Theorem nati 18142

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 2760	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
6		nati.o	. . . . 5 ⊢ · = (comp‘𝐷)
7	2	natrcl 18137	. . . . . . . 8 ⊢ (𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩) → (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) ∧ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷)))
8	1, 7	syl 18	. . . . . . 7 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) ∧ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷)))
9	8	simpld 500	. . . . . 6 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
10		df-br 5103	. . . . . 6 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
11	9, 10	sylibr 237	. . . . 5 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
12	8	simprd 501	. . . . . 6 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
13		df-br 5103	. . . . . 6 ⊢ (𝐾(𝐶 Func 𝐷)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
14	12, 13	sylibr 237	. . . . 5 ⊢ (𝜑 → 𝐾(𝐶 Func 𝐷)𝐿)
15	2, 3, 4, 5, 6, 11, 14	isnat 18134	. . . 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 235	. . 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 501	. 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 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑌 ∈ 𝐵)
21		nati.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))
22	21	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑅 ∈ (𝑋𝐻𝑌))
23		simplr 781	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
24		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
25	23, 24	oveq12d 7442	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
26	22, 25	eleqtrrd 2863	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑅 ∈ (𝑥𝐻𝑦))
27		simpllr 788	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → 𝑥 = 𝑋)
28	27	fveq2d 6894	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐹‘𝑥) = (𝐹‘𝑋))
29		simplr 781	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → 𝑦 = 𝑌)
30	29	fveq2d 6894	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐹‘𝑦) = (𝐹‘𝑌))
31	28, 30	opeq12d 4847	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ = ⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩)
32	29	fveq2d 6894	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐾‘𝑦) = (𝐾‘𝑌))
33	31, 32	oveq12d 7442	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦)) = (⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩ · (𝐾‘𝑌)))
34	29	fveq2d 6894	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐴‘𝑦) = (𝐴‘𝑌))
35	27, 29	oveq12d 7442	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝑥𝐺𝑦) = (𝑋𝐺𝑌))
36		simpr 490	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → 𝑓 = 𝑅)
37	35, 36	fveq12d 6897	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ((𝑥𝐺𝑦)‘𝑓) = ((𝑋𝐺𝑌)‘𝑅))
38	33, 34, 37	oveq123d 7445	. . . . . 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 6894	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐾‘𝑥) = (𝐾‘𝑋))
40	28, 39	opeq12d 4847	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ = ⟨(𝐹‘𝑋), (𝐾‘𝑋)⟩)
41	40, 32	oveq12d 7442	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦)) = (⟨(𝐹‘𝑋), (𝐾‘𝑋)⟩ · (𝐾‘𝑌)))
42	27, 29	oveq12d 7442	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝑥𝐿𝑦) = (𝑋𝐿𝑌))
43	42, 36	fveq12d 6897	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ((𝑥𝐿𝑦)‘𝑓) = ((𝑋𝐿𝑌)‘𝑅))
44	27	fveq2d 6894	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐴‘𝑥) = (𝐴‘𝑋))
45	41, 43, 44	oveq123d 7445	. . . . . 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 2776	. . . . 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 3568	. . . 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 3566	. . 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 3566	. 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 16	1 ⊢ (𝜑 → ((𝐴‘𝑌)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩ · (𝐾‘𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(⟨(𝐹‘𝑋), (𝐾‘𝑋)⟩ · (𝐾‘𝑌))(𝐴‘𝑋)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ⟨cop 4834   class class class wbr 5102  ‘cfv 6543  (class class class)co 7424  Xcixp 8933  Basecbs 17396  Hom chom 17448  compcco 17449   Func cfunc 18038   Nat cnat 18128

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-ixp 8934  df-func 18042  df-nat 18130

This theorem is used by:  fuccocl  18151  invfuc  18161  evlfcllem  18404  yonedalem3b  18462  yonedainv  18464  natoppf  50487  fuco22natlem1  50600  fuco22natlem2  50601  fuco23alem  50609  concom  50982  coccom  50983