Theorem arwlid 18139

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 2740	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		arwlid.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
4		arwlid.h	. . . . . . . 8 ⊢ 𝐻 = (Homa‘𝐶)
5	4	homarcl 18095	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
6	3, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		eqid 2740	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
8	4, 2	homarcl2 18102	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
9	3, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
10	9	simprd 495	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
11	1, 2, 6, 7, 10	ida2 18126	. . . . 5 ⊢ (𝜑 → (2nd ‘( 1 ‘𝑌)) = ((Id‘𝐶)‘𝑌))
12	11	oveq1d 7463	. . . 4 ⊢ (𝜑 → ((2nd ‘( 1 ‘𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹)))
13		eqid 2740	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
14	9	simpld 494	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
15		eqid 2740	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
16	4, 13	homahom 18106	. . . . . 6 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
17	3, 16	syl 17	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
18	2, 13, 7, 6, 14, 15, 10, 17	catlid 17741	. . . 4 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (2nd ‘𝐹))
19	12, 18	eqtrd 2780	. . 3 ⊢ (𝜑 → ((2nd ‘( 1 ‘𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (2nd ‘𝐹))
20	19	oteq3d 4911	. 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 18127	. . 3 ⊢ (𝜑 → ( 1 ‘𝑌) ∈ (𝑌𝐻𝑌))
23	21, 4, 3, 22, 15	coaval 18135	. 2 ⊢ (𝜑 → (( 1 ‘𝑌) · 𝐹) = ⟨𝑋, 𝑌, ((2nd ‘( 1 ‘𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹))⟩)
24	4	homadmcd 18109	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd ‘𝐹)⟩)
25	3, 24	syl 17	. 2 ⊢ (𝜑 → 𝐹 = ⟨𝑋, 𝑌, (2nd ‘𝐹)⟩)
26	20, 23, 25	3eqtr4d 2790	1 ⊢ (𝜑 → (( 1 ‘𝑌) · 𝐹) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ⟨cop 4654  ⟨cotp 4656  ‘cfv 6573  (class class class)co 7448  2^nd c2nd 8029  Basecbs 17258  Hom chom 17322  compcco 17323  Catccat 17722  Idccid 17723  Hom_achoma 18090  Id_acida 18120  comp_accoa 18121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-ot 4657  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-cat 17726  df-cid 17727  df-doma 18091  df-coda 18092  df-homa 18093  df-arw 18094  df-ida 18122  df-coa 18123

This theorem is referenced by: (None)