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

Theorem nati 16822
Description: Naturality property of a natural transformation. (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)
nati.h 𝐻 = (Hom ‘𝐶)
nati.o · = (comp‘𝐷)
nati.x (𝜑𝑋𝐵)
nati.y (𝜑𝑌𝐵)
nati.r (𝜑𝑅 ∈ (𝑋𝐻𝑌))
Assertion
Ref Expression
nati (𝜑 → ((𝐴𝑌)(⟨(𝐹𝑋), (𝐹𝑌)⟩ · (𝐾𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(⟨(𝐹𝑋), (𝐾𝑋)⟩ · (𝐾𝑌))(𝐴𝑋)))

Proof of Theorem nati
Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 natixp.2 . . . 4 (𝜑𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩))
2 natrcl.1 . . . . 5 𝑁 = (𝐶 Nat 𝐷)
3 natixp.b . . . . 5 𝐵 = (Base‘𝐶)
4 nati.h . . . . 5 𝐻 = (Hom ‘𝐶)
5 eqid 2771 . . . . 5 (Hom ‘𝐷) = (Hom ‘𝐷)
6 nati.o . . . . 5 · = (comp‘𝐷)
72natrcl 16817 . . . . . . . 8 (𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩) → (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) ∧ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷)))
81, 7syl 17 . . . . . . 7 (𝜑 → (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) ∧ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷)))
98simpld 482 . . . . . 6 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
10 df-br 4788 . . . . . 6 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
119, 10sylibr 224 . . . . 5 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
128simprd 483 . . . . . 6 (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
13 df-br 4788 . . . . . 6 (𝐾(𝐶 Func 𝐷)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
1412, 13sylibr 224 . . . . 5 (𝜑𝐾(𝐶 Func 𝐷)𝐿)
152, 3, 4, 5, 6, 11, 14isnat 16814 . . . 4 (𝜑 → (𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩) ↔ (𝐴X𝑥𝐵 ((𝐹𝑥)(Hom ‘𝐷)(𝐾𝑥)) ∧ ∀𝑥𝐵𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥)))))
161, 15mpbid 222 . . 3 (𝜑 → (𝐴X𝑥𝐵 ((𝐹𝑥)(Hom ‘𝐷)(𝐾𝑥)) ∧ ∀𝑥𝐵𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥))))
1716simprd 483 . 2 (𝜑 → ∀𝑥𝐵𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥)))
18 nati.x . . 3 (𝜑𝑋𝐵)
19 nati.y . . . . 5 (𝜑𝑌𝐵)
2019adantr 466 . . . 4 ((𝜑𝑥 = 𝑋) → 𝑌𝐵)
21 nati.r . . . . . . 7 (𝜑𝑅 ∈ (𝑋𝐻𝑌))
2221ad2antrr 705 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑅 ∈ (𝑋𝐻𝑌))
23 simplr 752 . . . . . . 7 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
24 simpr 471 . . . . . . 7 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
2523, 24oveq12d 6814 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
2622, 25eleqtrrd 2853 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑅 ∈ (𝑥𝐻𝑦))
27 simpllr 760 . . . . . . . . . 10 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → 𝑥 = 𝑋)
2827fveq2d 6337 . . . . . . . . 9 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐹𝑥) = (𝐹𝑋))
29 simplr 752 . . . . . . . . . 10 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → 𝑦 = 𝑌)
3029fveq2d 6337 . . . . . . . . 9 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐹𝑦) = (𝐹𝑌))
3128, 30opeq12d 4548 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ⟨(𝐹𝑥), (𝐹𝑦)⟩ = ⟨(𝐹𝑋), (𝐹𝑌)⟩)
