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Theorem evlfval 17455
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 17451 . . . 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 767 . . . . . 6 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → 𝑐 = 𝐶)
5 simprr 769 . . . . . 6 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → 𝑑 = 𝐷)
64, 5oveq12d 7163 . . . . 5 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (𝑐 Func 𝑑) = (𝐶 Func 𝐷))
74fveq2d 6667 . . . . . 6 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (Base‘𝑐) = (Base‘𝐶))
8 evlfval.b . . . . . 6 𝐵 = (Base‘𝐶)
97, 8syl6eqr 2871 . . . . 5 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (Base‘𝑐) = 𝐵)
10 eqidd 2819 . . . . 5 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → ((1st𝑓)‘𝑥) = ((1st𝑓)‘𝑥))
116, 9, 10mpoeq123dv 7218 . . . 4 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (Base‘𝑐) ↦ ((1st𝑓)‘𝑥)) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)))
126, 9xpeq12d 5579 . . . . 5 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → ((𝑐 Func 𝑑) × (Base‘𝑐)) = ((𝐶 Func 𝐷) × 𝐵))
134, 5oveq12d 7163 . . . . . . . . . 10 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (𝑐 Nat 𝑑) = (𝐶 Nat 𝐷))
14 evlfval.n . . . . . . . . . 10 𝑁 = (𝐶 Nat 𝐷)
1513, 14syl6eqr 2871 . . . . . . . . 9 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (𝑐 Nat 𝑑) = 𝑁)
1615oveqd 7162 . . . . . . . 8 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (𝑚(𝑐 Nat 𝑑)𝑛) = (𝑚𝑁𝑛))
174fveq2d 6667 . . . . . . . . . 10 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (Hom ‘𝑐) = (Hom ‘𝐶))
18 evlfval.h . . . . . . . . . 10 𝐻 = (Hom ‘𝐶)
1917, 18syl6eqr 2871 . . . . . . . . 9 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (Hom ‘𝑐) = 𝐻)
2019oveqd 7162 . . . . . . . 8 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) = ((2nd𝑥)𝐻(2nd𝑦)))
215fveq2d 6667 . . . . . . . . . . 11 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (comp‘𝑑) = (comp‘𝐷))
22 evlfval.o . . . . . . . . . . 11 · = (comp‘𝐷)
2321, 22syl6eqr 2871 . . . . . . . . . 10 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (comp‘𝑑) = · )
2423oveqd 7162 . . . . . . . . 9 ((𝜑 ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → (⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝑑)((1st𝑛)‘(2nd𝑦))) = (⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦))))
2524oveqd 7162 . . . . . . . 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 7218 . . . . . . 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 3887 . . . . . 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 3887 . . . . 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 7218 . . . 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 4803 . . 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 5347 . . . 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 7291 . 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, 35syl5eq 2865 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 396   = wceq 1528  wcel 2105  Vcvv 3492  csb 3880  cop 4563   × cxp 5546  cfv 6348  (class class class)co 7145  cmpo 7147  1st c1st 7676  2nd c2nd 7677  Basecbs 16471  Hom chom 16564  compcco 16565  Catccat 16923   Func cfunc 17112   Nat cnat 17199   evalF cevlf 17447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-evlf 17451
This theorem is referenced by:  evlf2  17456  evlf1  17458  evlfcl  17460
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