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Theorem evlfval 18262
Description: Value of the evaluation functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
evlfval.e 𝐸 = (𝐶 evalF 𝐷)
evlfval.c (𝜑𝐶 ∈ Cat)
evlfval.d (𝜑𝐷 ∈ Cat)
evlfval.b 𝐵 = (Base‘𝐶)
evlfval.h 𝐻 = (Hom ‘𝐶)
evlfval.o · = (comp‘𝐷)
evlfval.n 𝑁 = (𝐶 Nat 𝐷)
Assertion
Ref Expression
evlfval (𝜑𝐸 = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))⟩)
Distinct variable groups:   𝑓,𝑎,𝑔,𝑚,𝑛,𝑥,𝑦,𝐶   𝐷,𝑎,𝑓,𝑔,𝑚,𝑛,𝑥,𝑦   𝑔,𝐻,𝑚,𝑛,𝑥,𝑦   𝑁,𝑎,𝑔,𝑚,𝑛,𝑥,𝑦   𝜑,𝑎,𝑓,𝑔,𝑚,𝑛,𝑥,𝑦   · ,𝑎,𝑔,𝑚,𝑛,𝑥,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐵(𝑓,𝑔,𝑚,𝑛,𝑎)   · (𝑓)   𝐸(𝑥,𝑦,𝑓,𝑔,𝑚,𝑛,𝑎)   𝐻(𝑓,𝑎)   𝑁(𝑓)

Proof of Theorem evlfval
Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evlfval.e . 2 𝐸 = (𝐶 evalF 𝐷)
2 df-evlf 18258 . . . 4 evalF = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ ⟨(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (Base‘𝑐) ↦ ((1st𝑓)‘𝑥)), (𝑥 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)), 𝑦 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝑑)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))⟩)
32a1i 11 . . 3 (𝜑 → evalF = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ ⟨(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (Base‘𝑐) ↦ ((1st𝑓)‘𝑥)), (𝑥 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)), 𝑦 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝑑)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))⟩))
4 simprl 771 . . . . . 6 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → 𝑐 = 𝐶)
5 simprr 773 . . . . . 6 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → 𝑑 = 𝐷)
64, 5oveq12d 7449 . . . . 5 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (𝑐 Func 𝑑) = (𝐶 Func 𝐷))
74fveq2d 6910 . . . . . 6 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (Base‘𝑐) = (Base‘𝐶))
8 evlfval.b . . . . . 6 𝐵 = (Base‘𝐶)
97, 8eqtr4di 2795 . . . . 5 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (Base‘𝑐) = 𝐵)
10 eqidd 2738 . . . . 5 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → ((1st𝑓)‘𝑥) = ((1st𝑓)‘𝑥))
116, 9, 10mpoeq123dv 7508 . . . 4 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (Base‘𝑐) ↦ ((1st𝑓)‘𝑥)) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)))
126, 9xpeq12d 5716 . . . . 5 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → ((𝑐 Func 𝑑) × (Base‘𝑐)) = ((𝐶 Func 𝐷) × 𝐵))
134, 5oveq12d 7449 . . . . . . . . . 10 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (𝑐 Nat 𝑑) = (𝐶 Nat 𝐷))
14 evlfval.n . . . . . . . . . 10 𝑁 = (𝐶 Nat 𝐷)
1513, 14eqtr4di 2795 . . . . . . . . 9 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (𝑐 Nat 𝑑) = 𝑁)
1615oveqd 7448 . . . . . . . 8 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (𝑚(𝑐 Nat 𝑑)𝑛) = (𝑚𝑁𝑛))
174fveq2d 6910 . . . . . . . . . 10 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (Hom ‘𝑐) = (Hom ‘𝐶))
18 evlfval.h . . . . . . . . . 10 𝐻 = (Hom ‘𝐶)
1917, 18eqtr4di 2795 . . . . . . . . 9 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (Hom ‘𝑐) = 𝐻)
2019oveqd 7448 . . . . . . . 8 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) = ((2nd𝑥)𝐻(2nd𝑦)))
215fveq2d 6910 . . . . . . . . . . 11 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (comp‘𝑑) = (comp‘𝐷))
22 evlfval.o . . . . . . . . . . 11 · = (comp‘𝐷)
2321, 22eqtr4di 2795 . . . . . . . . . 10 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (comp‘𝑑) = · )
2423oveqd 7448 . . . . . . . . 9 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝑑)((1st𝑛)‘(2nd𝑦))) = (⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦))))
2524oveqd 7448 . . . . . . . 8 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝑑)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔)) = ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔)))
2616, 20, 25mpoeq123dv 7508 . . . . . . 7 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝑑)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))) = (𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))
2726csbeq2dv 3906 . . . . . 6 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝑑)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))) = (1st𝑦) / 𝑛(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))
2827csbeq2dv 3906 . . . . 5 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝑑)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))) = (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))
2912, 12, 28mpoeq123dv 7508 . . . 4 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (𝑥 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)), 𝑦 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝑑)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔)))) = (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔)))))
3011, 29opeq12d 4881 . . 3 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → ⟨(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (Base‘𝑐) ↦ ((1st𝑓)‘𝑥)), (𝑥 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)), 𝑦 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝑑)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))⟩ = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))⟩)
31 evlfval.c . . 3 (𝜑𝐶 ∈ Cat)
32 evlfval.d . . 3 (𝜑𝐷 ∈ Cat)
33 opex 5469 . . . 4 ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))⟩ ∈ V
3433a1i 11 . . 3 (𝜑 → ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))⟩ ∈ V)
353, 30, 31, 32, 34ovmpod 7585 . 2 (𝜑 → (𝐶 evalF 𝐷) = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))⟩)
361, 35eqtrid 2789 1 (𝜑𝐸 = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  csb 3899  cop 4632   × cxp 5683  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707   Func cfunc 17899   Nat cnat 17989   evalF cevlf 18254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-evlf 18258
This theorem is referenced by:  evlf2  18263  evlf1  18265  evlfcl  18267
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