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Theorem frec2uzrdg 8876
 Description: A helper lemma for the value of a recursive definition generator on upper integers (typically either ℕ or ℕ0) with characteristic function 𝐹(x, y) and initial value A. This lemma shows that evaluating 𝑅 at an element of 𝜔 gives an ordered pair whose first element is the index (translated from 𝜔 to (ℤ≥‘𝐶)). See comment in frec2uz0d 8866 which describes 𝐺 and the index translation. (Contributed by Jim Kingdon, 24-May-2020.)
Hypotheses
Ref Expression
frec2uz.1 (φ𝐶 ℤ)
frec2uz.2 𝐺 = frec((x ℤ ↦ (x + 1)), 𝐶)
uzrdg.s (φ𝑆 𝑉)
uzrdg.a (φA 𝑆)
uzrdg.f ((φ (x (ℤ𝐶) y 𝑆)) → (x𝐹y) 𝑆)
uzrdg.2 𝑅 = frec((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)
uzrdg.b (φB 𝜔)
Assertion
Ref Expression
frec2uzrdg (φ → (𝑅B) = ⟨(𝐺B), (2nd ‘(𝑅B))⟩)
Distinct variable groups:   y,A   x,𝐶,y   y,𝐺   x,𝐹,y   x,𝑆,y   φ,x,y
Allowed substitution hints:   A(x)   B(x,y)   𝑅(x,y)   𝐺(x)   𝑉(x,y)

Proof of Theorem frec2uzrdg
Dummy variables w z v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uzrdg.b . 2 (φB 𝜔)
2 fveq2 5121 . . . . 5 (z = B → (𝑅z) = (𝑅B))
3 fveq2 5121 . . . . . 6 (z = B → (𝐺z) = (𝐺B))
42fveq2d 5125 . . . . . 6 (z = B → (2nd ‘(𝑅z)) = (2nd ‘(𝑅B)))
53, 4opeq12d 3548 . . . . 5 (z = B → ⟨(𝐺z), (2nd ‘(𝑅z))⟩ = ⟨(𝐺B), (2nd ‘(𝑅B))⟩)
62, 5eqeq12d 2051 . . . 4 (z = B → ((𝑅z) = ⟨(𝐺z), (2nd ‘(𝑅z))⟩ ↔ (𝑅B) = ⟨(𝐺B), (2nd ‘(𝑅B))⟩))
76imbi2d 219 . . 3 (z = B → ((φ → (𝑅z) = ⟨(𝐺z), (2nd ‘(𝑅z))⟩) ↔ (φ → (𝑅B) = ⟨(𝐺B), (2nd ‘(𝑅B))⟩)))
8 fveq2 5121 . . . . 5 (z = ∅ → (𝑅z) = (𝑅‘∅))
9 fveq2 5121 . . . . . 6 (z = ∅ → (𝐺z) = (𝐺‘∅))
108fveq2d 5125 . . . . . 6 (z = ∅ → (2nd ‘(𝑅z)) = (2nd ‘(𝑅‘∅)))
119, 10opeq12d 3548 . . . . 5 (z = ∅ → ⟨(𝐺z), (2nd ‘(𝑅z))⟩ = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
128, 11eqeq12d 2051 . . . 4 (z = ∅ → ((𝑅z) = ⟨(𝐺z), (2nd ‘(𝑅z))⟩ ↔ (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩))
13 fveq2 5121 . . . . 5 (z = v → (𝑅z) = (𝑅v))
14 fveq2 5121 . . . . . 6 (z = v → (𝐺z) = (𝐺v))
1513fveq2d 5125 . . . . . 6 (z = v → (2nd ‘(𝑅z)) = (2nd ‘(𝑅v)))
1614, 15opeq12d 3548 . . . . 5 (z = v → ⟨(𝐺z), (2nd ‘(𝑅z))⟩ = ⟨(𝐺v), (2nd ‘(𝑅v))⟩)
1713, 16eqeq12d 2051 . . . 4 (z = v → ((𝑅z) = ⟨(𝐺z), (2nd ‘(𝑅z))⟩ ↔ (𝑅v) = ⟨(𝐺v), (2nd ‘(𝑅v))⟩))
18 fveq2 5121 . . . . 5 (z = suc v → (𝑅z) = (𝑅‘suc v))
19 fveq2 5121 . . . . . 6 (z = suc v → (𝐺z) = (𝐺‘suc v))
2018fveq2d 5125 . . . . . 6 (z = suc v → (2nd ‘(𝑅z)) = (2nd ‘(𝑅‘suc v)))
2119, 20opeq12d 3548 . . . . 5 (z = suc v → ⟨(𝐺z), (2nd ‘(𝑅z))⟩ = ⟨(𝐺‘suc v), (2nd ‘(𝑅‘suc v))⟩)
