Theorem rcaninv 17423

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 2738	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2738	. . . . . 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 2738	. . . . . . . 8 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
8	1, 2, 7, 4, 5, 6	isohom 17405	. . . . . . 7 ⊢ (𝜑 → (𝑌(Iso‘𝐶)𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
9		rcaninv.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑌(Iso‘𝐶)𝑋))
10	8, 9	sseldd 3918	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
11	1, 2, 7, 4, 6, 5	isohom 17405	. . . . . . 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 17397	. . . . . . . 8 ⊢ (𝜑 → (𝑌𝑁𝑋):(𝑌(Iso‘𝐶)𝑋)⟶(𝑋(Iso‘𝐶)𝑌))
14	13, 9	ffvelrnd 6944	. . . . . . 7 ⊢ (𝜑 → ((𝑌𝑁𝑋)‘𝐹) ∈ (𝑋(Iso‘𝐶)𝑌))
15	11, 14	sseldd 3918	. . . . . 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 17312	. . . . 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 2738	. . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶)
20		eqid 2738	. . . . . . . 8 ⊢ (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
21	1, 7, 12, 4, 5, 6, 9, 19, 20	invcoisoid 17421	. . . . . . 7 ⊢ (𝜑 → (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌))
22	21	eqcomd 2744	. . . . . 6 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) = (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹))
23	22	oveq2d 7271	. . . . 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 17310	. . . . 5 ⊢ (𝜑 → (𝐺(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)) = 𝐺)
25	18, 23, 24	3eqtr2rd 2785	. . . 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 2747	. . . . . . . 8 ⊢ (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍) = ⚬
29	28	a1i 11	. . . . . . 7 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍) = ⚬ )
30		eqidd 2739	. . . . . . 7 ⊢ (𝜑 → 𝐺 = 𝐺)
31		rcaninv.1	. . . . . . . . 9 ⊢ 𝑅 = ((𝑌𝑁𝑋)‘𝐹)
32	31	eqcomi 2747	. . . . . . . 8 ⊢ ((𝑌𝑁𝑋)‘𝐹) = 𝑅
33	32	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((𝑌𝑁𝑋)‘𝐹) = 𝑅)
34	29, 30, 33	oveq123d 7276	. . . . . 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 2778	. . . 4 ⊢ ((𝜑 ∧ (𝐺 ⚬ 𝑅) = (𝐻 ⚬ 𝑅)) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹)) = (𝐻 ⚬ 𝑅))
38	37	oveq1d 7270	. . 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 7268	. . . . . . 7 ⊢ (𝐻 ⚬ 𝑅) = (𝐻(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑅)
40	39	oveq1i 7265	. . . . . 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 17312	. . . . . 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 7265	. . . . . . . 8 ⊢ (𝑅(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)
46	45	oveq2i 7266	. . . . . . 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 7271	. . . . . 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 2782	. . . . 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 17310	. . . . 5 ⊢ (𝜑 → (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)) = 𝐻)
51	41, 49, 50	3eqtrd 2782	. . . 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 2782	. 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 1539   ∈ wcel 2108  ⟨cop 4564  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Hom chom 16899  compcco 16900  Catccat 17290  Idccid 17291  Invcinv 17374  Isociso 17375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-cat 17294  df-cid 17295  df-sect 17376  df-inv 17377  df-iso 17378

This theorem is referenced by:  initoeu2lem0  17644