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Theorem evlf2 18263
Description: Value of the evaluation functor at a morphism. (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)
evlf2.f (𝜑𝐹 ∈ (𝐶 Func 𝐷))
evlf2.g (𝜑𝐺 ∈ (𝐶 Func 𝐷))
evlf2.x (𝜑𝑋𝐵)
evlf2.y (𝜑𝑌𝐵)
evlf2.l 𝐿 = (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)
Assertion
Ref Expression
evlf2 (𝜑𝐿 = (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝑔))))
Distinct variable groups:   𝑔,𝑎,𝐶   𝐷,𝑎,𝑔   𝑔,𝐻   𝐹,𝑎,𝑔   𝑁,𝑎,𝑔   𝐺,𝑎,𝑔   𝜑,𝑎,𝑔   · ,𝑎,𝑔   𝑋,𝑎,𝑔   𝑌,𝑎,𝑔
Allowed substitution hints:   𝐵(𝑔,𝑎)   𝐸(𝑔,𝑎)   𝐻(𝑎)   𝐿(𝑔,𝑎)

Proof of Theorem evlf2
Dummy variables 𝑓 𝑚 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evlf2.l . 2 𝐿 = (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)
2 evlfval.e . . . . 5 𝐸 = (𝐶 evalF 𝐷)
3 evlfval.c . . . . 5 (𝜑𝐶 ∈ Cat)
4 evlfval.d . . . . 5 (𝜑𝐷 ∈ Cat)
5 evlfval.b . . . . 5 𝐵 = (Base‘𝐶)
6 evlfval.h . . . . 5 𝐻 = (Hom ‘𝐶)
7 evlfval.o . . . . 5 · = (comp‘𝐷)
8 evlfval.n . . . . 5 𝑁 = (𝐶 Nat 𝐷)
92, 3, 4, 5, 6, 7, 8evlfval 18262 . . . 4 (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))))⟩)
10 ovex 7464 . . . . . 6 (𝐶 Func 𝐷) ∈ V
115fvexi 6920 . . . . . 6 𝐵 ∈ V
1210, 11mpoex 8104 . . . . 5 (𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)) ∈ V
1310, 11xpex 7773 . . . . . 6 ((𝐶 Func 𝐷) × 𝐵) ∈ V
1413, 13mpoex 8104 . . . . 5 (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
1512, 14op2ndd 8025 . . . 4 (𝐸 = ⟨(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))))⟩ → (2nd𝐸) = (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)))))
169, 15syl 17 . . 3 (𝜑 → (2nd𝐸) = (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)))))
17 fvexd 6921 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) → (1st𝑥) ∈ V)
18 simprl 771 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) → 𝑥 = ⟨𝐹, 𝑋⟩)
1918fveq2d 6910 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) → (1st𝑥) = (1st ‘⟨𝐹, 𝑋⟩))
20 evlf2.f . . . . . . 7 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
21 evlf2.x . . . . . . 7 (𝜑𝑋𝐵)
22 op1stg 8026 . . . . . . 7 ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋𝐵) → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
2320, 21, 22syl2anc 584 . . . . . 6 (𝜑 → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
2423adantr 480 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
2519, 24eqtrd 2777 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) → (1st𝑥) = 𝐹)
26 fvexd 6921 . . . . 5 (((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) → (1st𝑦) ∈ V)
27 simplrr 778 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) → 𝑦 = ⟨𝐺, 𝑌⟩)
2827fveq2d 6910 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) → (1st𝑦) = (1st ‘⟨𝐺, 𝑌⟩))
29 evlf2.g . . . . . . . 8 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
30 evlf2.y . . . . . . . 8 (𝜑𝑌𝐵)
31 op1stg 8026 . . . . . . . 8 ((𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝑌𝐵) → (1st ‘⟨𝐺, 𝑌⟩) = 𝐺)
3229, 30, 31syl2anc 584 . . . . . . 7 (𝜑 → (1st ‘⟨𝐺, 𝑌⟩) = 𝐺)
3332ad2antrr 726 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) → (1st ‘⟨𝐺, 𝑌⟩) = 𝐺)
3428, 33eqtrd 2777 . . . . 5 (((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) → (1st𝑦) = 𝐺)
35 simplr 769 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → 𝑚 = 𝐹)
36 simpr 484 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → 𝑛 = 𝐺)
3735, 36oveq12d 7449 . . . . . 6 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (𝑚𝑁𝑛) = (𝐹𝑁𝐺))
3818ad2antrr 726 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → 𝑥 = ⟨𝐹, 𝑋⟩)
3938fveq2d 6910 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd𝑥) = (2nd ‘⟨𝐹, 𝑋⟩))
40 op2ndg 8027 . . . . . . . . . 10 ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋𝐵) → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
