Theorem rcaninv 17807

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 2735	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2735	. . . . . 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 2735	. . . . . . . 8 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
8	1, 2, 7, 4, 5, 6	isohom 17789	. . . . . . 7 ⊢ (𝜑 → (𝑌(Iso‘𝐶)𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
9		rcaninv.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑌(Iso‘𝐶)𝑋))
10	8, 9	sseldd 3959	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
11	1, 2, 7, 4, 6, 5	isohom 17789	. . . . . . 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 17781	. . . . . . . 8 ⊢ (𝜑 → (𝑌𝑁𝑋):(𝑌(Iso‘𝐶)𝑋)⟶(𝑋(Iso‘𝐶)𝑌))
14	13, 9	ffvelcdmd 7075	. . . . . . 7 ⊢ (𝜑 → ((𝑌𝑁𝑋)‘𝐹) ∈ (𝑋(Iso‘𝐶)𝑌))
15	11, 14	sseldd 3959	. . . . . 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 17698	. . . . 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 2735	. . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶)
20		eqid 2735	. . . . . . . 8 ⊢ (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
21	1, 7, 12, 4, 5, 6, 9, 19, 20	invcoisoid 17805	. . . . . . 7 ⊢ (𝜑 → (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌))
22	21	eqcomd 2741	. . . . . 6 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) = (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹))
23	22	oveq2d 7421	. . . . 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 17696	. . . . 5 ⊢ (𝜑 → (𝐺(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)) = 𝐺)
25	18, 23, 24	3eqtr2rd 2777	. . . 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 2744	. . . . . . . 8 ⊢ (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍) = ⚬
29	28	a1i 11	. . . . . . 7 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍) = ⚬ )
30		eqidd 2736	. . . . . . 7 ⊢ (𝜑 → 𝐺 = 𝐺)
31		rcaninv.1	. . . . . . . . 9 ⊢ 𝑅 = ((𝑌𝑁𝑋)‘𝐹)
32	31	eqcomi 2744	. . . . . . . 8 ⊢ ((𝑌𝑁𝑋)‘𝐹) = 𝑅
33	32	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((𝑌𝑁𝑋)‘𝐹) = 𝑅)
34	29, 30, 33	oveq123d 7426	. . . . . 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 2770	. . . 4 ⊢ ((𝜑 ∧ (𝐺 ⚬ 𝑅) = (𝐻 ⚬ 𝑅)) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹)) = (𝐻 ⚬ 𝑅))
38	37	oveq1d 7420	. . 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 7418	. . . . . . 7 ⊢ (𝐻 ⚬ 𝑅) = (𝐻(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑅)
40	39	oveq1i 7415	. . . . . 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 17698	. . . . . 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 7415	. . . . . . . 8 ⊢ (𝑅(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)
46	45	oveq2i 7416	. . . . . . 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 7421	. . . . . 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 2774	. . . . 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 17696	. . . . 5 ⊢ (𝜑 → (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)) = 𝐻)
51	41, 49, 50	3eqtrd 2774	. . . 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 2774	. 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 1540   ∈ wcel 2108  ⟨cop 4607  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  Hom chom 17282  compcco 17283  Catccat 17676  Idccid 17677  Invcinv 17758  Isociso 17759

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-cat 17680  df-cid 17681  df-sect 17760  df-inv 17761  df-iso 17762

This theorem is referenced by:  initoeu2lem0  18026