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Theorem stoweidlem39 44840
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 4400 . . . . . 6 ((𝐷 𝑟𝐷 ≠ ∅) → 𝑟 ≠ ∅)
5 unieq 4919 . . . . . . . 8 (𝑟 = ∅ → 𝑟 = ∅)
6 uni0 4939 . . . . . . . 8 ∅ = ∅
75, 6eqtrdi 2788 . . . . . . 7 (𝑟 = ∅ → 𝑟 = ∅)
87necon3i 2973 . . . . . 6 ( 𝑟 ≠ ∅ → 𝑟 ≠ ∅)
93, 4, 83syl 18 . . . . 5 (𝜑𝑟 ≠ ∅)
109neneqd 2945 . . . 4 (𝜑 → ¬ 𝑟 = ∅)
11 stoweidlem39.7 . . . . . 6 (𝜑𝑟 ∈ (𝒫 𝑊 ∩ Fin))
12 elinel2 4196 . . . . . 6 (𝑟 ∈ (𝒫 𝑊 ∩ Fin) → 𝑟 ∈ Fin)
1311, 12syl 17 . . . . 5 (𝜑𝑟 ∈ Fin)
14 fz1f1o 15658 . . . . 5 (𝑟 ∈ Fin → (𝑟 = ∅ ∨ ((♯‘𝑟) ∈ ℕ ∧ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto𝑟)))
15 pm2.53 849 . . . . 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 7419 . . . . . 6 (𝑚 = (♯‘𝑟) → (1...𝑚) = (1...(♯‘𝑟)))
1918f1oeq2d 6829 . . . . 5 (𝑚 = (♯‘𝑟) → (𝑣:(1...𝑚)–1-1-onto𝑟𝑣:(1...(♯‘𝑟))–1-1-onto𝑟))
2019exbidv 1924 . . . 4 (𝑚 = (♯‘𝑟) → (∃𝑣 𝑣:(1...𝑚)–1-1-onto𝑟 ↔ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto𝑟))
2120rspcev 3612 . . 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 6833 . . . . . . . 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 765 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝜑)
26 elinel1 4195 . . . . . . . . 9 (𝑟 ∈ (𝒫 𝑊 ∩ Fin) → 𝑟 ∈ 𝒫 𝑊)
2726elpwid 4611 . . . . . . . 8 (𝑟 ∈ (𝒫 𝑊 ∩ Fin) → 𝑟𝑊)
2825, 11, 273syl 18 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝑟𝑊)
2924, 28fssd 6735 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝑣:(1...𝑚)⟶𝑊)
301ad2antrr 724 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝐷 𝑟)
31 dff1o2 6838 . . . . . . . . . 10 (𝑣:(1...𝑚)–1-1-onto𝑟 ↔ (𝑣 Fn (1...𝑚) ∧ Fun 𝑣 ∧ ran 𝑣 = 𝑟))
3231simp3bi 1147 . . . . . . . . 9 (𝑣:(1...𝑚)–1-1-onto𝑟 → ran 𝑣 = 𝑟)
3332unieqd 4922 . . . . . . . 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 4023 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝐷 ran 𝑣)
36 stoweidlem39.1 . . . . . . . . 9 𝜑
37 nfv 1917 . . . . . . . . 9 𝑚 ∈ ℕ
3836, 37nfan 1902 . . . . . . . 8 (𝜑𝑚 ∈ ℕ)
39 nfv 1917 . . . . . . . 8 𝑣:(1...𝑚)–1-1-onto𝑟
4038, 39nfan 1902 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟)
41 stoweidlem39.2 . . . . . . . . 9 𝑡𝜑
42 nfv 1917 . . . . . . . . 9 𝑡 𝑚 ∈ ℕ
4341, 42nfan 1902 . . . . . . . 8 𝑡(𝜑𝑚 ∈ ℕ)
44 nfv 1917 . . . . . . . 8 𝑡 𝑣:(1...𝑚)–1-1-onto𝑟
4543, 44nfan 1902 . . . . . . 7 𝑡((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟)
46 stoweidlem39.3 . . . . . . . . 9 𝑤𝜑
47 nfv 1917 . . . . . . . . 9 𝑤 𝑚 ∈ ℕ
4846, 47nfan 1902 . . . . . . . 8 𝑤(𝜑𝑚 ∈ ℕ)
49 nfv 1917 . . . . . . . 8 𝑤 𝑣:(1...𝑚)–1-1-onto𝑟
