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Theorem fucco 16823
Description: Value of the composition of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fucco.q 𝑄 = (𝐶 FuncCat 𝐷)
fucco.n 𝑁 = (𝐶 Nat 𝐷)
fucco.a 𝐴 = (Base‘𝐶)
fucco.o · = (comp‘𝐷)
fucco.x = (comp‘𝑄)
fucco.f (𝜑𝑅 ∈ (𝐹𝑁𝐺))
fucco.g (𝜑𝑆 ∈ (𝐺𝑁𝐻))
Assertion
Ref Expression
fucco (𝜑 → (𝑆(⟨𝐹, 𝐺 𝐻)𝑅) = (𝑥𝐴 ↦ ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑅𝑥))))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝑅   𝑥,𝑆   𝑥,𝐶   𝑥,𝐷   𝑥, ·   𝑥,𝐹   𝑥,𝐺   𝑥,𝐻
Allowed substitution hints:   𝑄(𝑥)   (𝑥)   𝑁(𝑥)

Proof of Theorem fucco
Dummy variables 𝑎 𝑏 𝑓 𝑔 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fucco.q . . . 4 𝑄 = (𝐶 FuncCat 𝐷)
2 eqid 2760 . . . 4 (𝐶 Func 𝐷) = (𝐶 Func 𝐷)
3 fucco.n . . . 4 𝑁 = (𝐶 Nat 𝐷)
4 fucco.a . . . 4 𝐴 = (Base‘𝐶)
5 fucco.o . . . 4 · = (comp‘𝐷)
6 fucco.f . . . . . . . 8 (𝜑𝑅 ∈ (𝐹𝑁𝐺))
73natrcl 16811 . . . . . . . 8 (𝑅 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
86, 7syl 17 . . . . . . 7 (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
98simpld 477 . . . . . 6 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
10 funcrcl 16724 . . . . . 6 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
119, 10syl 17 . . . . 5 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1211simpld 477 . . . 4 (𝜑𝐶 ∈ Cat)
1311simprd 482 . . . 4 (𝜑𝐷 ∈ Cat)
14 fucco.x . . . 4 = (comp‘𝑄)
151, 2, 3, 4, 5, 12, 13, 14fuccofval 16820 . . 3 (𝜑 = (𝑣 ∈ ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)), ∈ (𝐶 Func 𝐷) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))))))
16 fvexd 6364 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) → (1st𝑣) ∈ V)
17 simprl 811 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) → 𝑣 = ⟨𝐹, 𝐺⟩)
1817fveq2d 6356 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) → (1st𝑣) = (1st ‘⟨𝐹, 𝐺⟩))
19 op1stg 7345 . . . . . . 7 ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
208, 19syl 17 . . . . . 6 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
2120adantr 472 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
2218, 21eqtrd 2794 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) → (1st𝑣) = 𝐹)
23 fvexd 6364 . . . . 5 (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd𝑣) ∈ V)
2417adantr 472 . . . . . . 7 (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) → 𝑣 = ⟨𝐹, 𝐺⟩)
2524fveq2d 6356 . . . . . 6 (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd𝑣) = (2nd ‘⟨𝐹, 𝐺⟩))
26 op2ndg 7346 . . . . . . . 8 ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
278, 26syl 17 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
2827ad2antrr 764 . . . . . 6 (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
2925, 28eqtrd 2794 . . . . 5 (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd𝑣) = 𝐺)
30 simpr 479 . . . . . . 7 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
31 simprr 813 . . . . . . . 8 ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) → = 𝐻)
3231ad2antrr 764 . . . . . . 7 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → = 𝐻)
3330, 32oveq12d 6831 . . . . . 6 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑔𝑁) = (𝐺𝑁𝐻))
34 simplr 809 . . . . . . 7 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
3534, 30oveq12d 6831 . . . . . 6 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑓𝑁𝑔) = (𝐹𝑁𝐺))
3634fveq2d 6356 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st𝑓) = (1st𝐹))
3736fveq1d 6354 . . . . . . . . . 10 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st𝑓)‘𝑥) = ((1st𝐹)‘𝑥))
3830fveq2d 6356 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st𝑔) = (1st𝐺))
3938fveq1d 6354 . . . . . . . . . 10 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st𝑔)‘𝑥) = ((1st𝐺)‘𝑥))
