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Theorem pfx2 40076
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 11152 . . . 4 2 ∈ ℕ0
21a1i 11 . . 3 ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → 2 ∈ ℕ0)
3 lencl 13121 . . . 4 (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0)
43adantr 479 . . 3 ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → (#‘𝑊) ∈ ℕ0)
5 simpr 475 . . 3 ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → 2 ≤ (#‘𝑊))
6 elfz2nn0 12251 . . 3 (2 ∈ (0...(#‘𝑊)) ↔ (2 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0 ∧ 2 ≤ (#‘𝑊)))
72, 4, 5, 6syl3anbrc 1238 . 2 ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → 2 ∈ (0...(#‘𝑊)))
8 pfxlen 40055 . . . 4 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → (#‘(𝑊 prefix 2)) = 2)
9 s2len 13426 . . . . . . 7 (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) = 2
109eqcomi 2614 . . . . . 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 471 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → 𝑊 ∈ Word 𝑉)
13 simpr 475 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → 2 ∈ (0...(#‘𝑊)))
14 2nn 11028 . . . . . . . . . . . 12 2 ∈ ℕ
15 lbfzo0 12326 . . . . . . . . . . . 12 (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
1614, 15mpbir 219 . . . . . . . . . . 11 0 ∈ (0..^2)
1716a1i 11 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → 0 ∈ (0..^2))
1812, 13, 173jca 1234 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → (𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊)) ∧ 0 ∈ (0..^2)))
1918adantr 479 . . . . . . . 8 (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → (𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊)) ∧ 0 ∈ (0..^2)))
20 pfxfv 40063 . . . . . . . 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 6094 . . . . . . . 8 (𝑊‘0) ∈ V
23 s2fv0 13424 . . . . . . . 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 2657 . . . . . 6 (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → ((𝑊 prefix 2)‘0) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0))
26 1nn0 11151 . . . . . . . . . 10 1 ∈ ℕ0
27 1lt2 11037 . . . . . . . . . 10 1 < 2
28 elfzo0 12327 . . . . . . . . . 10 (1 ∈ (0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1 < 2))
2926, 14, 27, 28mpbir3an 1236 . . . . . . . . 9 1 ∈ (0..^2)
30 pfxfv 40063 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊)) ∧ 1 ∈ (0..^2)) → ((𝑊 prefix 2)‘1) = (𝑊‘1))
3129, 30mp3an3 1404 . . . . . . . 8 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → ((𝑊 prefix 2)‘1) = (𝑊‘1))
32 fvex 6094 . . . . . . . . 9 (𝑊‘1) ∈ V
33 s2fv1 13425 . . . . . . . . 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 2657 . . . . . . 7 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → ((𝑊 prefix 2)‘1) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1))
3635adantr 479 . . . . . 6 (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → ((𝑊 prefix 2)‘1) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1))
37 0nn0 11150 . . . . . . . . 9 0 ∈ ℕ0
3837, 26pm3.2i 469 . . . . . . . 8 (0 ∈ ℕ0 ∧ 1 ∈ ℕ0)
3938a1i 11 . . . . . . 7 (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → (0 ∈ ℕ0 ∧ 1 ∈ ℕ0))
40 fveq2 6084 . . . . . . . . 9 (𝑖 = 0 → ((𝑊 prefix 2)‘𝑖) = ((𝑊 prefix 2)‘0))
41 fveq2 6084 . . . . . . . . 9 (𝑖 = 0 → (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0))
4240, 41eqeq12d 2620 . . . . . . . 8 (𝑖 = 0 → (((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) ↔ ((𝑊 prefix 2)‘0) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0)))
43 fveq2 6084 . . . . . . . . 9 (𝑖 = 1 → ((𝑊 prefix 2)‘𝑖) = ((𝑊 prefix 2)‘1))
44 fveq2 6084 . . . . . . . . 9 (𝑖 = 1 → (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1))
4543, 44eqeq12d 2620 . . . . . . . 8 (𝑖 = 1 → (((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) ↔ ((𝑊 prefix 2)‘1) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1)))
4642, 45ralprg 4176 . . . . . . 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 958 . . . . 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 2609 . . . . . . 7 ((#‘(𝑊 prefix 2)) = 2 → ((#‘(𝑊 prefix 2)) = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ↔ 2 = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩)))
50 oveq2 6531 . . . . . . . . 9 ((#‘(𝑊 prefix 2)) = 2 → (0..^(#‘(𝑊 prefix 2))) = (0..^2))
51 fzo0to2pr 12371 . . . . . . . . 9 (0..^2) = {0, 1}