5048, 49nfan 1902 . . . . . . 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 2732 . . . . . . 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 767 . . . . . . 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 724 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝐸 ∈ ℝ+)
58 stoweidlem39.11 . . . . . . . . . . . 12 (𝜑𝐵𝑇)
5958sselda 3982 . . . . . . . . . . 11 ((𝜑𝑏𝐵) → 𝑏𝑇)
60 notnot 142 . . . . . . . . . . . . . . 15 (𝑏𝐵 → ¬ ¬ 𝑏𝐵)
6160intnand 489 . . . . . . . . . . . . . 14 (𝑏𝐵 → ¬ (𝑏𝑇 ∧ ¬ 𝑏𝐵))
6261adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑏𝐵) → ¬ (𝑏𝑇 ∧ ¬ 𝑏𝐵))
63 eldif 3958 . . . . . . . . . . . . 13 (𝑏 ∈ (𝑇𝐵) ↔ (𝑏𝑇 ∧ ¬ 𝑏𝐵))
6462, 63sylnibr 328 . . . . . . . . . . . 12 ((𝜑𝑏𝐵) → ¬ 𝑏 ∈ (𝑇𝐵))
65 stoweidlem39.4 . . . . . . . . . . . . 13 𝑈 = (𝑇𝐵)
6665eleq2i 2825 . . . . . . . . . . . 12 (𝑏𝑈𝑏 ∈ (𝑇𝐵))
6764, 66sylnibr 328 . . . . . . . . . . 11 ((𝜑𝑏𝐵) → ¬ 𝑏𝑈)
6859, 67eldifd 3959 . . . . . . . . . 10 ((𝜑𝑏𝐵) → 𝑏 ∈ (𝑇𝑈))
6968ralrimiva 3146 . . . . . . . . 9 (𝜑 → ∀𝑏𝐵 𝑏 ∈ (𝑇𝑈))
70 dfss3 3970 . . . . . . . . 9 (𝐵 ⊆ (𝑇𝑈) ↔ ∀𝑏𝐵 𝑏 ∈ (𝑇𝑈))
7169, 70sylibr 233 . . . . . . . 8 (𝜑𝐵 ⊆ (𝑇𝑈))
7271ad2antrr 724 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝐵 ⊆ (𝑇𝑈))
73 stoweidlem39.12 . . . . . . . 8 (𝜑𝑊 ∈ V)
7473ad2antrr 724 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝑊 ∈ V)
75 stoweidlem39.13 . . . . . . . 8 (𝜑𝐴 ∈ V)
7675ad2antrr 724 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝐴 ∈ V)
7713ad2antrr 724 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto𝑟) → 𝑟 ∈ Fin)
78 mptfi 9353 . . . . . . . 8 (𝑟 ∈ Fin → (𝑤𝑟 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑚)) < (𝑡))}) ∈ Fin)
79 rnfi 9337 . . . . . . . 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 44832 . . . . . 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 1128 . . . . 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 1920 . . 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 3168 . 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 845  w3a 1087   = wceq 1541  wex 1781  wnf 1785  wcel 2106  wne 2940  wral 3061  wrex 3070  {crab 3432  Vcvv 3474  cdif 3945  cin 3947  wss 3948  c0 4322  𝒫 cpw 4602   cuni 4908   class class class wbr 5148  cmpt 5231  ccnv 5675  ran crn 5677  Fun wfun 6537   Fn wfn 6538  wf 6539  1-1-ontowf1o 6542  cfv 6543  (class class class)co 7411  Fincfn 8941  0cc0 11112  1c1 11113   < clt 11250  cle 11251  cmin 11446   / cdiv 11873  cn 12214  +crp 12976  ...cfz 13486  chash 14292
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-n0 12475  df-z 12561  df-uz 12825  df-rp 12977  df-fz 13487  df-hash 14293
This theorem is referenced by:  stoweidlem57  44858
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