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Theorem evlf2 17460
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 17459 . . . 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 7181 . . . . . 6 (𝐶 Func 𝐷) ∈ V
115fvexi 6677 . . . . . 6 𝐵 ∈ V
1210, 11mpoex 7769 . . . . 5 (𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)) ∈ V
1310, 11xpex 7468 . . . . . 6 ((𝐶 Func 𝐷) × 𝐵) ∈ V
1413, 13mpoex 7769 . . . . 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 7692 . . . 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 6678 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) → (1st𝑥) ∈ V)
18 simprl 769 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) → 𝑥 = ⟨𝐹, 𝑋⟩)
1918fveq2d 6667 . . . . 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 7693 . . . . . . 7 ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋𝐵) → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
2320, 21, 22syl2anc 586 . . . . . 6 (𝜑 → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
2423adantr 483 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
2519, 24eqtrd 2854 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) → (1st𝑥) = 𝐹)
26 fvexd 6678 . . . . 5 (((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) → (1st𝑦) ∈ V)
27 simplrr 776 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) → 𝑦 = ⟨𝐺, 𝑌⟩)
2827fveq2d 6667 . . . . . 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 7693 . . . . . . . 8 ((𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝑌𝐵) → (1st ‘⟨𝐺, 𝑌⟩) = 𝐺)
3229, 30, 31syl2anc 586 . . . . . . 7 (𝜑 → (1st ‘⟨𝐺, 𝑌⟩) = 𝐺)
3332ad2antrr 724 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) → (1st ‘⟨𝐺, 𝑌⟩) = 𝐺)
3428, 33eqtrd 2854 . . . . 5 (((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) → (1st𝑦) = 𝐺)
35 simplr 767 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → 𝑚 = 𝐹)
36 simpr 487 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → 𝑛 = 𝐺)
3735, 36oveq12d 7166 . . . . . 6 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (𝑚𝑁𝑛) = (𝐹𝑁𝐺))
3818ad2antrr 724 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → 𝑥 = ⟨𝐹, 𝑋⟩)
3938fveq2d 6667 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd𝑥) = (2nd ‘⟨𝐹, 𝑋⟩))
40 op2ndg 7694 . . . . . . . . . 10 ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋𝐵) → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
4120, 21, 40syl2anc 586 . . . . . . . . 9 (𝜑 → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
4241ad3antrrr 728 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
4339, 42eqtrd 2854 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd𝑥) = 𝑋)
4427adantr 483 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → 𝑦 = ⟨𝐺, 𝑌⟩)
4544fveq2d 6667 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd𝑦) = (2nd ‘⟨𝐺, 𝑌⟩))
46 op2ndg 7694 . . . . . . . . . 10 ((𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝑌𝐵) → (2nd ‘⟨𝐺, 𝑌⟩) = 𝑌)
4729, 30, 46syl2anc 586 . . . . . . . . 9 (𝜑 → (2nd ‘⟨𝐺, 𝑌⟩) = 𝑌)
4847ad3antrrr 728 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd ‘⟨𝐺, 𝑌⟩) = 𝑌)
4945, 48eqtrd 2854 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd𝑦) = 𝑌)
5043, 49oveq12d 7166 . . . . . 6 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((2nd𝑥)𝐻(2nd𝑦)) = (𝑋𝐻𝑌))
5135fveq2d 6667 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (1st𝑚) = (1st𝐹))
5251, 43fveq12d 6670 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((1st𝑚)‘(2nd𝑥)) = ((1st𝐹)‘𝑋))
5351, 49fveq12d 6670 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((1st𝑚)‘(2nd𝑦)) = ((1st𝐹)‘𝑌))
5452, 53opeq12d 4803 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ = ⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩)
5536fveq2d 6667 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (1st𝑛) = (1st𝐺))
5655, 49fveq12d 6670 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((1st𝑛)‘(2nd𝑦)) = ((1st𝐺)‘𝑌))
5754, 56oveq12d 7166 . . . . . . 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 6667 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (𝑎‘(2nd𝑦)) = (𝑎𝑌))
5935fveq2d 6667 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd𝑚) = (2nd𝐹))
6059, 43, 49oveq123d 7169 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((2nd𝑥)(2nd𝑚)(2nd𝑦)) = (𝑋(2nd𝐹)𝑌))
6160fveq1d 6665 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔) = ((𝑋(2nd𝐹)𝑌)‘𝑔))
6257, 58, 61oveq123d 7169 . . . . . 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 7221 . . . . 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 3918 . . . 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 3918 . . 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 5586 . . 3 (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝐶 Func 𝐷) × 𝐵))
6729, 30opelxpd 5586 . . 3 (𝜑 → ⟨𝐺, 𝑌⟩ ∈ ((𝐶 Func 𝐷) × 𝐵))
68 ovex 7181 . . . . 5 (𝐹𝑁𝐺) ∈ V
69 ovex 7181 . . . . 5 (𝑋𝐻𝑌) ∈ V
7068, 69mpoex 7769 . . . 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 7294 . 2 (𝜑 → (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩) = (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝑔))))
731, 72syl5eq 2866 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 398   = wceq 1530  wcel 2107  Vcvv 3493  csb 3881  cop 4565   × cxp 5546  cfv 6348  (class class class)co 7148  cmpo 7150  1st c1st 7679  2nd c2nd 7680  Basecbs 16475  Hom chom 16568  compcco 16569  Catccat 16927   Func cfunc 17116   Nat cnat 17203   evalF cevlf 17451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-evlf 17455
This theorem is referenced by:  evlf2val  17461  evlfcl  17464
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