Theorem arwrid 18280

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 2763	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		arwlid.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
4		arwlid.h	. . . . . . . 8 ⊢ 𝐻 = (Homa‘𝐶)
5	4	homarcl 18235	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
6	3, 5	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		eqid 2763	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
8	4, 2	homarcl2 18242	. . . . . . . 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 18266	. . . . 5 ⊢ (𝜑 → (2nd ‘( 1 ‘𝑋)) = ((Id‘𝐶)‘𝑋))
12	11	oveq2d 7462	. . . 4 ⊢ (𝜑 → ((2nd ‘𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(2nd ‘( 1 ‘𝑋))) = ((2nd ‘𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)))
13		eqid 2763	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
14		eqid 2763	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
15	9	simprd 501	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
16	4, 13	homahom 18246	. . . . . 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 17890	. . . 4 ⊢ (𝜑 → ((2nd ‘𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = (2nd ‘𝐹))
19	12, 18	eqtrd 2798	. . 3 ⊢ (𝜑 → ((2nd ‘𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(2nd ‘( 1 ‘𝑋))) = (2nd ‘𝐹))
20	19	oteq3d 4868	. 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 18267	. . 3 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋))
23	21, 4, 22, 3, 14	coaval 18275	. 2 ⊢ (𝜑 → (𝐹 · ( 1 ‘𝑋)) = ⟨𝑋, 𝑌, ((2nd ‘𝐹)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(2nd ‘( 1 ‘𝑋)))⟩)
24	4	homadmcd 18249	. . 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 2808	1 ⊢ (𝜑 → (𝐹 · ( 1 ‘𝑋)) = 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ⟨cop 4838  ⟨cotp 4861  ‘cfv 6561  (class class class)co 7446  2^nd c2nd 8027  Basecbs 17419  Hom chom 17471  compcco 17472  Catccat 17870  Idccid 17871  Hom_achoma 18230  Id_acida 18260  comp_accoa 18261

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-ot 4862  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-cat 17874  df-cid 17875  df-doma 18231  df-coda 18232  df-homa 18233  df-arw 18234  df-ida 18262  df-coa 18263

This theorem is used by: (None)