Theorem nati 17976

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	⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))
natixp.b	⊢ 𝐵 = (Base‘𝐶)
nati.h	⊢ 𝐻 = (Hom ‘𝐶)
nati.o	⊢ · = (comp‘𝐷)
nati.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
nati.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
nati.r	⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))

Assertion

Ref	Expression
nati	⊢ (𝜑 → ((𝐴‘𝑌)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩ · (𝐾‘𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(⟨(𝐹‘𝑋), (𝐾‘𝑋)⟩ · (𝐾‘𝑌))(𝐴‘𝑋)))

Proof of Theorem nati

Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		natixp.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 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 2736	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
6		nati.o	. . . . 5 ⊢ · = (comp‘𝐷)
7	2	natrcl 17971	. . . . . . . 8 ⊢ (𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩) → (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) ∧ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷)))
8	1, 7	syl 17	. . . . . . 7 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) ∧ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷)))
9	8	simpld 494	. . . . . 6 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
10		df-br 5125	. . . . . 6 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
11	9, 10	sylibr 234	. . . . 5 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
12	8	simprd 495	. . . . . 6 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
13		df-br 5125	. . . . . 6 ⊢ (𝐾(𝐶 Func 𝐷)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
14	12, 13	sylibr 234	. . . . 5 ⊢ (𝜑 → 𝐾(𝐶 Func 𝐷)𝐿)
15	2, 3, 4, 5, 6, 11, 14	isnat 17968	. . . 4 ⊢ (𝜑 → (𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩) ↔ (𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)(Hom ‘𝐷)(𝐾‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝐴‘𝑥)))))
16	1, 15	mpbid 232	. . 3 ⊢ (𝜑 → (𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)(Hom ‘𝐷)(𝐾‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝐴‘𝑥))))
17	16	simprd 495	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝐴‘𝑥)))
18		nati.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
19		nati.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
20	19	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑌 ∈ 𝐵)
21		nati.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))
22	21	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑅 ∈ (𝑋𝐻𝑌))
23		simplr 768	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
24		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
25	23, 24	oveq12d 7428	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
26	22, 25	eleqtrrd 2838	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑅 ∈ (𝑥𝐻𝑦))
27		simpllr 775	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → 𝑥 = 𝑋)
28	27	fveq2d 6885	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐹‘𝑥) = (𝐹‘𝑋))
29		simplr 768	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → 𝑦 = 𝑌)
30	29	fveq2d 6885	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐹‘𝑦) = (𝐹‘𝑌))
31	28, 30	opeq12d 4862	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ = ⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩)
32	29	fveq2d 6885	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐾‘𝑦) = (𝐾‘𝑌))
33	31, 32	oveq12d 7428	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦)) = (⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩ · (𝐾‘𝑌)))
34	29	fveq2d 6885	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐴‘𝑦) = (𝐴‘𝑌))
35	27, 29	oveq12d 7428	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝑥𝐺𝑦) = (𝑋𝐺𝑌))
36		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → 𝑓 = 𝑅)
37	35, 36	fveq12d 6888	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ((𝑥𝐺𝑦)‘𝑓) = ((𝑋𝐺𝑌)‘𝑅))
38	33, 34, 37	oveq123d 7431	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ((𝐴‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘𝑓)) = ((𝐴‘𝑌)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩ · (𝐾‘𝑌))((𝑋𝐺𝑌)‘𝑅)))
39	27	fveq2d 6885	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐾‘𝑥) = (𝐾‘𝑋))
40	28, 39	opeq12d 4862	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ = ⟨(𝐹‘𝑋), (𝐾‘𝑋)⟩)
41	40, 32	oveq12d 7428	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦)) = (⟨(𝐹‘𝑋), (𝐾‘𝑋)⟩ · (𝐾‘𝑌)))
42	27, 29	oveq12d 7428	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝑥𝐿𝑦) = (𝑋𝐿𝑌))
43	42, 36	fveq12d 6888	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ((𝑥𝐿𝑦)‘𝑓) = ((𝑋𝐿𝑌)‘𝑅))
44	27	fveq2d 6885	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐴‘𝑥) = (𝐴‘𝑋))
45	41, 43, 44	oveq123d 7431	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (((𝑥𝐿𝑦)‘𝑓)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝐴‘𝑥)) = (((𝑋𝐿𝑌)‘𝑅)(⟨(𝐹‘𝑋), (𝐾‘𝑋)⟩ · (𝐾‘𝑌))(𝐴‘𝑋)))
46	38, 45	eqeq12d 2752	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (((𝐴‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝐴‘𝑥)) ↔ ((𝐴‘𝑌)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩ · (𝐾‘𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(⟨(𝐹‘𝑋), (𝐾‘𝑋)⟩ · (𝐾‘𝑌))(𝐴‘𝑋))))
47	26, 46	rspcdv 3598	. . . 4 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (∀𝑓 ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝐴‘𝑥)) → ((𝐴‘𝑌)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩ · (𝐾‘𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(⟨(𝐹‘𝑋), (𝐾‘𝑋)⟩ · (𝐾‘𝑌))(𝐴‘𝑋))))
48	20, 47	rspcimdv 3596	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝐴‘𝑥)) → ((𝐴‘𝑌)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩ · (𝐾‘𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(⟨(𝐹‘𝑋), (𝐾‘𝑋)⟩ · (𝐾‘𝑌))(𝐴‘𝑋))))
49	18, 48	rspcimdv 3596	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘𝑓)) = (((𝑥𝐿𝑦)‘𝑓)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝐴‘𝑥)) → ((𝐴‘𝑌)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩ · (𝐾‘𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(⟨(𝐹‘𝑋), (𝐾‘𝑋)⟩ · (𝐾‘𝑌))(𝐴‘𝑋))))
50	17, 49	mpd 15	1 ⊢ (𝜑 → ((𝐴‘𝑌)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩ · (𝐾‘𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(⟨(𝐹‘𝑋), (𝐾‘𝑋)⟩ · (𝐾‘𝑌))(𝐴‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ⟨cop 4612   class class class wbr 5124  ‘cfv 6536  (class class class)co 7410  Xcixp 8916  Basecbs 17233  Hom chom 17287  compcco 17288   Func cfunc 17872   Nat cnat 17962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-ixp 8917  df-func 17876  df-nat 17964

This theorem is referenced by:  fuccocl  17985  invfuc  17995  evlfcllem  18238  yonedalem3b  18296  yonedainv  18298  fuco22natlem1  49220  fuco22natlem2  49221  fuco23alem  49229  concom  49500  coccom  49501