Theorem rcaninv 17064

Description: Right cancellation of an inverse of an isomorphism. (Contributed by AV, 5-Apr-2020.)

Hypotheses

Ref	Expression
rcaninv.b	⊢ 𝐵 = (Base‘𝐶)
rcaninv.n	⊢ 𝑁 = (Inv‘𝐶)
rcaninv.c	⊢ (𝜑 → 𝐶 ∈ Cat)
rcaninv.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
rcaninv.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
rcaninv.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
rcaninv.f	⊢ (𝜑 → 𝐹 ∈ (𝑌(Iso‘𝐶)𝑋))
rcaninv.g	⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑍))
rcaninv.h	⊢ (𝜑 → 𝐻 ∈ (𝑌(Hom ‘𝐶)𝑍))
rcaninv.1	⊢ 𝑅 = ((𝑌𝑁𝑋)‘𝐹)
rcaninv.o	⊢ ⚬ = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)

Assertion

Ref	Expression
rcaninv	⊢ (𝜑 → ((𝐺 ⚬ 𝑅) = (𝐻 ⚬ 𝑅) → 𝐺 = 𝐻))

Proof of Theorem rcaninv

Step	Hyp	Ref	Expression
1		rcaninv.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
2		eqid 2821	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2821	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		rcaninv.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		rcaninv.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6		rcaninv.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		eqid 2821	. . . . . . . 8 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
8	1, 2, 7, 4, 5, 6	isohom 17046	. . . . . . 7 ⊢ (𝜑 → (𝑌(Iso‘𝐶)𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
9		rcaninv.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑌(Iso‘𝐶)𝑋))
10	8, 9	sseldd 3968	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
11	1, 2, 7, 4, 6, 5	isohom 17046	. . . . . . 7 ⊢ (𝜑 → (𝑋(Iso‘𝐶)𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
12		rcaninv.n	. . . . . . . . 9 ⊢ 𝑁 = (Inv‘𝐶)
13	1, 12, 4, 5, 6, 7	invf 17038	. . . . . . . 8 ⊢ (𝜑 → (𝑌𝑁𝑋):(𝑌(Iso‘𝐶)𝑋)⟶(𝑋(Iso‘𝐶)𝑌))
14	13, 9	ffvelrnd 6852	. . . . . . 7 ⊢ (𝜑 → ((𝑌𝑁𝑋)‘𝐹) ∈ (𝑋(Iso‘𝐶)𝑌))
15	11, 14	sseldd 3968	. . . . . 6 ⊢ (𝜑 → ((𝑌𝑁𝑋)‘𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
16		rcaninv.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
17		rcaninv.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑍))
18	1, 2, 3, 4, 5, 6, 5, 10, 15, 16, 17	catass 16957	. . . . 5 ⊢ (𝜑 → ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹))(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = (𝐺(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)(((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)))
19		eqid 2821	. . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶)
20		eqid 2821	. . . . . . . 8 ⊢ (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
21	1, 7, 12, 4, 5, 6, 9, 19, 20	invcoisoid 17062	. . . . . . 7 ⊢ (𝜑 → (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌))
22	21	eqcomd 2827	. . . . . 6 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) = (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹))
23	22	oveq2d 7172	. . . . 5 ⊢ (𝜑 → (𝐺(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)) = (𝐺(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)(((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)))
24	1, 2, 19, 4, 5, 3, 16, 17	catrid 16955	. . . . 5 ⊢ (𝜑 → (𝐺(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)) = 𝐺)
25	18, 23, 24	3eqtr2rd 2863	. . . 4 ⊢ (𝜑 → 𝐺 = ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹))(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹))
26	25	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝐺 ⚬ 𝑅) = (𝐻 ⚬ 𝑅)) → 𝐺 = ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹))(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹))
27		rcaninv.o	. . . . . . . . 9 ⊢ ⚬ = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)
