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Theorem pfx2 40741
 Description: A prefix of length 2. (Contributed by AV, 15-May-2020.)
Assertion
Ref Expression
pfx2 ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → (𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩)

Proof of Theorem pfx2
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 2nn0 11269 . . . 4 2 ∈ ℕ0
21a1i 11 . . 3 ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → 2 ∈ ℕ0)
3 lencl 13279 . . . 4 (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0)
43adantr 481 . . 3 ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → (#‘𝑊) ∈ ℕ0)
5 simpr 477 . . 3 ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → 2 ≤ (#‘𝑊))
6 elfz2nn0 12388 . . 3 (2 ∈ (0...(#‘𝑊)) ↔ (2 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0 ∧ 2 ≤ (#‘𝑊)))
72, 4, 5, 6syl3anbrc 1244 . 2 ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → 2 ∈ (0...(#‘𝑊)))
8 pfxlen 40720 . . . 4 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → (#‘(𝑊 prefix 2)) = 2)
9 s2len 13586 . . . . . . 7 (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) = 2
109eqcomi 2630 . . . . . 6 2 = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩)
1110a1i 11 . . . . 5 (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → 2 = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩))
12 simpl 473 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → 𝑊 ∈ Word 𝑉)
13 simpr 477 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → 2 ∈ (0...(#‘𝑊)))
14 2nn 11145 . . . . . . . . . . . 12 2 ∈ ℕ
15 lbfzo0 12464 . . . . . . . . . . . 12 (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
1614, 15mpbir 221 . . . . . . . . . . 11 0 ∈ (0..^2)
1716a1i 11 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → 0 ∈ (0..^2))
1812, 13, 173jca 1240 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → (𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊)) ∧ 0 ∈ (0..^2)))
1918adantr 481 . . . . . . . 8 (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → (𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊)) ∧ 0 ∈ (0..^2)))
20 pfxfv 40728 . . . . . . . 8 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊)) ∧ 0 ∈ (0..^2)) → ((𝑊 prefix 2)‘0) = (𝑊‘0))
2119, 20syl 17 . . . . . . 7 (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → ((𝑊 prefix 2)‘0) = (𝑊‘0))
22 fvex 6168 . . . . . . . 8 (𝑊‘0) ∈ V
23 s2fv0 13584 . . . . . . . 8 ((𝑊‘0) ∈ V → (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0) = (𝑊‘0))
2422, 23ax-mp 5 . . . . . . 7 (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0) = (𝑊‘0)
2521, 24syl6eqr 2673 . . . . . 6 (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → ((𝑊 prefix 2)‘0) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0))
26 1nn0 11268 . . . . . . . . . 10 1 ∈ ℕ0
27 1lt2 11154 . . . . . . . . . 10 1 < 2
28 elfzo0 12465 . . . . . . . . . 10 (1 ∈ (0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1 < 2))
2926, 14, 27, 28mpbir3an 1242 . . . . . . . . 9 1 ∈ (0..^2)
30 pfxfv 40728 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊)) ∧ 1 ∈ (0..^2)) → ((𝑊 prefix 2)‘1) = (𝑊‘1))
3129, 30mp3an3 1410 . . . . . . . 8 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → ((𝑊 prefix 2)‘1) = (𝑊‘1))
32 fvex 6168 . . . . . . . . 9 (𝑊‘1) ∈ V
33 s2fv1 13585 . . . . . . . . 9 ((𝑊‘1) ∈ V → (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1) = (𝑊‘1))
3432, 33ax-mp 5 . . . . . . . 8 (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1) = (𝑊‘1)
3531, 34syl6eqr 2673 . . . . . . 7 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → ((𝑊 prefix 2)‘1) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1))
3635adantr 481 . . . . . 6 (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → ((𝑊 prefix 2)‘1) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1))
37 0nn0 11267 . . . . . . . . 9 0 ∈ ℕ0
3837, 26pm3.2i 471 . . . . . . . 8 (0 ∈ ℕ0 ∧ 1 ∈ ℕ0)