2218, 21eqeq12d 2051 . . . 4 (z = suc v → ((𝑅z) = ⟨(𝐺z), (2nd ‘(𝑅z))⟩ ↔ (𝑅‘suc v) = ⟨(𝐺‘suc v), (2nd ‘(𝑅‘suc v))⟩))
23 uzrdg.2 . . . . . . 7 𝑅 = frec((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)
2423fveq1i 5122 . . . . . 6 (𝑅‘∅) = (frec((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)‘∅)
25 frec2uz.1 . . . . . . . 8 (φ𝐶 ℤ)
26 uzrdg.a . . . . . . . 8 (φA 𝑆)
27 opexg 3955 . . . . . . . 8 ((𝐶 A 𝑆) → ⟨𝐶, A V)
2825, 26, 27syl2anc 391 . . . . . . 7 (φ → ⟨𝐶, A V)
29 frec0g 5922 . . . . . . 7 (⟨𝐶, A V → (frec((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)‘∅) = ⟨𝐶, A⟩)
3028, 29syl 14 . . . . . 6 (φ → (frec((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)‘∅) = ⟨𝐶, A⟩)
3124, 30syl5eq 2081 . . . . 5 (φ → (𝑅‘∅) = ⟨𝐶, A⟩)
32 frec2uz.2 . . . . . . 7 𝐺 = frec((x ℤ ↦ (x + 1)), 𝐶)
3325, 32frec2uz0d 8866 . . . . . 6 (φ → (𝐺‘∅) = 𝐶)
3431fveq2d 5125 . . . . . . 7 (φ → (2nd ‘(𝑅‘∅)) = (2nd ‘⟨𝐶, A⟩))
35 uzid 8263 . . . . . . . . 9 (𝐶 ℤ → 𝐶 (ℤ𝐶))
3625, 35syl 14 . . . . . . . 8 (φ𝐶 (ℤ𝐶))
37 op2ndg 5720 . . . . . . . 8 ((𝐶 (ℤ𝐶) A 𝑆) → (2nd ‘⟨𝐶, A⟩) = A)
3836, 26, 37syl2anc 391 . . . . . . 7 (φ → (2nd ‘⟨𝐶, A⟩) = A)
3934, 38eqtrd 2069 . . . . . 6 (φ → (2nd ‘(𝑅‘∅)) = A)
4033, 39opeq12d 3548 . . . . 5 (φ → ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩ = ⟨𝐶, A⟩)
4131, 40eqtr4d 2072 . . . 4 (φ → (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
42 zex 8030 . . . . . . . . . . . . . . . 16 V
43 uzssz 8268 . . . . . . . . . . . . . . . 16 (ℤ𝐶) ⊆ ℤ
4442, 43ssexi 3886 . . . . . . . . . . . . . . 15 (ℤ𝐶) V
4544a1i 9 . . . . . . . . . . . . . 14 ((φ v 𝜔) → (ℤ𝐶) V)
46 uzrdg.s . . . . . . . . . . . . . . 15 (φ𝑆 𝑉)
4746adantr 261 . . . . . . . . . . . . . 14 ((φ v 𝜔) → 𝑆 𝑉)
48 mpt2exga 5777 . . . . . . . . . . . . . 14 (((ℤ𝐶) V 𝑆 𝑉) → (x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩) V)
4945, 47, 48syl2anc 391 . . . . . . . . . . . . 13 ((φ v 𝜔) → (x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩) V)
50 vex 2554 . . . . . . . . . . . . . 14 z V
5150a1i 9 . . . . . . . . . . . . 13 ((φ v 𝜔) → z V)
52 fvexg 5137 . . . . . . . . . . . . 13 (((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩) V z V) → ((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘z) V)
5349, 51, 52syl2anc 391 . . . . . . . . . . . 12 ((φ v 𝜔) → ((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘z) V)
5453alrimiv 1751 . . . . . . . . . . 11 ((φ v 𝜔) → z((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘z) V)
5528adantr 261 . . . . . . . . . . 11 ((φ v 𝜔) → ⟨𝐶, A V)