3229fveq2d 6337 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐾𝑦) = (𝐾𝑌))
3331, 32oveq12d 6814 . . . . . . 7 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦)) = (⟨(𝐹𝑋), (𝐹𝑌)⟩ · (𝐾𝑌)))
3429fveq2d 6337 . . . . . . 7 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐴𝑦) = (𝐴𝑌))
3527, 29oveq12d 6814 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝑥𝐺𝑦) = (𝑋𝐺𝑌))
36 simpr 471 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → 𝑓 = 𝑅)
3735, 36fveq12d 6340 . . . . . . 7 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ((𝑥𝐺𝑦)‘𝑓) = ((𝑋𝐺𝑌)‘𝑅))
3833, 34, 37oveq123d 6817 . . . . . 6 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘𝑓)) = ((𝐴𝑌)(⟨(𝐹𝑋), (𝐹𝑌)⟩ · (𝐾𝑌))((𝑋𝐺𝑌)‘𝑅)))
3927fveq2d 6337 . . . . . . . . 9 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐾𝑥) = (𝐾𝑋))
4028, 39opeq12d 4548 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ⟨(𝐹𝑥), (𝐾𝑥)⟩ = ⟨(𝐹𝑋), (𝐾𝑋)⟩)
4140, 32oveq12d 6814 . . . . . . 7 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦)) = (⟨(𝐹𝑋), (𝐾𝑋)⟩ · (𝐾𝑌)))
4227, 29oveq12d 6814 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝑥𝐿𝑦) = (𝑋𝐿𝑌))
4342, 36fveq12d 6340 . . . . . . 7 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ((𝑥𝐿𝑦)‘𝑓) = ((𝑋𝐿𝑌)‘𝑅))
4427fveq2d 6337 . . . . . . 7 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐴𝑥) = (𝐴𝑋))
4541, 43, 44oveq123d 6817 . . . . . 6 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (((𝑥𝐿𝑦)‘𝑓)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥)) = (((𝑋𝐿𝑌)‘𝑅)(⟨(𝐹𝑋), (𝐾𝑋)⟩ · (𝐾𝑌))(𝐴𝑋)))
4638, 45eqeq12d 2786 . . . . 5 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥)) ↔ ((𝐴𝑌)(⟨(𝐹𝑋), (𝐹𝑌)⟩ · (𝐾𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(⟨(𝐹𝑋), (𝐾𝑋)⟩ · (𝐾𝑌))(𝐴𝑋))))
4726, 46rspcdv 3463 . . . 4 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (∀𝑓 ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥)) → ((𝐴𝑌)(⟨(𝐹𝑋), (𝐹𝑌)⟩ · (𝐾𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(⟨(𝐹𝑋), (𝐾𝑋)⟩ · (𝐾𝑌))(𝐴𝑋))))
4820, 47rspcimdv 3461 . . 3 ((𝜑𝑥 = 𝑋) → (∀𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥)) → ((𝐴𝑌)(⟨(𝐹𝑋), (𝐹𝑌)⟩ · (𝐾𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(⟨(𝐹𝑋), (𝐾𝑋)⟩ · (𝐾𝑌))(𝐴𝑋))))
4918, 48rspcimdv 3461 . 2 (𝜑 → (∀𝑥𝐵𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥)) → ((𝐴𝑌)(⟨(𝐹𝑋), (𝐹𝑌)⟩ · (𝐾𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(⟨(𝐹𝑋), (𝐾𝑋)⟩ · (𝐾𝑌))(𝐴𝑋))))
5017, 49mpd 15 1 (𝜑 → ((𝐴𝑌)(⟨(𝐹𝑋), (𝐹𝑌)⟩ · (𝐾𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(⟨(𝐹𝑋), (𝐾𝑋)⟩ · (𝐾𝑌))(𝐴𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  cop 4323   class class class wbr 4787  cfv 6030  (class class class)co 6796  Xcixp 8066  Basecbs 16064  Hom chom 16160  compcco 16161   Func cfunc 16721   Nat cnat 16808
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 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-ixp 8067  df-func 16725  df-nat 16810
This theorem is referenced by:  fuccocl  16831  invfuc  16841  evlfcllem  17069  yonedalem3b  17127  yonedainv  17129
  Copyright terms: Public domain W3C validator