (𝜑𝐴 ∈ V)
7675ad2antrr 722 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝐴 ∈ V)
7713ad2antrr 722 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝑟 ∈ Fin)
78 mptfi 8811 . . . . . . . 8 (𝑟 ∈ Fin → (𝑤𝑟 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑚)) < (𝑡))}) ∈ Fin)
79 rnfi 8795 . . . . . . . 8 ((𝑤𝑟 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑚)) < (𝑡))}) ∈ Fin → ran (𝑤𝑟 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑚)) < (𝑡))}) ∈ Fin)
8077, 78, 793syl 18 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → ran (𝑤𝑟 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑚)) < (𝑡))}) ∈ Fin)
8140, 45, 50, 51, 52, 53, 28, 54, 55, 57, 72, 74, 76, 80stoweidlem31 42193 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → ∃𝑥(𝑥:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣𝑖)((𝑥𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑚)) < ((𝑥𝑖)‘𝑡))))
8229, 35, 813jca 1120 . . . . 5 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → (𝑣:(1...𝑚)⟶𝑊𝐷 ran 𝑣 ∧ ∃𝑥(𝑥:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣𝑖)((𝑥𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑚)) < ((𝑥𝑖)‘𝑡)))))
8382ex 413 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑣:(1...𝑚)–1-1-onto𝑟 → (𝑣:(1...𝑚)⟶𝑊𝐷 ran 𝑣 ∧ ∃𝑥(𝑥:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣𝑖)((𝑥𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑚)) < ((𝑥𝑖)‘𝑡))))))
8483eximdv 1909 . . 3 ((𝜑𝑚 ∈ ℕ) → (∃𝑣 𝑣:(1...𝑚)–1-1-onto𝑟 → ∃𝑣(𝑣:(1...𝑚)⟶𝑊𝐷 ran 𝑣 ∧ ∃𝑥(𝑥:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣𝑖)((𝑥𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑚)) < ((𝑥𝑖)‘𝑡))))))
8584reximdva 3271 . 2 (𝜑 → (∃𝑚 ∈ ℕ ∃𝑣 𝑣:(1...𝑚)–1-1-onto𝑟 → ∃𝑚 ∈ ℕ ∃𝑣(𝑣:(1...𝑚)⟶𝑊𝐷 ran 𝑣 ∧ ∃𝑥(𝑥:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣𝑖)((𝑥𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑚)) < ((𝑥𝑖)‘𝑡))))))
8622, 85mpd 15 1 (𝜑 → ∃𝑚 ∈ ℕ ∃𝑣(𝑣:(1...𝑚)⟶𝑊𝐷 ran 𝑣 ∧ ∃𝑥(𝑥:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣𝑖)((𝑥𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑚)) < ((𝑥𝑖)‘𝑡)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 841  w3a 1079   = wceq 1528  wex 1771  wnf 1775  wcel 2105  wne 3013  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  cdif 3930  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535   cuni 4830   class class class wbr 5057  cmpt 5137  ccnv 5547  ran crn 5549  Fun wfun 6342   Fn wfn 6343  wf 6344  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  Fincfn 8497  0cc0 10525  1c1 10526   < clt 10663  cle 10664  cmin 10858   / cdiv 11285  cn 11626  +crp 12377  ...cfz 12880  chash 13678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-hash 13679
This theorem is referenced by:  stoweidlem57  42219
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