56 simpr 103 . . . . . . . . . . 11 ((φ v 𝜔) → v 𝜔)
57 frecsuc 5930 . . . . . . . . . . 11 ((z((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘z) V 𝐶, A V v 𝜔) → (frec((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)‘suc v) = ((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘(frec((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)‘v)))
5854, 55, 56, 57syl3anc 1134 . . . . . . . . . 10 ((φ v 𝜔) → (frec((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)‘suc v) = ((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘(frec((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)‘v)))
5923fveq1i 5122 . . . . . . . . . 10 (𝑅‘suc v) = (frec((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)‘suc v)
6023fveq1i 5122 . . . . . . . . . . 11 (𝑅v) = (frec((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)‘v)
6160fveq2i 5124 . . . . . . . . . 10 ((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘(𝑅v)) = ((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘(frec((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)‘v))
6258, 59, 613eqtr4g 2094 . . . . . . . . 9 ((φ v 𝜔) → (𝑅‘suc v) = ((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘(𝑅v)))
6362adantr 261 . . . . . . . 8 (((φ v 𝜔) (𝑅v) = ⟨(𝐺v), (2nd ‘(𝑅v))⟩) → (𝑅‘suc v) = ((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘(𝑅v)))
64 fveq2 5121 . . . . . . . . 9 ((𝑅v) = ⟨(𝐺v), (2nd ‘(𝑅v))⟩ → ((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘(𝑅v)) = ((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘⟨(𝐺v), (2nd ‘(𝑅v))⟩))
65 df-ov 5458 . . . . . . . . . 10 ((𝐺v)(x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)(2nd ‘(𝑅v))) = ((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘⟨(𝐺v), (2nd ‘(𝑅v))⟩)
6625adantr 261 . . . . . . . . . . . 12 ((φ v 𝜔) → 𝐶 ℤ)
6766, 32, 56frec2uzuzd 8869 . . . . . . . . . . 11 ((φ v 𝜔) → (𝐺v) (ℤ𝐶))
68 uzrdg.f . . . . . . . . . . . . 13 ((φ (x (ℤ𝐶) y 𝑆)) → (x𝐹y) 𝑆)
6925, 32, 46, 26, 68, 23frecuzrdgrrn 8875 . . . . . . . . . . . 12 ((φ v 𝜔) → (𝑅v) ((ℤ𝐶) × 𝑆))
70 xp2nd 5735 . . . . . . . . . . . 12 ((𝑅v) ((ℤ𝐶) × 𝑆) → (2nd ‘(𝑅v)) 𝑆)
7169, 70syl 14 . . . . . . . . . . 11 ((φ v 𝜔) → (2nd ‘(𝑅v)) 𝑆)
72 peano2uz 8302 . . . . . . . . . . . . 13 ((𝐺v) (ℤ𝐶) → ((𝐺v) + 1) (ℤ𝐶))
7367, 72syl 14 . . . . . . . . . . . 12 ((φ v 𝜔) → ((𝐺v) + 1) (ℤ𝐶))
7468caovclg 5595 . . . . . . . . . . . . . 14 ((φ (z (ℤ𝐶) w 𝑆)) → (z𝐹w) 𝑆)
7574adantlr 446 . . . . . . . . . . . . 13 (((φ v 𝜔) (z (ℤ𝐶) w 𝑆)) → (z𝐹w) 𝑆)
