Theorem nati 18010

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 2763	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
6		nati.o	. . . . 5 ⊢ · = (comp‘𝐷)
7	2	natrcl 18005	. . . . . . . 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 499	. . . . . 6 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
10		df-br 5110	. . . . . 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 500	. . . . . 6 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
13		df-br 5110	. . . . . 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 18002	. . . 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 500	. 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 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑌 ∈ 𝐵)
21		nati.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))
22	21	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑅 ∈ (𝑋𝐻𝑌))
23		simplr 780	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
24		simpr 489	. . . . . . 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 2866	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑅 ∈ (𝑥𝐻𝑦))
27		simpllr 787	. . . . . . . . . 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 780	. . . . . . . . . 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 4846	. . . . . . . 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 489	. . . . . . . 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 4846	. . . . . . . 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 2779	. . . . 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 3573	. . . 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 3571	. . 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 3571	. 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)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ⟨cop 4595   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410  Xcixp 8891  Basecbs 17264  Hom chom 17316  compcco 17317   Func cfunc 17906   Nat cnat 17996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  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 7982  df-2nd 7983  df-ixp 8892  df-func 17910  df-nat 17998

This theorem is referenced by:  fuccocl  18019  invfuc  18029  evlfcllem  18272  yonedalem3b  18330  yonedainv  18332  natoppf  50027  fuco22natlem1  50140  fuco22natlem2  50141  fuco23alem  50149  concom  50461  coccom  50462