Theorem arwlid 18129

Description: Left 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
arwlid	⊢ (𝜑 → (( 1 ‘𝑌) · 𝐹) = 𝐹)

Proof of Theorem arwlid

Step	Hyp	Ref	Expression
1		arwlid.a	. . . . . 6 ⊢ 1 = (Ida‘𝐶)
2		eqid 2769	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		arwlid.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
4		arwlid.h	. . . . . . . 8 ⊢ 𝐻 = (Homa‘𝐶)
5	4	homarcl 18085	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
6	3, 5	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		eqid 2769	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
8	4, 2	homarcl2 18092	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
9	3, 8	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
10	9	simprd 500	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
11	1, 2, 6, 7, 10	ida2 18116	. . . . 5 ⊢ (𝜑 → (2nd ‘( 1 ‘𝑌)) = ((Id‘𝐶)‘𝑌))
12	11	oveq1d 7426	. . . 4 ⊢ (𝜑 → ((2nd ‘( 1 ‘𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹)))
13		eqid 2769	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
14	9	simpld 499	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
15		eqid 2769	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
16	4, 13	homahom 18096	. . . . . 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, 14, 15, 10, 17	catlid 17739	. . . 4 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (2nd ‘𝐹))
19	12, 18	eqtrd 2804	. . 3 ⊢ (𝜑 → ((2nd ‘( 1 ‘𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (2nd ‘𝐹))
20	19	oteq3d 4856	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌, ((2nd ‘( 1 ‘𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹))⟩ = ⟨𝑋, 𝑌, (2nd ‘𝐹)⟩)
21		arwlid.o	. . 3 ⊢ · = (compa‘𝐶)
22	1, 2, 6, 10, 4	idahom 18117	. . 3 ⊢ (𝜑 → ( 1 ‘𝑌) ∈ (𝑌𝐻𝑌))
23	21, 4, 3, 22, 15	coaval 18125	. 2 ⊢ (𝜑 → (( 1 ‘𝑌) · 𝐹) = ⟨𝑋, 𝑌, ((2nd ‘( 1 ‘𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹))⟩)
24	4	homadmcd 18099	. . 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 2814	1 ⊢ (𝜑 → (( 1 ‘𝑌) · 𝐹) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ⟨cop 4600  ⟨cotp 4602  ‘cfv 6537  (class class class)co 7411  2^nd c2nd 7985  Basecbs 17269  Hom chom 17321  compcco 17322  Catccat 17720  Idccid 17721  Hom_achoma 18080  Id_acida 18110  comp_accoa 18111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-ot 4603  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-cat 17724  df-cid 17725  df-doma 18081  df-coda 18082  df-homa 18083  df-arw 18084  df-ida 18112  df-coa 18113

This theorem is referenced by: (None)