7675, 67, 71caovcld 5596 . . . . . . . . . . . 12 ((φ v 𝜔) → ((𝐺v)𝐹(2nd ‘(𝑅v))) 𝑆)
77 opelxp 4317 . . . . . . . . . . . 12 (⟨((𝐺v) + 1), ((𝐺v)𝐹(2nd ‘(𝑅v)))⟩ ((ℤ𝐶) × 𝑆) ↔ (((𝐺v) + 1) (ℤ𝐶) ((𝐺v)𝐹(2nd ‘(𝑅v))) 𝑆))
7873, 76, 77sylanbrc 394 . . . . . . . . . . 11 ((φ v 𝜔) → ⟨((𝐺v) + 1), ((𝐺v)𝐹(2nd ‘(𝑅v)))⟩ ((ℤ𝐶) × 𝑆))
79 oveq1 5462 . . . . . . . . . . . . 13 (w = (𝐺v) → (w + 1) = ((𝐺v) + 1))
80 oveq1 5462 . . . . . . . . . . . . 13 (w = (𝐺v) → (w𝐹z) = ((𝐺v)𝐹z))
8179, 80opeq12d 3548 . . . . . . . . . . . 12 (w = (𝐺v) → ⟨(w + 1), (w𝐹z)⟩ = ⟨((𝐺v) + 1), ((𝐺v)𝐹z)⟩)
82 oveq2 5463 . . . . . . . . . . . . 13 (z = (2nd ‘(𝑅v)) → ((𝐺v)𝐹z) = ((𝐺v)𝐹(2nd ‘(𝑅v))))
8382opeq2d 3547 . . . . . . . . . . . 12 (z = (2nd ‘(𝑅v)) → ⟨((𝐺v) + 1), ((𝐺v)𝐹z)⟩ = ⟨((𝐺v) + 1), ((𝐺v)𝐹(2nd ‘(𝑅v)))⟩)
84 oveq1 5462 . . . . . . . . . . . . . 14 (x = w → (x + 1) = (w + 1))
85 oveq1 5462 . . . . . . . . . . . . . 14 (x = w → (x𝐹y) = (w𝐹y))
8684, 85opeq12d 3548 . . . . . . . . . . . . 13 (x = w → ⟨(x + 1), (x𝐹y)⟩ = ⟨(w + 1), (w𝐹y)⟩)
87 oveq2 5463 . . . . . . . . . . . . . 14 (y = z → (w𝐹y) = (w𝐹z))
8887opeq2d 3547 . . . . . . . . . . . . 13 (y = z → ⟨(w + 1), (w𝐹y)⟩ = ⟨(w + 1), (w𝐹z)⟩)
8986, 88cbvmpt2v 5526 . . . . . . . . . . . 12 (x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩) = (w (ℤ𝐶), z 𝑆 ↦ ⟨(w + 1), (w𝐹z)⟩)
9081, 83, 89ovmpt2g 5577 . . . . . . . . . . 11 (((𝐺v) (ℤ𝐶) (2nd ‘(𝑅v)) 𝑆 ⟨((𝐺v) + 1), ((𝐺v)𝐹(2nd ‘(𝑅v)))⟩ ((ℤ𝐶) × 𝑆)) → ((𝐺v)(x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)(2nd ‘(𝑅v))) = ⟨((𝐺v) + 1), ((𝐺v)𝐹(2nd ‘(𝑅v)))⟩)
9167, 71, 78, 90syl3anc 1134 . . . . . . . . . 10 ((φ v 𝜔) → ((𝐺v)(x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)(2nd ‘(𝑅v))) = ⟨((𝐺v) + 1), ((𝐺v)𝐹(2nd ‘(𝑅v)))⟩)
9265, 91syl5eqr 2083 . . . . . . . . 9 ((φ v 𝜔) → ((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘⟨(𝐺v), (2nd ‘(𝑅v))⟩) = ⟨((𝐺v) + 1), ((𝐺v)𝐹(2nd ‘(𝑅v)))⟩)
9364, 92sylan9eqr 2091 . . . . . . . 8 (((φ v 𝜔) (𝑅v) = ⟨(𝐺v), (2nd ‘(𝑅v))⟩) → ((x (ℤ𝐶), y 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘(𝑅v)) = ⟨((𝐺v) + 1), ((𝐺v)𝐹(2nd ‘(𝑅v)))⟩)
9463, 93eqtrd 2069 . . . . . . 7 (((φ v 𝜔) (𝑅v) = ⟨(𝐺v), (2nd ‘(𝑅v))⟩) → (𝑅‘suc v) = ⟨((𝐺v) + 1), ((𝐺v)𝐹(2nd ‘(𝑅v)))⟩)
9566, 32, 56frec2uzsucd 8868 . . . . . . . . 9 ((φ v 𝜔) → (𝐺‘suc v) = ((𝐺v) + 1))
9695adantr 261 . . . . . . . 8 (((φ v 𝜔) (𝑅v) = ⟨(𝐺v), (2nd ‘(𝑅v))⟩) → (𝐺‘suc v) = ((𝐺v) + 1))
9794fveq2d 5125 . . . . . . . . 9 (((φ v 𝜔) (𝑅v) = ⟨(𝐺v), (2nd ‘(𝑅v))⟩) → (2nd ‘(𝑅‘suc v)) = (2nd ‘⟨((𝐺v) + 1), ((𝐺v)𝐹(2nd ‘(𝑅v)))⟩))
9866, 32, 56frec2uzzd 8867 . . . . . . . . . . . 12 ((φ v 𝜔) → (𝐺v) ℤ)
