Theorem arwlid 17979

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 2731	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		arwlid.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
4		arwlid.h	. . . . . . . 8 ⊢ 𝐻 = (Homa‘𝐶)
5	4	homarcl 17935	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
6	3, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		eqid 2731	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
8	4, 2	homarcl2 17942	. . . . . . . 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 17966	. . . . 5 ⊢ (𝜑 → (2nd ‘( 1 ‘𝑌)) = ((Id‘𝐶)‘𝑌))
12	11	oveq1d 7361	. . . 4 ⊢ (𝜑 → ((2nd ‘( 1 ‘𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹)))
13		eqid 2731	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
14	9	simpld 494	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
15		eqid 2731	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
16	4, 13	homahom 17946	. . . . . 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 17589	. . . 4 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (2nd ‘𝐹))
19	12, 18	eqtrd 2766	. . 3 ⊢ (𝜑 → ((2nd ‘( 1 ‘𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (2nd ‘𝐹))
20	19	oteq3d 4836	. 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 17967	. . 3 ⊢ (𝜑 → ( 1 ‘𝑌) ∈ (𝑌𝐻𝑌))
23	21, 4, 3, 22, 15	coaval 17975	. 2 ⊢ (𝜑 → (( 1 ‘𝑌) · 𝐹) = ⟨𝑋, 𝑌, ((2nd ‘( 1 ‘𝑌))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)(2nd ‘𝐹))⟩)
24	4	homadmcd 17949	. . 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 2776	1 ⊢ (𝜑 → (( 1 ‘𝑌) · 𝐹) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ⟨cop 4579  ⟨cotp 4581  ‘cfv 6481  (class class class)co 7346  2^nd c2nd 7920  Basecbs 17120  Hom chom 17172  compcco 17173  Catccat 17570  Idccid 17571  Hom_achoma 17930  Id_acida 17960  comp_accoa 17961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-ot 4582  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-cat 17574  df-cid 17575  df-doma 17931  df-coda 17932  df-homa 17933  df-arw 17934  df-ida 17962  df-coa 17963

This theorem is referenced by: (None)