Theorem rcaninv 17842

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 17824	. . . . . . 7 ⊢ (𝜑 → (𝑌(Iso‘𝐶)𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
9		rcaninv.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑌(Iso‘𝐶)𝑋))
10	8, 9	sseldd 3996	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
11	1, 2, 7, 4, 6, 5	isohom 17824	. . . . . . 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 17816	. . . . . . . 8 ⊢ (𝜑 → (𝑌𝑁𝑋):(𝑌(Iso‘𝐶)𝑋)⟶(𝑋(Iso‘𝐶)𝑌))
14	13, 9	ffvelcdmd 7105	. . . . . . 7 ⊢ (𝜑 → ((𝑌𝑁𝑋)‘𝐹) ∈ (𝑋(Iso‘𝐶)𝑌))
15	11, 14	sseldd 3996	. . . . . 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 17731	. . . . 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 17840	. . . . . . 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 7447	. . . . 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 17729	. . . . 5 ⊢ (𝜑 → (𝐺(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)) = 𝐺)
25	18, 23, 24	3eqtr2rd 2782	. . . 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 7452	. . . . . 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 2775	. . . 4 ⊢ ((𝜑 ∧ (𝐺 ⚬ 𝑅) = (𝐻 ⚬ 𝑅)) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹)) = (𝐻 ⚬ 𝑅))
38	37	oveq1d 7446	. . 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 7444	. . . . . . 7 ⊢ (𝐻 ⚬ 𝑅) = (𝐻(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑅)
40	39	oveq1i 7441	. . . . . 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 2843	. . . . . . 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 17731	. . . . . 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 7441	. . . . . . . 8 ⊢ (𝑅(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)
46	45	oveq2i 7442	. . . . . . 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 7447	. . . . . 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 2779	. . . . 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 17729	. . . . 5 ⊢ (𝜑 → (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)) = 𝐻)
51	41, 49, 50	3eqtrd 2779	. . . 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 2779	. 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 1537   ∈ wcel 2106  ⟨cop 4637  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  Hom chom 17309  compcco 17310  Catccat 17709  Idccid 17710  Invcinv 17793  Isociso 17794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-cat 17713  df-cid 17714  df-sect 17795  df-inv 17796  df-iso 17797

This theorem is referenced by:  initoeu2lem0  18067