Theorem rcaninv 17716

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 2734	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2734	. . . . . 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 2734	. . . . . . . 8 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
8	1, 2, 7, 4, 5, 6	isohom 17698	. . . . . . 7 ⊢ (𝜑 → (𝑌(Iso‘𝐶)𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
9		rcaninv.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑌(Iso‘𝐶)𝑋))
10	8, 9	sseldd 3932	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
11	1, 2, 7, 4, 6, 5	isohom 17698	. . . . . . 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 17690	. . . . . . . 8 ⊢ (𝜑 → (𝑌𝑁𝑋):(𝑌(Iso‘𝐶)𝑋)⟶(𝑋(Iso‘𝐶)𝑌))
14	13, 9	ffvelcdmd 7028	. . . . . . 7 ⊢ (𝜑 → ((𝑌𝑁𝑋)‘𝐹) ∈ (𝑋(Iso‘𝐶)𝑌))
15	11, 14	sseldd 3932	. . . . . 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 17607	. . . . 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 2734	. . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶)
20		eqid 2734	. . . . . . . 8 ⊢ (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
21	1, 7, 12, 4, 5, 6, 9, 19, 20	invcoisoid 17714	. . . . . . 7 ⊢ (𝜑 → (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌))
22	21	eqcomd 2740	. . . . . 6 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) = (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹))
23	22	oveq2d 7372	. . . . 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 17605	. . . . 5 ⊢ (𝜑 → (𝐺(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)) = 𝐺)
25	18, 23, 24	3eqtr2rd 2776	. . . 4 ⊢ (𝜑 → 𝐺 = ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹))(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹))
26	25	adantr 480	. . 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 2743	. . . . . . . 8 ⊢ (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍) = ⚬
29	28	a1i 11	. . . . . . 7 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍) = ⚬ )
30		eqidd 2735	. . . . . . 7 ⊢ (𝜑 → 𝐺 = 𝐺)
31		rcaninv.1	. . . . . . . . 9 ⊢ 𝑅 = ((𝑌𝑁𝑋)‘𝐹)
32	31	eqcomi 2743	. . . . . . . 8 ⊢ ((𝑌𝑁𝑋)‘𝐹) = 𝑅
33	32	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((𝑌𝑁𝑋)‘𝐹) = 𝑅)
34	29, 30, 33	oveq123d 7377	. . . . . 6 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹)) = (𝐺 ⚬ 𝑅))
35	34	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 ⚬ 𝑅) = (𝐻 ⚬ 𝑅)) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹)) = (𝐺 ⚬ 𝑅))
36		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 ⚬ 𝑅) = (𝐻 ⚬ 𝑅)) → (𝐺 ⚬ 𝑅) = (𝐻 ⚬ 𝑅))
37	35, 36	eqtrd 2769	. . . 4 ⊢ ((𝜑 ∧ (𝐺 ⚬ 𝑅) = (𝐻 ⚬ 𝑅)) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹)) = (𝐻 ⚬ 𝑅))
38	37	oveq1d 7371	. . 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 7369	. . . . . . 7 ⊢ (𝐻 ⚬ 𝑅) = (𝐻(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑅)
40	39	oveq1i 7366	. . . . . 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 2838	. . . . . . 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 17607	. . . . . 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 7366	. . . . . . . 8 ⊢ (𝑅(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)
46	45	oveq2i 7367	. . . . . . 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 7372	. . . . . 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 2773	. . . . 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 17605	. . . . 5 ⊢ (𝜑 → (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)) = 𝐻)
51	41, 49, 50	3eqtrd 2773	. . . 4 ⊢ (𝜑 → ((𝐻 ⚬ 𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = 𝐻)
52	51	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐺 ⚬ 𝑅) = (𝐻 ⚬ 𝑅)) → ((𝐻 ⚬ 𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = 𝐻)
53	26, 38, 52	3eqtrd 2773	. 2 ⊢ ((𝜑 ∧ (𝐺 ⚬ 𝑅) = (𝐻 ⚬ 𝑅)) → 𝐺 = 𝐻)
54	53	ex 412	1 ⊢ (𝜑 → ((𝐺 ⚬ 𝑅) = (𝐻 ⚬ 𝑅) → 𝐺 = 𝐻))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ⟨cop 4584  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  Hom chom 17186  compcco 17187  Catccat 17585  Idccid 17586  Invcinv 17667  Isociso 17668

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-cat 17589  df-cid 17590  df-sect 17669  df-inv 17670  df-iso 17671

This theorem is referenced by:  initoeu2lem0  17935