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

Theorem natixp 17947
Description: A natural transformation is a function from the objects of 𝐶 to homomorphisms from 𝐹(𝑥) to 𝐺(𝑥). (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
natrcl.1 𝑁 = (𝐶 Nat 𝐷)
natixp.2 (𝜑𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩))
natixp.b 𝐵 = (Base‘𝐶)
natixp.j 𝐽 = (Hom ‘𝐷)
Assertion
Ref Expression
natixp (𝜑𝐴X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝑥,𝐶   𝑥,𝐾   𝜑,𝑥   𝑥,𝐷   𝑥,𝐿   𝑥,𝐵   𝑥,𝐽
Allowed substitution hint:   𝑁(𝑥)

Proof of Theorem natixp
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 natixp.2 . . 3 (𝜑𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩))
2 natrcl.1 . . . 4 𝑁 = (𝐶 Nat 𝐷)
3 natixp.b . . . 4 𝐵 = (Base‘𝐶)
4 eqid 2727 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
5 natixp.j . . . 4 𝐽 = (Hom ‘𝐷)
6 eqid 2727 . . . 4 (comp‘𝐷) = (comp‘𝐷)
72natrcl 17945 . . . . . . 7 (𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩) → (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) ∧ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷)))
81, 7syl 17 . . . . . 6 (𝜑 → (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) ∧ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷)))
98simpld 493 . . . . 5 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
10 df-br 5151 . . . . 5 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
119, 10sylibr 233 . . . 4 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
128simprd 494 . . . . 5 (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
13 df-br 5151 . . . . 5 (𝐾(𝐶 Func 𝐷)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
1412, 13sylibr 233 . . . 4 (𝜑𝐾(𝐶 Func 𝐷)𝐿)
152, 3, 4, 5, 6, 11, 14isnat 17942 . . 3 (𝜑 → (𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩) ↔ (𝐴X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∧ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐷)(𝐾𝑦))((𝑥𝐺𝑦)‘𝑧)) = (((𝑥𝐿𝑦)‘𝑧)(⟨(𝐹𝑥), (𝐾𝑥)⟩(comp‘𝐷)(𝐾𝑦))(𝐴𝑥)))))
161, 15mpbid 231 . 2 (𝜑 → (𝐴X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∧ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐷)(𝐾𝑦))((𝑥𝐺𝑦)‘𝑧)) = (((𝑥𝐿𝑦)‘𝑧)(⟨(𝐹𝑥), (𝐾𝑥)⟩(comp‘𝐷)(𝐾𝑦))(𝐴𝑥))))
1716simpld 493 1 (𝜑𝐴X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3057  cop 4636   class class class wbr 5150  cfv 6551  (class class class)co 7424  Xcixp 8920  Basecbs 17185  Hom chom 17249  compcco 17250   Func cfunc 17845   Nat cnat 17936
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 7997  df-2nd 7998  df-ixp 8921  df-func 17849  df-nat 17938
This theorem is referenced by:  natcl  17948  natfn  17949
  Copyright terms: Public domain W3C validator