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Theorem evlf2 16681
 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 16680 . . . 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 6577 . . . . . 6 (𝐶 Func 𝐷) ∈ V
11 fvex 6113 . . . . . . 7 (Base‘𝐶) ∈ V
125, 11eqeltri 2684 . . . . . 6 𝐵 ∈ V
1310, 12mpt2ex 7136 . . . . 5 (𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)) ∈ V
1410, 12xpex 6860 . . . . . 6 ((𝐶 Func 𝐷) × 𝐵) ∈ V
1514, 14mpt2ex 7136 . . . . 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
1613, 15op2ndd 7070 . . . 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)))))
179, 16syl 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)))))
18 fvex 6113 . . . . 5 (1st𝑥) ∈ V
1918a1i 11 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) → (1st𝑥) ∈ V)
20 simprl 790 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) → 𝑥 = ⟨𝐹, 𝑋⟩)
2120fveq2d 6107 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) → (1st𝑥) = (1st ‘⟨𝐹, 𝑋⟩))
22 evlf2.f . . . . . . 7 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
23 evlf2.x . . . . . . 7 (𝜑𝑋𝐵)
24 op1stg 7071 . . . . . . 7 ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋𝐵) → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
2522, 23, 24syl2anc 691 . . . . . 6 (𝜑 → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
2625adantr 480 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
2721, 26eqtrd 2644 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) → (1st𝑥) = 𝐹)
28 fvex 6113 . . . . . 6 (1st𝑦) ∈ V
2928a1i 11 . . . . 5 (((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) → (1st𝑦) ∈ V)
30 simplrr 797 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) → 𝑦 = ⟨𝐺, 𝑌⟩)
3130fveq2d 6107 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) → (1st𝑦) = (1st ‘⟨𝐺, 𝑌⟩))
32 evlf2.g . . . . . . . 8 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
33 evlf2.y . . . . . . . 8 (𝜑𝑌𝐵)
34 op1stg 7071 . . . . . . . 8 ((𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝑌𝐵) → (1st ‘⟨𝐺, 𝑌⟩) = 𝐺)
3532, 33, 34syl2anc 691 . . . . . . 7 (𝜑 → (1st ‘⟨𝐺, 𝑌⟩) = 𝐺)
3635ad2antrr 758 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) → (1st ‘⟨𝐺, 𝑌⟩) = 𝐺)
3731, 36eqtrd 2644 . . . . 5 (((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) → (1st𝑦) = 𝐺)
38 simplr 788 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → 𝑚 = 𝐹)
39 simpr 476 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → 𝑛 = 𝐺)
4038, 39oveq12d 6567 . . . . . 6 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (𝑚𝑁𝑛) = (𝐹𝑁𝐺))
4120ad2antrr 758 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → 𝑥 = ⟨𝐹, 𝑋⟩)
4241fveq2d 6107 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd𝑥) = (2nd ‘⟨𝐹, 𝑋⟩))
43 op2ndg 7072 . . . . . . . . . 10 ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋𝐵) → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
4422, 23, 43syl2anc 691 . . . . . . . . 9 (𝜑 → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
4544ad3antrrr 762 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
4642, 45eqtrd 2644 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd𝑥) = 𝑋)
4730adantr 480 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → 𝑦 = ⟨𝐺, 𝑌⟩)
4847fveq2d 6107 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd𝑦) = (2nd ‘⟨𝐺, 𝑌⟩))
49 op2ndg 7072 . . . . . . . . . 10 ((𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝑌𝐵) → (2nd ‘⟨𝐺, 𝑌⟩) = 𝑌)
5032, 33, 49syl2anc 691 . . . . . . . . 9 (𝜑 → (2nd ‘⟨𝐺, 𝑌⟩) = 𝑌)
5150ad3antrrr 762 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd ‘⟨𝐺, 𝑌⟩) = 𝑌)
5248, 51eqtrd 2644 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd𝑦) = 𝑌)
5346, 52oveq12d 6567 . . . . . 6 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((2nd𝑥)𝐻(2nd𝑦)) = (𝑋𝐻𝑌))
5438fveq2d 6107 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (1st𝑚) = (1st𝐹))
5554, 46fveq12d 6109 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((1st𝑚)‘(2nd𝑥)) = ((1st𝐹)‘𝑋))
5654, 52fveq12d 6109 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((1st𝑚)‘(2nd𝑦)) = ((1st𝐹)‘𝑌))
5755, 56opeq12d 4348 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ = ⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩)
5839fveq2d 6107 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (1st𝑛) = (1st𝐺))
5958, 52fveq12d 6109 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((1st𝑛)‘(2nd𝑦)) = ((1st𝐺)‘𝑌))
6057, 59oveq12d 6567 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦))) = (⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌)))
6152fveq2d 6107 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (𝑎‘(2nd𝑦)) = (𝑎𝑌))
6238fveq2d 6107 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd𝑚) = (2nd𝐹))
6362, 46, 52oveq123d 6570 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((2nd𝑥)(2nd𝑚)(2nd𝑦)) = (𝑋(2nd𝐹)𝑌))
6463fveq1d 6105 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔) = ((𝑋(2nd𝐹)𝑌)‘𝑔))
6560, 61, 64oveq123d 6570 . . . . . 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)))
6640, 53, 65mpt2eq123dv 6615 . . . . 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))))
6729, 37, 66csbied2 3527 . . . 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))))
6819, 27, 67csbied2 3527 . . 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))))
69 opelxpi 5072 . . . 4 ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋𝐵) → ⟨𝐹, 𝑋⟩ ∈ ((𝐶 Func 𝐷) × 𝐵))
7022, 23, 69syl2anc 691 . . 3 (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝐶 Func 𝐷) × 𝐵))
71 opelxpi 5072 . . . 4 ((𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝑌𝐵) → ⟨𝐺, 𝑌⟩ ∈ ((𝐶 Func 𝐷) × 𝐵))
7232, 33, 71syl2anc 691 . . 3 (𝜑 → ⟨𝐺, 𝑌⟩ ∈ ((𝐶 Func 𝐷) × 𝐵))
73 ovex 6577 . . . . 5 (𝐹𝑁𝐺) ∈ V
74 ovex 6577 . . . . 5 (𝑋𝐻𝑌) ∈ V
7573, 74mpt2ex 7136 . . . 4 (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝑔))) ∈ V
7675a1i 11 . . 3 (𝜑 → (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝑔))) ∈ V)
7717, 68, 70, 72, 76ovmpt2d 6686 . 2 (𝜑 → (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩) = (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝑔))))
781, 77syl5eq 2656 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 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⦋csb 3499  ⟨cop 4131   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148   Func cfunc 16337   Nat cnat 16424   evalF cevlf 16672 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-evlf 16676 This theorem is referenced by:  evlf2val  16682  evlfcl  16685
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