MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isnat2 Structured version   Visualization version   GIF version

Theorem isnat2 16380
Description: Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
natfval.1 𝑁 = (𝐶 Nat 𝐷)
natfval.b 𝐵 = (Base‘𝐶)
natfval.h 𝐻 = (Hom ‘𝐶)
natfval.j 𝐽 = (Hom ‘𝐷)
natfval.o · = (comp‘𝐷)
isnat2.f (𝜑𝐹 ∈ (𝐶 Func 𝐷))
isnat2.g (𝜑𝐺 ∈ (𝐶 Func 𝐷))
Assertion
Ref Expression
isnat2 (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ (𝐴X𝑥𝐵 (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) ∧ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩ · ((1st𝐺)‘𝑦))((𝑥(2nd𝐹)𝑦)‘)) = (((𝑥(2nd𝐺)𝑦)‘)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐺)‘𝑦))(𝐴𝑥)))))
Distinct variable groups:   𝑥,,𝑦,𝐴   𝑥,𝐵,𝑦   𝐶,,𝑥,𝑦   ,𝐹,𝑥,𝑦   ,𝐺,𝑥,𝑦   ,𝐻   𝜑,,𝑥,𝑦   𝐷,,𝑥,𝑦
Allowed substitution hints:   𝐵()   · (𝑥,𝑦,)   𝐻(𝑥,𝑦)   𝐽(𝑥,𝑦,)   𝑁(𝑥,𝑦,)

Proof of Theorem isnat2
StepHypRef Expression
1 relfunc 16294 . . . . 5 Rel (𝐶 Func 𝐷)
2 isnat2.f . . . . 5 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
3 1st2nd 7083 . . . . 5 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
41, 2, 3sylancr 694 . . . 4 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
5 isnat2.g . . . . 5 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
6 1st2nd 7083 . . . . 5 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
71, 5, 6sylancr 694 . . . 4 (𝜑𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
84, 7oveq12d 6545 . . 3 (𝜑 → (𝐹𝑁𝐺) = (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
98eleq2d 2673 . 2 (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ 𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩)))
10 natfval.1 . . 3 𝑁 = (𝐶 Nat 𝐷)
11 natfval.b . . 3 𝐵 = (Base‘𝐶)
12 natfval.h . . 3 𝐻 = (Hom ‘𝐶)
13 natfval.j . . 3 𝐽 = (Hom ‘𝐷)
14 natfval.o . . 3 · = (comp‘𝐷)
15 1st2ndbr 7086 . . . 4 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
161, 2, 15sylancr 694 . . 3 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
17 1st2ndbr 7086 . . . 4 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
181, 5, 17sylancr 694 . . 3 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
1910, 11, 12, 13, 14, 16, 18isnat 16379 . 2 (𝜑 → (𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩) ↔ (𝐴X𝑥𝐵 (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) ∧ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩ · ((1st𝐺)‘𝑦))((𝑥(2nd𝐹)𝑦)‘)) = (((𝑥(2nd𝐺)𝑦)‘)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐺)‘𝑦))(𝐴𝑥)))))
209, 19bitrd 267 1 (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ (𝐴X𝑥𝐵 (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) ∧ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩ · ((1st𝐺)‘𝑦))((𝑥(2nd𝐹)𝑦)‘)) = (((𝑥(2nd𝐺)𝑦)‘)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐺)‘𝑦))(𝐴𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  cop 4131   class class class wbr 4578  Rel wrel 5033  cfv 5790  (class class class)co 6527  1st c1st 7035  2nd c2nd 7036  Xcixp 7772  Basecbs 15644  Hom chom 15728  compcco 15729   Func cfunc 16286   Nat cnat 16373
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 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
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 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7037  df-2nd 7038  df-ixp 7773  df-func 16290  df-nat 16375
This theorem is referenced by:  fuccocl  16396  fucidcl  16397  invfuc  16406  curf2cl  16643  yonedalem4c  16689  yonedalem3  16692
  Copyright terms: Public domain W3C validator