3938a1i 11 . . . . . . 7 (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → (0 ∈ ℕ0 ∧ 1 ∈ ℕ0))
40 fveq2 6158 . . . . . . . . 9 (𝑖 = 0 → ((𝑊 prefix 2)‘𝑖) = ((𝑊 prefix 2)‘0))
41 fveq2 6158 . . . . . . . . 9 (𝑖 = 0 → (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0))
4240, 41eqeq12d 2636 . . . . . . . 8 (𝑖 = 0 → (((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) ↔ ((𝑊 prefix 2)‘0) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0)))
43 fveq2 6158 . . . . . . . . 9 (𝑖 = 1 → ((𝑊 prefix 2)‘𝑖) = ((𝑊 prefix 2)‘1))
44 fveq2 6158 . . . . . . . . 9 (𝑖 = 1 → (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1))
4543, 44eqeq12d 2636 . . . . . . . 8 (𝑖 = 1 → (((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) ↔ ((𝑊 prefix 2)‘1) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1)))
4642, 45ralprg 4212 . . . . . . 7 ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → (∀𝑖 ∈ {0, 1} ((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) ↔ (((𝑊 prefix 2)‘0) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0) ∧ ((𝑊 prefix 2)‘1) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1))))
4739, 46syl 17 . . . . . 6 (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → (∀𝑖 ∈ {0, 1} ((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) ↔ (((𝑊 prefix 2)‘0) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0) ∧ ((𝑊 prefix 2)‘1) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1))))
4825, 36, 47mpbir2and 956 . . . . 5 (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → ∀𝑖 ∈ {0, 1} ((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖))
49 eqeq1 2625 . . . . . . 7 ((#‘(𝑊 prefix 2)) = 2 → ((#‘(𝑊 prefix 2)) = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ↔ 2 = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩)))
50 oveq2 6623 . . . . . . . . 9 ((#‘(𝑊 prefix 2)) = 2 → (0..^(#‘(𝑊 prefix 2))) = (0..^2))
51 fzo0to2pr 12510 . . . . . . . . 9 (0..^2) = {0, 1}
5250, 51syl6eq 2671 . . . . . . . 8 ((#‘(𝑊 prefix 2)) = 2 → (0..^(#‘(𝑊 prefix 2))) = {0, 1})
5352raleqdv 3137 . . . . . . 7 ((#‘(𝑊 prefix 2)) = 2 → (∀𝑖 ∈ (0..^(#‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) ↔ ∀𝑖 ∈ {0, 1} ((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖)))
5449, 53anbi12d 746 . . . . . 6 ((#‘(𝑊 prefix 2)) = 2 → (((#‘(𝑊 prefix 2)) = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ (0..^(#‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖)) ↔ (2 = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ {0, 1} ((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖))))
5554adantl 482 . . . . 5 (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → (((#‘(𝑊 prefix 2)) = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ (0..^(#‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖)) ↔ (2 = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ {0, 1} ((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖))))
5611, 48, 55mpbir2and 956 . . . 4 (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → ((#‘(𝑊 prefix 2)) = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ (0..^(#‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖)))
578, 56mpdan 701 . . 3 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → ((#‘(𝑊 prefix 2)) = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ (0..^(#‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖)))
58 pfxcl 40715 . . . . 5 (𝑊 ∈ Word 𝑉 → (𝑊 prefix 2) ∈ Word 𝑉)
5958adantr 481 . . . 4 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → (𝑊 prefix 2) ∈ Word 𝑉)
60 simp2 1060 . . . . . . . . . 10 ((2 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0 ∧ 2 ≤ (#‘𝑊)) → (#‘𝑊) ∈ ℕ0)
61 1red 10015 . . . . . . . . . . . . . . 15 ((#‘𝑊) ∈ ℕ0 → 1 ∈ ℝ)
62 2re 11050 . . . . . . . . . . . . . . . 16 2 ∈ ℝ
6362a1i 11 . . . . . . . . . . . . . . 15 ((#‘𝑊) ∈ ℕ0 → 2 ∈ ℝ)
64 nn0re 11261 . . . . . . . . . . . . . . 15 ((#‘𝑊) ∈ ℕ0 → (#‘𝑊) ∈ ℝ)
6561, 63, 643jca 1240 . . . . . . . . . . . . . 14 ((#‘𝑊) ∈ ℕ0 → (1 ∈ ℝ ∧ 2 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ))