9998peano2zd 8139 . . . . . . . . . . 11 ((φ v 𝜔) → ((𝐺v) + 1) ℤ)
10099adantr 261 . . . . . . . . . 10 (((φ v 𝜔) (𝑅v) = ⟨(𝐺v), (2nd ‘(𝑅v))⟩) → ((𝐺v) + 1) ℤ)
10176adantr 261 . . . . . . . . . 10 (((φ v 𝜔) (𝑅v) = ⟨(𝐺v), (2nd ‘(𝑅v))⟩) → ((𝐺v)𝐹(2nd ‘(𝑅v))) 𝑆)
102 op2ndg 5720 . . . . . . . . . 10 ((((𝐺v) + 1) ((𝐺v)𝐹(2nd ‘(𝑅v))) 𝑆) → (2nd ‘⟨((𝐺v) + 1), ((𝐺v)𝐹(2nd ‘(𝑅v)))⟩) = ((𝐺v)𝐹(2nd ‘(𝑅v))))
103100, 101, 102syl2anc 391 . . . . . . . . 9 (((φ v 𝜔) (𝑅v) = ⟨(𝐺v), (2nd ‘(𝑅v))⟩) → (2nd ‘⟨((𝐺v) + 1), ((𝐺v)𝐹(2nd ‘(𝑅v)))⟩) = ((𝐺v)𝐹(2nd ‘(𝑅v))))
10497, 103eqtrd 2069 . . . . . . . 8 (((φ v 𝜔) (𝑅v) = ⟨(𝐺v), (2nd ‘(𝑅v))⟩) → (2nd ‘(𝑅‘suc v)) = ((𝐺v)𝐹(2nd ‘(𝑅v))))
10596, 104opeq12d 3548 . . . . . . 7 (((φ v 𝜔) (𝑅v) = ⟨(𝐺v), (2nd ‘(𝑅v))⟩) → ⟨(𝐺‘suc v), (2nd ‘(𝑅‘suc v))⟩ = ⟨((𝐺v) + 1), ((𝐺v)𝐹(2nd ‘(𝑅v)))⟩)
10694, 105eqtr4d 2072 . . . . . 6 (((φ v 𝜔) (𝑅v) = ⟨(𝐺v), (2nd ‘(𝑅v))⟩) → (𝑅‘suc v) = ⟨(𝐺‘suc v), (2nd ‘(𝑅‘suc v))⟩)
107106ex 108 . . . . 5 ((φ v 𝜔) → ((𝑅v) = ⟨(𝐺v), (2nd ‘(𝑅v))⟩ → (𝑅‘suc v) = ⟨(𝐺‘suc v), (2nd ‘(𝑅‘suc v))⟩))
108107expcom 109 . . . 4 (v 𝜔 → (φ → ((𝑅v) = ⟨(𝐺v), (2nd ‘(𝑅v))⟩ → (𝑅‘suc v) = ⟨(𝐺‘suc v), (2nd ‘(𝑅‘suc v))⟩)))
10912, 17, 22, 41, 108finds2 4267 . . 3 (z 𝜔 → (φ → (𝑅z) = ⟨(𝐺z), (2nd ‘(𝑅z))⟩))
1107, 109vtoclga 2613 . 2 (B 𝜔 → (φ → (𝑅B) = ⟨(𝐺B), (2nd ‘(𝑅B))⟩))
1111, 110mpcom 32 1 (φ → (𝑅B) = ⟨(𝐺B), (2nd ‘(𝑅B))⟩)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240   = wceq 1242   ∈ wcel 1390  Vcvv 2551  ∅c0 3218  ⟨cop 3370   ↦ cmpt 3809  suc csuc 4068  𝜔com 4256   × cxp 4286  ‘cfv 4845  (class class class)co 5455   ↦ cmpt2 5457  2nd c2nd 5708  freccfrec 5917  1c1 6712   + caddc 6714  ℤcz 8021  ℤ≥cuz 8249 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-1re 6777  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-addcom 6783  ax-addass 6785  ax-distr 6787  ax-i2m1 6788  ax-0id 6791  ax-rnegex 6792  ax-cnre 6794  ax-pre-ltirr 6795  ax-pre-ltwlin 6796  ax-pre-lttrn 6797  ax-pre-ltadd 6799 This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-frec 5918  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-i1p 6450  df-iplp 6451  df-iltp 6453  df-enr 6654  df-nr 6655  df-ltr 6658  df-0r 6659  df-1r 6660  df-0 6718  df-1 6719  df-r 6721  df-lt 6724  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862  df-le 6863  df-sub 6981  df-neg 6982  df-inn 7696  df-n0 7958  df-z 8022  df-uz 8250 This theorem is referenced by:  frecuzrdglem  8878  frecuzrdgfn  8879  frecuzrdgsuc  8882
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