4120, 21, 40syl2anc 584 . . . . . . . . 9 (𝜑 → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
4241ad3antrrr 730 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
4339, 42eqtrd 2777 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd𝑥) = 𝑋)
4427adantr 480 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → 𝑦 = ⟨𝐺, 𝑌⟩)
4544fveq2d 6910 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd𝑦) = (2nd ‘⟨𝐺, 𝑌⟩))
46 op2ndg 8027 . . . . . . . . . 10 ((𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝑌𝐵) → (2nd ‘⟨𝐺, 𝑌⟩) = 𝑌)
4729, 30, 46syl2anc 584 . . . . . . . . 9 (𝜑 → (2nd ‘⟨𝐺, 𝑌⟩) = 𝑌)
4847ad3antrrr 730 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd ‘⟨𝐺, 𝑌⟩) = 𝑌)
4945, 48eqtrd 2777 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd𝑦) = 𝑌)
5043, 49oveq12d 7449 . . . . . 6 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((2nd𝑥)𝐻(2nd𝑦)) = (𝑋𝐻𝑌))
5135fveq2d 6910 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (1st𝑚) = (1st𝐹))
5251, 43fveq12d 6913 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((1st𝑚)‘(2nd𝑥)) = ((1st𝐹)‘𝑋))
5351, 49fveq12d 6913 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((1st𝑚)‘(2nd𝑦)) = ((1st𝐹)‘𝑌))
5452, 53opeq12d 4881 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ = ⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩)
5536fveq2d 6910 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (1st𝑛) = (1st𝐺))
5655, 49fveq12d 6913 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((1st𝑛)‘(2nd𝑦)) = ((1st𝐺)‘𝑌))
5754, 56oveq12d 7449 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦))) = (⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌)))
5849fveq2d 6910 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (𝑎‘(2nd𝑦)) = (𝑎𝑌))
5935fveq2d 6910 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd𝑚) = (2nd𝐹))
6059, 43, 49oveq123d 7452 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((2nd𝑥)(2nd𝑚)(2nd𝑦)) = (𝑋(2nd𝐹)𝑌))
6160fveq1d 6908 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔) = ((𝑋(2nd𝐹)𝑌)‘𝑔))
6257, 58, 61oveq123d 7452 . . . . . 6 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔)) = ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝑔)))
6337, 50, 62mpoeq123dv 7508 . . . . 5 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))) = (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝑔))))
6426, 34, 63csbied2 3936 . . . 4 (((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) → (1st𝑦) / 𝑛(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))) = (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝑔))))
6517, 25, 64csbied2 3936 . . 3 ((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) → (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))) = (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝑔))))
6620, 21opelxpd 5724 . . 3 (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝐶 Func 𝐷) × 𝐵))
6729, 30opelxpd 5724 . . 3 (𝜑 → ⟨𝐺, 𝑌⟩ ∈ ((𝐶 Func 𝐷) × 𝐵))
68 ovex 7464 . . . . 5 (𝐹𝑁𝐺) ∈ V
69 ovex 7464 . . . . 5 (𝑋𝐻𝑌) ∈ V
7068, 69mpoex 8104 . . . 4 (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝑔))) ∈ V
7170a1i 11 . . 3 (𝜑 → (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝑔))) ∈ V)
7216, 65, 66, 67, 71ovmpod 7585 . 2 (𝜑 → (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩) = (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝑔))))
731, 72eqtrid 2789 1 (𝜑𝐿 = (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘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-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
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-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-evlf 18258
This theorem is referenced by:  evlf2val  18264  evlfcl  18267
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