66 ltleletr 10090 . . . . . . . . . . . . . 14 ((1 ∈ ℝ ∧ 2 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ) → ((1 < 2 ∧ 2 ≤ (#‘𝑊)) → 1 ≤ (#‘𝑊)))
6765, 66syl 17 . . . . . . . . . . . . 13 ((#‘𝑊) ∈ ℕ0 → ((1 < 2 ∧ 2 ≤ (#‘𝑊)) → 1 ≤ (#‘𝑊)))
6827, 67mpani 711 . . . . . . . . . . . 12 ((#‘𝑊) ∈ ℕ0 → (2 ≤ (#‘𝑊) → 1 ≤ (#‘𝑊)))
6968imp 445 . . . . . . . . . . 11 (((#‘𝑊) ∈ ℕ0 ∧ 2 ≤ (#‘𝑊)) → 1 ≤ (#‘𝑊))
70693adant1 1077 . . . . . . . . . 10 ((2 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0 ∧ 2 ≤ (#‘𝑊)) → 1 ≤ (#‘𝑊))
7160, 70jca 554 . . . . . . . . 9 ((2 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0 ∧ 2 ≤ (#‘𝑊)) → ((#‘𝑊) ∈ ℕ0 ∧ 1 ≤ (#‘𝑊)))
72 elnnnn0c 11298 . . . . . . . . 9 ((#‘𝑊) ∈ ℕ ↔ ((#‘𝑊) ∈ ℕ0 ∧ 1 ≤ (#‘𝑊)))
7371, 72sylibr 224 . . . . . . . 8 ((2 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0 ∧ 2 ≤ (#‘𝑊)) → (#‘𝑊) ∈ ℕ)
746, 73sylbi 207 . . . . . . 7 (2 ∈ (0...(#‘𝑊)) → (#‘𝑊) ∈ ℕ)
75 lbfzo0 12464 . . . . . . 7 (0 ∈ (0..^(#‘𝑊)) ↔ (#‘𝑊) ∈ ℕ)
7674, 75sylibr 224 . . . . . 6 (2 ∈ (0...(#‘𝑊)) → 0 ∈ (0..^(#‘𝑊)))
77 wrdsymbcl 13273 . . . . . 6 ((𝑊 ∈ Word 𝑉 ∧ 0 ∈ (0..^(#‘𝑊))) → (𝑊‘0) ∈ 𝑉)
7876, 77sylan2 491 . . . . 5 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → (𝑊‘0) ∈ 𝑉)
7926a1i 11 . . . . . . 7 (2 ∈ (0...(#‘𝑊)) → 1 ∈ ℕ0)
8065adantl 482 . . . . . . . . . . 11 ((2 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0) → (1 ∈ ℝ ∧ 2 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ))
81 ltletr 10089 . . . . . . . . . . 11 ((1 ∈ ℝ ∧ 2 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ) → ((1 < 2 ∧ 2 ≤ (#‘𝑊)) → 1 < (#‘𝑊)))
8280, 81syl 17 . . . . . . . . . 10 ((2 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0) → ((1 < 2 ∧ 2 ≤ (#‘𝑊)) → 1 < (#‘𝑊)))
8327, 82mpani 711 . . . . . . . . 9 ((2 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0) → (2 ≤ (#‘𝑊) → 1 < (#‘𝑊)))
84833impia 1258 . . . . . . . 8 ((2 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0 ∧ 2 ≤ (#‘𝑊)) → 1 < (#‘𝑊))
856, 84sylbi 207 . . . . . . 7 (2 ∈ (0...(#‘𝑊)) → 1 < (#‘𝑊))
86 elfzo0 12465 . . . . . . 7 (1 ∈ (0..^(#‘𝑊)) ↔ (1 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ ∧ 1 < (#‘𝑊)))
8779, 74, 85, 86syl3anbrc 1244 . . . . . 6 (2 ∈ (0...(#‘𝑊)) → 1 ∈ (0..^(#‘𝑊)))
88 wrdsymbcl 13273 . . . . . 6 ((𝑊 ∈ Word 𝑉 ∧ 1 ∈ (0..^(#‘𝑊))) → (𝑊‘1) ∈ 𝑉)
8987, 88sylan2 491 . . . . 5 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → (𝑊‘1) ∈ 𝑉)
9078, 89s2cld 13568 . . . 4 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → ⟨“(𝑊‘0)(𝑊‘1)”⟩ ∈ Word 𝑉)
91 eqwrd 13301 . . . 4 (((𝑊 prefix 2) ∈ Word 𝑉 ∧ ⟨“(𝑊‘0)(𝑊‘1)”⟩ ∈ Word 𝑉) → ((𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩ ↔ ((#‘(𝑊 prefix 2)) = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ (0..^(#‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖))))
9259, 90, 91syl2anc 692 . . 3 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → ((𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩ ↔ ((#‘(𝑊 prefix 2)) = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ (0..^(#‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖))))
9357, 92mpbird 247 . 2 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → (𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩)
947, 93syldan 487 1 ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → (𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2908  Vcvv 3190  {cpr 4157   class class class wbr 4623  ‘cfv 5857  (class class class)co 6615  ℝcr 9895  0cc0 9896  1c1 9897   < clt 10034   ≤ cle 10035  ℕcn 10980  2c2 11030  ℕ0cn0 11252  ...cfz 12284  ..^cfzo 12422  #chash 13073  Word cword 13246  ⟨“cs2 13539   prefix cpfx 40710 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-fzo 12423  df-hash 13074  df-word 13254  df-concat 13256  df-s1 13257  df-substr 13258  df-s2 13546  df-pfx 40711 This theorem is referenced by: (None)
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