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Theorem evlf2 17066
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 17065 . . . 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 6823 . . . . . 6 (𝐶 Func 𝐷) ∈ V
11 fvex 6342 . . . . . . 7 (Base‘𝐶) ∈ V
125, 11eqeltri 2846 . . . . . 6 𝐵 ∈ V
1310, 12mpt2ex 7397 . . . . 5 (𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)) ∈ V
1410, 12xpex 7109 . . . . . 6 ((𝐶 Func 𝐷) × 𝐵) ∈ V
1514, 14mpt2ex 7397 . . . . 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 7326 . . . 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 fvexd 6344 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) → (1st𝑥) ∈ V)
19 simprl 754 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) → 𝑥 = ⟨𝐹, 𝑋⟩)
2019fveq2d 6336 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) → (1st𝑥) = (1st ‘⟨𝐹, 𝑋⟩))
21 evlf2.f . . . . . . 7 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
22 evlf2.x . . . . . . 7 (𝜑𝑋𝐵)
23 op1stg 7327 . . . . . . 7 ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋𝐵) → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
2421, 22, 23syl2anc 573 . . . . . 6 (𝜑 → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
2524adantr 466 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
2620, 25eqtrd 2805 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) → (1st𝑥) = 𝐹)
27 fvexd 6344 . . . . 5 (((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) → (1st𝑦) ∈ V)
28 simplrr 763 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) → 𝑦 = ⟨𝐺, 𝑌⟩)
2928fveq2d 6336 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) → (1st𝑦) = (1st ‘⟨𝐺, 𝑌⟩))
30 evlf2.g . . . . . . . 8 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
31 evlf2.y . . . . . . . 8 (𝜑𝑌𝐵)
32 op1stg 7327 . . . . . . . 8 ((𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝑌𝐵) → (1st ‘⟨𝐺, 𝑌⟩) = 𝐺)
3330, 31, 32syl2anc 573 . . . . . . 7 (𝜑 → (1st ‘⟨𝐺, 𝑌⟩) = 𝐺)
3433ad2antrr 705 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) → (1st ‘⟨𝐺, 𝑌⟩) = 𝐺)
3529, 34eqtrd 2805 . . . . 5 (((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) → (1st𝑦) = 𝐺)
36 simplr 752 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → 𝑚 = 𝐹)
37 simpr 471 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → 𝑛 = 𝐺)
3836, 37oveq12d 6811 . . . . . 6 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (𝑚𝑁𝑛) = (𝐹𝑁𝐺))
3919ad2antrr 705 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → 𝑥 = ⟨𝐹, 𝑋⟩)
4039fveq2d 6336 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd𝑥) = (2nd ‘⟨𝐹, 𝑋⟩))
41 op2ndg 7328 . . . . . . . . . 10 ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋𝐵) → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
4221, 22, 41syl2anc 573 . . . . . . . . 9 (𝜑 → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
4342ad3antrrr 709 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
4440, 43eqtrd 2805 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd𝑥) = 𝑋)
4528adantr 466 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → 𝑦 = ⟨𝐺, 𝑌⟩)
4645fveq2d 6336 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd𝑦) = (2nd ‘⟨𝐺, 𝑌⟩))
47 op2ndg 7328 . . . . . . . . . 10 ((𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝑌𝐵) → (2nd ‘⟨𝐺, 𝑌⟩) = 𝑌)
4830, 31, 47syl2anc 573 . . . . . . . . 9 (𝜑 → (2nd ‘⟨𝐺, 𝑌⟩) = 𝑌)
4948ad3antrrr 709 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd ‘⟨𝐺, 𝑌⟩) = 𝑌)
5046, 49eqtrd 2805 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd𝑦) = 𝑌)
5144, 50oveq12d 6811 . . . . . 6 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((2nd𝑥)𝐻(2nd𝑦)) = (𝑋𝐻𝑌))
5236fveq2d 6336 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (1st𝑚) = (1st𝐹))
5352, 44fveq12d 6338 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((1st𝑚)‘(2nd𝑥)) = ((1st𝐹)‘𝑋))
5452, 50fveq12d 6338 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((1st𝑚)‘(2nd𝑦)) = ((1st𝐹)‘𝑌))
5553, 54opeq12d 4547 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ = ⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩)
5637fveq2d 6336 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (1st𝑛) = (1st𝐺))
5756, 50fveq12d 6338 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((1st𝑛)‘(2nd𝑦)) = ((1st𝐺)‘𝑌))
5855, 57oveq12d 6811 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦))) = (⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌)))
5950fveq2d 6336 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (𝑎‘(2nd𝑦)) = (𝑎𝑌))
6036fveq2d 6336 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (2nd𝑚) = (2nd𝐹))
6160, 44, 50oveq123d 6814 . . . . . . . 8 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → ((2nd𝑥)(2nd𝑚)(2nd𝑦)) = (𝑋(2nd𝐹)𝑌))
6261fveq1d 6334 . . . . . . 7 ((((𝜑 ∧ (𝑥 = ⟨𝐹, 𝑋⟩ ∧ 𝑦 = ⟨𝐺, 𝑌⟩)) ∧ 𝑚 = 𝐹) ∧ 𝑛 = 𝐺) → (((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔) = ((𝑋(2nd𝐹)𝑌)‘𝑔))
6358, 59, 62oveq123d 6814 . . . . . 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)))
6438, 51, 63mpt2eq123dv 6864 . . . . 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))))
6527, 35, 64csbied2 3710 . . . 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))))
6618, 26, 65csbied2 3710 . . 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))))
67 opelxpi 5288 . . . 4 ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋𝐵) → ⟨𝐹, 𝑋⟩ ∈ ((𝐶 Func 𝐷) × 𝐵))
6821, 22, 67syl2anc 573 . . 3 (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝐶 Func 𝐷) × 𝐵))
69 opelxpi 5288 . . . 4 ((𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝑌𝐵) → ⟨𝐺, 𝑌⟩ ∈ ((𝐶 Func 𝐷) × 𝐵))
7030, 31, 69syl2anc 573 . . 3 (𝜑 → ⟨𝐺, 𝑌⟩ ∈ ((𝐶 Func 𝐷) × 𝐵))
71 ovex 6823 . . . . 5 (𝐹𝑁𝐺) ∈ V
72 ovex 6823 . . . . 5 (𝑋𝐻𝑌) ∈ V
7371, 72mpt2ex 7397 . . . 4 (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝑔))) ∈ V
7473a1i 11 . . 3 (𝜑 → (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝑔))) ∈ V)
7517, 66, 68, 70, 74ovmpt2d 6935 . 2 (𝜑 → (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩) = (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝑔))))
761, 75syl5eq 2817 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 382   = wceq 1631  wcel 2145  Vcvv 3351  csb 3682  cop 4322   × cxp 5247  cfv 6031  (class class class)co 6793  cmpt2 6795  1st c1st 7313  2nd c2nd 7314  Basecbs 16064  Hom chom 16160  compcco 16161  Catccat 16532   Func cfunc 16721   Nat cnat 16808   evalF cevlf 17057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-evlf 17061
This theorem is referenced by:  evlf2val  17067  evlfcl  17070
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