Theorem grptcepi 49438

Description: All morphisms in a category converted from a group are epimorphisms. (Contributed by Zhi Wang, 23-Sep-2024.)

Hypotheses

Ref	Expression
grptcmon.c	⊢ (𝜑 → 𝐶 = (MndToCat‘𝐺))
grptcmon.g	⊢ (𝜑 → 𝐺 ∈ Grp)
grptcmon.b	⊢ (𝜑 → 𝐵 = (Base‘𝐶))
grptcmon.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
grptcmon.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
grptcmon.h	⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
grptcepi.e	⊢ (𝜑 → 𝐸 = (Epi‘𝐶))

Assertion

Ref	Expression
grptcepi	⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌))

Proof of Theorem grptcepi

Dummy variables 𝑓 𝑔 ℎ 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2736	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2736	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		eqid 2736	. . . . 5 ⊢ (Epi‘𝐶) = (Epi‘𝐶)
5		grptcmon.c	. . . . . 6 ⊢ (𝜑 → 𝐶 = (MndToCat‘𝐺))
6		grptcmon.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Grp)
7	6	grpmndd 18934	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd)
8	5, 7	mndtccat 49432	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
9		grptcmon.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10		grptcmon.b	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
11	9, 10	eleqtrd 2837	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
12		grptcmon.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
13	12, 10	eleqtrd 2837	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
14	1, 2, 3, 4, 8, 11, 13	isepi2 17759	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝑋(Epi‘𝐶)𝑌) ↔ (𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) = (ℎ(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) → 𝑔 = ℎ))))
15	5	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝐶 = (MndToCat‘𝐺))
16	7	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝐺 ∈ Mnd)
17	10	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝐵 = (Base‘𝐶))
18	9	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑋 ∈ 𝐵)
19	12	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑌 ∈ 𝐵)
20		simpr1 1195	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑧 ∈ (Base‘𝐶))
21	20, 17	eleqtrrd 2838	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑧 ∈ 𝐵)
22		eqidd 2737	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → (comp‘𝐶) = (comp‘𝐶))
23		eqidd 2737	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧) = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧))
24	15, 16, 17, 18, 19, 21, 22, 23	mndtcco2 49430	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(+g‘𝐺)𝑓))
25	15, 16, 17, 18, 19, 21, 22, 23	mndtcco2 49430	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → (ℎ(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) = (ℎ(+g‘𝐺)𝑓))
26	24, 25	eqeq12d 2752	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) = (ℎ(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) ↔ (𝑔(+g‘𝐺)𝑓) = (ℎ(+g‘𝐺)𝑓)))
27	6	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝐺 ∈ Grp)
28		simpr2 1196	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧))
29		eqidd 2737	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → (Hom ‘𝐶) = (Hom ‘𝐶))
30	15, 16, 17, 19, 21, 29	mndtchom 49428	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → (𝑌(Hom ‘𝐶)𝑧) = (Base‘𝐺))
31	28, 30	eleqtrd 2837	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑔 ∈ (Base‘𝐺))
32		simpr3 1197	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))
33	32, 30	eleqtrd 2837	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → ℎ ∈ (Base‘𝐺))
34		simplr 768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌))
35	15, 16, 17, 18, 19, 29	mndtchom 49428	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → (𝑋(Hom ‘𝐶)𝑌) = (Base‘𝐺))
36	34, 35	eleqtrd 2837	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (Base‘𝐺))
37		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺)
38		eqid 2736	. . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝐺)
39	37, 38	grprcan 18961	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑔 ∈ (Base‘𝐺) ∧ ℎ ∈ (Base‘𝐺) ∧ 𝑓 ∈ (Base‘𝐺))) → ((𝑔(+g‘𝐺)𝑓) = (ℎ(+g‘𝐺)𝑓) ↔ 𝑔 = ℎ))
40	27, 31, 33, 36, 39	syl13anc 1374	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → ((𝑔(+g‘𝐺)𝑓) = (ℎ(+g‘𝐺)𝑓) ↔ 𝑔 = ℎ))
41	26, 40	bitrd 279	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) = (ℎ(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) ↔ 𝑔 = ℎ))
42	41	biimpd 229	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) = (ℎ(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) → 𝑔 = ℎ))
43	42	ralrimivvva 3191	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) → ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) = (ℎ(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) → 𝑔 = ℎ))
44	14, 43	mpbiran3d 48743	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑋(Epi‘𝐶)𝑌) ↔ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)))
45	44	eqrdv 2734	. 2 ⊢ (𝜑 → (𝑋(Epi‘𝐶)𝑌) = (𝑋(Hom ‘𝐶)𝑌))
46		grptcepi.e	. . 3 ⊢ (𝜑 → 𝐸 = (Epi‘𝐶))
47	46	oveqd 7427	. 2 ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑋(Epi‘𝐶)𝑌))
48		grptcmon.h	. . 3 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
49	48	oveqd 7427	. 2 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
50	45, 47, 49	3eqtr4d 2781	1 ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ⟨cop 4612  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  +_gcplusg 17276  Hom chom 17287  compcco 17288  Epicepi 17747  Mndcmnd 18717  Grpcgrp 18921  MndToCatcmndtc 49421

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-ot 4615  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-tpos 8230  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12858  df-fz 13530  df-struct 17171  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-hom 17300  df-cco 17301  df-0g 17460  df-cat 17685  df-cid 17686  df-oppc 17729  df-mon 17748  df-epi 17749  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-grp 18924  df-mndtc 49422

This theorem is referenced by: (None)