5250, 51syl6eq 2655 . . . . . . . 8 ((#‘(𝑊 prefix 2)) = 2 → (0..^(#‘(𝑊 prefix 2))) = {0, 1})
5352raleqdv 3116 . . . . . . 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 742 . . . . . 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 480 . . . . 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 958 . . . 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 698 . . 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 40050 . . . . 5 (𝑊 ∈ Word 𝑉 → (𝑊 prefix 2) ∈ Word 𝑉)
5958adantr 479 . . . 4 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → (𝑊 prefix 2) ∈ Word 𝑉)
60 simp2 1054 . . . . . . . . . 10 ((2 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0 ∧ 2 ≤ (#‘𝑊)) → (#‘𝑊) ∈ ℕ0)
61 1red 9907 . . . . . . . . . . . . . . 15 ((#‘𝑊) ∈ ℕ0 → 1 ∈ ℝ)
62 2re 10933 . . . . . . . . . . . . . . . 16 2 ∈ ℝ
6362a1i 11 . . . . . . . . . . . . . . 15 ((#‘𝑊) ∈ ℕ0 → 2 ∈ ℝ)
64 nn0re 11144 . . . . . . . . . . . . . . 15 ((#‘𝑊) ∈ ℕ0 → (#‘𝑊) ∈ ℝ)
6561, 63, 643jca 1234 . . . . . . . . . . . . . 14 ((#‘𝑊) ∈ ℕ0 → (1 ∈ ℝ ∧ 2 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ))
66 ltleletr 9977 . . . . . . . . . . . . . 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 707 . . . . . . . . . . . 12 ((#‘𝑊) ∈ ℕ0 → (2 ≤ (#‘𝑊) → 1 ≤ (#‘𝑊)))
6968imp 443 . . . . . . . . . . 11 (((#‘𝑊) ∈ ℕ0 ∧ 2 ≤ (#‘𝑊)) → 1 ≤ (#‘𝑊))
70693adant1 1071 . . . . . . . . . 10 ((2 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0 ∧ 2 ≤ (#‘𝑊)) → 1 ≤ (#‘𝑊))
7160, 70jca 552 . . . . . . . . 9 ((2 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0 ∧ 2 ≤ (#‘𝑊)) → ((#‘𝑊) ∈ ℕ0 ∧ 1 ≤ (#‘𝑊)))
72 elnnnn0c 11181 . . . . . . . . 9 ((#‘𝑊) ∈ ℕ ↔ ((#‘𝑊) ∈ ℕ0 ∧ 1 ≤ (#‘𝑊)))
7371, 72sylibr 222 . . . . . . . 8 ((2 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0 ∧ 2 ≤ (#‘𝑊)) → (#‘𝑊) ∈ ℕ)
746, 73sylbi 205 . . . . . . 7 (2 ∈ (0...(#‘𝑊)) → (#‘𝑊) ∈ ℕ)
75 lbfzo0 12326 . . . . . . 7 (0 ∈ (0..^(#‘𝑊)) ↔ (#‘𝑊) ∈ ℕ)
7674, 75sylibr 222 . . . . . 6 (2 ∈ (0...(#‘𝑊)) → 0 ∈ (0..^(#‘𝑊)))
77 wrdsymbcl 13115 . . . . . 6 ((𝑊 ∈ Word 𝑉 ∧ 0 ∈ (0..^(#‘𝑊))) → (𝑊‘0) ∈ 𝑉)
7876, 77sylan2 489 . . . . 5 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → (𝑊‘0) ∈ 𝑉)
7926a1i 11 . . . . . . 7 (2 ∈ (0...(#‘𝑊)) → 1 ∈ ℕ0)
8065adantl 480 . . . . . . . . . . 11 ((2 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0) → (1 ∈ ℝ ∧ 2 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ))
81 ltletr 9976 . . . . . . . . . . 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 707 . . . . . . . . 9 ((2 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0) → (2 ≤ (#‘𝑊) → 1 < (#‘𝑊)))
84833impia 1252 . . . . . . . 8 ((2 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0 ∧ 2 ≤ (#‘𝑊)) → 1 < (#‘𝑊))
856, 84sylbi 205 . . . . . . 7 (2 ∈ (0...(#‘𝑊)) → 1 < (#‘𝑊))
86 elfzo0 12327 . . . . . . 7 (1 ∈ (0..^(#‘𝑊)) ↔ (1 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ ∧ 1 < (#‘𝑊)))
8779, 74, 85, 86syl3anbrc 1238 . . . . . 6 (2 ∈ (0...(#‘𝑊)) → 1 ∈ (0..^(#‘𝑊)))
88 wrdsymbcl 13115 . . . . . 6 ((𝑊 ∈ Word 𝑉 ∧ 1 ∈ (0..^(#‘𝑊))) → (𝑊‘1) ∈ 𝑉)
8987, 88sylan2 489 . . . . 5 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → (𝑊‘1) ∈ 𝑉)
9078, 89s2cld 13408 . . . 4 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → ⟨“(𝑊‘0)(𝑊‘1)”⟩ ∈ Word 𝑉)
91 eqwrd 13143 . . . 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 690 . . 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 245 . 2 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → (𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩)
947, 93syldan 485 1 ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → (𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wral 2891  Vcvv 3168  {cpr 4122   class class class wbr 4573  cfv 5786  (class class class)co 6523  cr 9787  0cc0 9788  1c1 9789   < clt 9926  cle 9927  cn 10863  2c2 10913  0cn0 11135  ...cfz 12148  ..^cfzo 12285  #chash 12930  Word cword 13088  ⟨“cs2 13379   prefix cpfx 40045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-er 7602  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-card 8621  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-2 10922  df-n0 11136  df-z 11207  df-uz 11516  df-fz 12149  df-fzo 12286  df-hash 12931  df-word 13096  df-concat 13098  df-s1 13099  df-substr 13100  df-s2 13386  df-pfx 40046
This theorem is referenced by: (None)
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