Theorem arwrid 18155

Description: Right identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
arwlid.h	⊢ 𝐻 = (Homa‘𝐶)
arwlid.o	⊢ · = (compa‘𝐶)
arwlid.a	⊢ 1 = (Ida‘𝐶)
arwlid.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))

Assertion

Ref	Expression
arwrid	⊢ (𝜑 → (𝐹 · ( 1 ‘𝑋)) = 𝐹)

Proof of Theorem arwrid

Step	Hyp	Ref	Expression
1		arwlid.a	. . . . . 6 ⊢ 1 = (Ida‘𝐶)
2		eqid 2766	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		arwlid.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
4		arwlid.h	. . . . . . . 8 ⊢ 𝐻 = (Homa‘𝐶)
5	4	homarcl 18110	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
6	3, 5	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		eqid 2766	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
8	4, 2	homarcl2 18117	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
9	3, 8	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
10	9	simpld 500	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
11	1, 2, 6, 7, 10	ida2 18141	. . . . 5 ⊢ (𝜑 → (2nd ‘( 1 ‘𝑋)) = ((Id‘𝐶)‘𝑋))
12	11	oveq2d 7439	. . . 4 ⊢ (𝜑 → ((2nd ‘𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(2nd ‘( 1 ‘𝑋))) = ((2nd ‘𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)))
13		eqid 2766	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
14		eqid 2766	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
15	9	simprd 501	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
16	4, 13	homahom 18121	. . . . . 6 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
17	3, 16	syl 18	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
18	2, 13, 7, 6, 10, 14, 15, 17	catrid 17765	. . . 4 ⊢ (𝜑 → ((2nd ‘𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = (2nd ‘𝐹))
19	12, 18	eqtrd 2801	. . 3 ⊢ (𝜑 → ((2nd ‘𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(2nd ‘( 1 ‘𝑋))) = (2nd ‘𝐹))
20	19	oteq3d 4857	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌, ((2nd ‘𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(2nd ‘( 1 ‘𝑋)))⟩ = ⟨𝑋, 𝑌, (2nd ‘𝐹)⟩)
21		arwlid.o	. . 3 ⊢ · = (compa‘𝐶)
22	1, 2, 6, 10, 4	idahom 18142	. . 3 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋))
23	21, 4, 22, 3, 14	coaval 18150	. 2 ⊢ (𝜑 → (𝐹 · ( 1 ‘𝑋)) = ⟨𝑋, 𝑌, ((2nd ‘𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(2nd ‘( 1 ‘𝑋)))⟩)
24	4	homadmcd 18124	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd ‘𝐹)⟩)
25	3, 24	syl 18	. 2 ⊢ (𝜑 → 𝐹 = ⟨𝑋, 𝑌, (2nd ‘𝐹)⟩)
26	20, 23, 25	3eqtr4d 2811	1 ⊢ (𝜑 → (𝐹 · ( 1 ‘𝑋)) = 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ⟨cop 4600  ⟨cotp 4602  ‘cfv 6543  (class class class)co 7423  2^nd c2nd 7994  Basecbs 17294  Hom chom 17346  compcco 17347  Catccat 17745  Idccid 17746  Hom_achoma 18105  Id_acida 18135  comp_accoa 18136

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-ot 4603  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996  df-cat 17749  df-cid 17750  df-doma 18106  df-coda 18107  df-homa 18108  df-arw 18109  df-ida 18137  df-coa 18138

This theorem is used by: (None)