28	27	eqcomi 2830	. . . . . . . 8 ⊢ (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍) = ⚬
29	28	a1i 11	. . . . . . 7 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍) = ⚬ )
30		eqidd 2822	. . . . . . 7 ⊢ (𝜑 → 𝐺 = 𝐺)
31		rcaninv.1	. . . . . . . . 9 ⊢ 𝑅 = ((𝑌𝑁𝑋)‘𝐹)
32	31	eqcomi 2830	. . . . . . . 8 ⊢ ((𝑌𝑁𝑋)‘𝐹) = 𝑅
33	32	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((𝑌𝑁𝑋)‘𝐹) = 𝑅)
34	29, 30, 33	oveq123d 7177	. . . . . 6 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹)) = (𝐺 ⚬ 𝑅))
35	34	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 ⚬ 𝑅) = (𝐻 ⚬ 𝑅)) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹)) = (𝐺 ⚬ 𝑅))
36		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 ⚬ 𝑅) = (𝐻 ⚬ 𝑅)) → (𝐺 ⚬ 𝑅) = (𝐻 ⚬ 𝑅))
37	35, 36	eqtrd 2856	. . . 4 ⊢ ((𝜑 ∧ (𝐺 ⚬ 𝑅) = (𝐻 ⚬ 𝑅)) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹)) = (𝐻 ⚬ 𝑅))
38	37	oveq1d 7171	. . 3 ⊢ ((𝜑 ∧ (𝐺 ⚬ 𝑅) = (𝐻 ⚬ 𝑅)) → ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹))(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = ((𝐻 ⚬ 𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹))
39	27	oveqi 7169	. . . . . . 7 ⊢ (𝐻 ⚬ 𝑅) = (𝐻(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑅)
40	39	oveq1i 7166	. . . . . 6 ⊢ ((𝐻 ⚬ 𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = ((𝐻(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹)
41	40	a1i 11	. . . . 5 ⊢ (𝜑 → ((𝐻 ⚬ 𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = ((𝐻(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹))
42	31, 15	eqeltrid 2917	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (𝑋(Hom ‘𝐶)𝑌))
43		rcaninv.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ (𝑌(Hom ‘𝐶)𝑍))
44	1, 2, 3, 4, 5, 6, 5, 10, 42, 16, 43	catass 16957	. . . . . 6 ⊢ (𝜑 → ((𝐻(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)(𝑅(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)))
45	31	oveq1i 7166	. . . . . . . 8 ⊢ (𝑅(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)
46	45	oveq2i 7167	. . . . . . 7 ⊢ (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)(𝑅(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)) = (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)(((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹))
47	46	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)(𝑅(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)) = (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)(((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)))
48	21	oveq2d 7172	. . . . . 6 ⊢ (𝜑 → (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)(((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)) = (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)))
49	44, 47, 48	3eqtrd 2860	. . . . 5 ⊢ (𝜑 → ((𝐻(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)))
50	1, 2, 19, 4, 5, 3, 16, 43	catrid 16955	. . . . 5 ⊢ (𝜑 → (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)) = 𝐻)
51	41, 49, 50	3eqtrd 2860	. . . 4 ⊢ (𝜑 → ((𝐻 ⚬ 𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = 𝐻)
52	51	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝐺 ⚬ 𝑅) = (𝐻 ⚬ 𝑅)) → ((𝐻 ⚬ 𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = 𝐻)
53	26, 38, 52	3eqtrd 2860	. 2 ⊢ ((𝜑 ∧ (𝐺 ⚬ 𝑅) = (𝐻 ⚬ 𝑅)) → 𝐺 = 𝐻)
54	53	ex 415	1 ⊢ (𝜑 → ((𝐺 ⚬ 𝑅) = (𝐻 ⚬ 𝑅) → 𝐺 = 𝐻))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ⟨cop 4573  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  Hom chom 16576  compcco 16577  Catccat 16935  Idccid 16936  Invcinv 17015  Isociso 17016

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-cat 16939  df-cid 16940  df-sect 17017  df-inv 17018  df-iso 17019

This theorem is referenced by:  initoeu2lem0  17273