Theorem nati 18048

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 18043	. . . . . . . 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 5104	. . . . . 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 5104	. . . . . 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 18040	. . . 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 7432	. . . . . 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 6883	. . . . . . . . 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 6883	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐹‘𝑦) = (𝐹‘𝑌))
31	28, 30	opeq12d 4841	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ = ⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩)
32	29	fveq2d 6883	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐾‘𝑦) = (𝐾‘𝑌))
33	31, 32	oveq12d 7432	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦)) = (⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩ · (𝐾‘𝑌)))
34	29	fveq2d 6883	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐴‘𝑦) = (𝐴‘𝑌))
35	27, 29	oveq12d 7432	. . . . . . . 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 6886	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ((𝑥𝐺𝑦)‘𝑓) = ((𝑋𝐺𝑌)‘𝑅))
38	33, 34, 37	oveq123d 7435	. . . . . 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 6883	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐾‘𝑥) = (𝐾‘𝑋))
40	28, 39	opeq12d 4841	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ = ⟨(𝐹‘𝑋), (𝐾‘𝑋)⟩)
41	40, 32	oveq12d 7432	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦)) = (⟨(𝐹‘𝑋), (𝐾‘𝑋)⟩ · (𝐾‘𝑌)))
42	27, 29	oveq12d 7432	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝑥𝐿𝑦) = (𝑋𝐿𝑌))
43	42, 36	fveq12d 6886	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → ((𝑥𝐿𝑦)‘𝑓) = ((𝑋𝐿𝑌)‘𝑅))
44	27	fveq2d 6883	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑓 = 𝑅) → (𝐴‘𝑥) = (𝐴‘𝑋))
45	41, 43, 44	oveq123d 7435	. . . . . 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 1570   ∈ wcel 2145  ∀wral 3076  ⟨cop 4590   class class class wbr 5103  ‘cfv 6533  (class class class)co 7414  Xcixp 8905  Basecbs 17302  Hom chom 17354  compcco 17355   Func cfunc 17944   Nat cnat 18034

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-ixp 8906  df-func 17948  df-nat 18036

This theorem is used by:  fuccocl  18057  invfuc  18067  evlfcllem  18310  yonedalem3b  18368  yonedainv  18370  natoppf  50156  fuco22natlem1  50269  fuco22natlem2  50270  fuco23alem  50278  concom  50590  coccom  50591