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Theorem nati 16836
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 2760 . . . . 5 (Hom ‘𝐷) = (Hom ‘𝐷)
6 nati.o . . . . 5 · = (comp‘𝐷)
72natrcl 16831 . . . . . . . 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 477 . . . . . 6 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
10 df-br 4805 . . . . . 6 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
119, 10sylibr 224 . . . . 5 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
128simprd 482 . . . . . 6 (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
13 df-br 4805 . . . . . 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 16828 . . . 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 482 . 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 472 . . . 4 ((𝜑𝑥 = 𝑋) → 𝑌𝐵)
21 nati.r . . . . . . 7 (𝜑𝑅 ∈ (𝑋𝐻𝑌))
2221ad2antrr 764 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑅 ∈ (𝑋𝐻𝑌))
23 simplr 809 . . . . . . 7 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
24 simpr 479 . . . . . . 7 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
2523, 24oveq12d 6832 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
2622, 25eleqtrrd 2842 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑅 ∈ (𝑥𝐻𝑦))
27 simpllr 817 . . . . . . . . . 10 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → 𝑥 = 𝑋)
2827fveq2d 6357 . . . . . . . . 9 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐹𝑥) = (𝐹𝑋))
29 simplr 809 . . . . . . . . . 10 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → 𝑦 = 𝑌)
3029fveq2d 6357 . . . . . . . . 9 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐹𝑦) = (𝐹𝑌))
3128, 30opeq12d 4561 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ⟨(𝐹𝑥), (𝐹𝑦)⟩ = ⟨(𝐹𝑋), (𝐹𝑌)⟩)
3229fveq2d 6357 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐾𝑦) = (𝐾𝑌))
3331, 32oveq12d 6832 . . . . . . 7 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦)) = (⟨(𝐹𝑋), (𝐹𝑌)⟩ · (𝐾𝑌)))
3429fveq2d 6357 . . . . . . 7 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐴𝑦) = (𝐴𝑌))
3527, 29oveq12d 6832 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝑥𝐺𝑦) = (𝑋𝐺𝑌))
36 simpr 479 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → 𝑓 = 𝑅)
3735, 36fveq12d 6359 . . . . . . 7 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ((𝑥𝐺𝑦)‘𝑓) = ((𝑋𝐺𝑌)‘𝑅))
3833, 34, 37oveq123d 6835 . . . . . 6 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘𝑓)) = ((𝐴𝑌)(⟨(𝐹𝑋), (𝐹𝑌)⟩ · (𝐾𝑌))((𝑋𝐺𝑌)‘𝑅)))
3927fveq2d 6357 . . . . . . . . 9 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐾𝑥) = (𝐾𝑋))
4028, 39opeq12d 4561 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ⟨(𝐹𝑥), (𝐾𝑥)⟩ = ⟨(𝐹𝑋), (𝐾𝑋)⟩)
4140, 32oveq12d 6832 . . . . . . 7 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦)) = (⟨(𝐹𝑋), (𝐾𝑋)⟩ · (𝐾𝑌)))
4227, 29oveq12d 6832 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝑥𝐿𝑦) = (𝑋𝐿𝑌))
4342, 36fveq12d 6359 . . . . . . 7 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ((𝑥𝐿𝑦)‘𝑓) = ((𝑋𝐿𝑌)‘𝑅))
4427fveq2d 6357 . . . . . . 7 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐴𝑥) = (𝐴𝑋))
4541, 43, 44oveq123d 6835 . . . . . 6 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (((𝑥𝐿𝑦)‘𝑓)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥)) = (((𝑋𝐿𝑌)‘𝑅)(⟨(𝐹𝑋), (𝐾𝑋)⟩ · (𝐾𝑌))(𝐴𝑋)))
4638, 45eqeq12d 2775 . . . . 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 3452 . . . 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 3450 . . 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 3450 . 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 383   = wceq 1632  wcel 2139  wral 3050  cop 4327   class class class wbr 4804  cfv 6049  (class class class)co 6814  Xcixp 8076  Basecbs 16079  Hom chom 16174  compcco 16175   Func cfunc 16735   Nat cnat 16822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-ixp 8077  df-func 16739  df-nat 16824
This theorem is referenced by:  fuccocl  16845  invfuc  16855  evlfcllem  17082  yonedalem3b  17140  yonedainv  17142
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