4037, 39opeq12d 4561 . . . . . . . . 9 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ = ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)
4132fveq2d 6356 . . . . . . . . . 10 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st) = (1st𝐻))
4241fveq1d 6354 . . . . . . . . 9 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st)‘𝑥) = ((1st𝐻)‘𝑥))
4340, 42oveq12d 6831 . . . . . . . 8 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥)) = (⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥)))
4443oveqd 6830 . . . . . . 7 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥)) = ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥)))
4544mpteq2dv 4897 . . . . . 6 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))) = (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥))))
4633, 35, 45mpt2eq123dv 6882 . . . . 5 ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥)))) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥)))))
4723, 29, 46csbied2 3702 . . . 4 (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥)))) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥)))))
4816, 22, 47csbied2 3702 . . 3 ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ = 𝐻)) → (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥)))) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥)))))
49 opelxpi 5305 . . . 4 ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → ⟨𝐹, 𝐺⟩ ∈ ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)))
508, 49syl 17 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)))
51 fucco.g . . . . 5 (𝜑𝑆 ∈ (𝐺𝑁𝐻))
523natrcl 16811 . . . . 5 (𝑆 ∈ (𝐺𝑁𝐻) → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)))
5351, 52syl 17 . . . 4 (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)))
5453simprd 482 . . 3 (𝜑𝐻 ∈ (𝐶 Func 𝐷))
55 ovex 6841 . . . . 5 (𝐺𝑁𝐻) ∈ V
56 ovex 6841 . . . . 5 (𝐹𝑁𝐺) ∈ V
5755, 56mpt2ex 7415 . . . 4 (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥)))) ∈ V
5857a1i 11 . . 3 (𝜑 → (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥)))) ∈ V)
5915, 48, 50, 54, 58ovmpt2d 6953 . 2 (𝜑 → (⟨𝐹, 𝐺 𝐻) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥)))))
60 simprl 811 . . . . 5 ((𝜑 ∧ (𝑏 = 𝑆𝑎 = 𝑅)) → 𝑏 = 𝑆)
6160fveq1d 6354 . . . 4 ((𝜑 ∧ (𝑏 = 𝑆𝑎 = 𝑅)) → (𝑏𝑥) = (𝑆𝑥))
62 simprr 813 . . . . 5 ((𝜑 ∧ (𝑏 = 𝑆𝑎 = 𝑅)) → 𝑎 = 𝑅)
6362fveq1d 6354 . . . 4 ((𝜑 ∧ (𝑏 = 𝑆𝑎 = 𝑅)) → (𝑎𝑥) = (𝑅𝑥))
6461, 63oveq12d 6831 . . 3 ((𝜑 ∧ (𝑏 = 𝑆𝑎 = 𝑅)) → ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥)) = ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑅𝑥)))
6564mpteq2dv 4897 . 2 ((𝜑 ∧ (𝑏 = 𝑆𝑎 = 𝑅)) → (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑎𝑥))) = (𝑥𝐴 ↦ ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑅𝑥))))
66 fvex 6362 . . . . 5 (Base‘𝐶) ∈ V
674, 66eqeltri 2835 . . . 4 𝐴 ∈ V
6867mptex 6650 . . 3 (𝑥𝐴 ↦ ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑅𝑥))) ∈ V
6968a1i 11 . 2 (𝜑 → (𝑥𝐴 ↦ ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑅𝑥))) ∈ V)
7059, 65, 51, 6, 69ovmpt2d 6953 1 (𝜑 → (𝑆(⟨𝐹, 𝐺 𝐻)𝑅) = (𝑥𝐴 ↦ ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑅𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  csb 3674  cop 4327  cmpt 4881   × cxp 5264  cfv 6049  (class class class)co 6813  cmpt2 6815  1st c1st 7331  2nd c2nd 7332  Basecbs 16059  compcco 16155  Catccat 16526   Func cfunc 16715   Nat cnat 16802   FuncCat cfuc 16803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-fz 12520  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-hom 16168  df-cco 16169  df-func 16719  df-nat 16804  df-fuc 16805
This theorem is referenced by:  fuccoval  16824  fuccocl  16825  fuclid  16827  fucrid  16828  fucass  16829  fucsect  16833  curfcl  17073
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