Theorem grptcepi 50339

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 2761	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2761	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2761	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		eqid 2761	. . . . 5 ⊢ (Epi‘𝐶) = (Epi‘𝐶)
5		grptcmon.c	. . . . . 6 ⊢ (𝜑 → 𝐶 = (MndToCat‘𝐺))
6		grptcmon.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Grp)
7	6	grpmndd 19012	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd)
8	5, 7	mndtccat 50333	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
9		grptcmon.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10		grptcmon.b	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
11	9, 10	eleqtrd 2863	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
12		grptcmon.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
13	12, 10	eleqtrd 2863	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
14	1, 2, 3, 4, 8, 11, 13	isepi2 17797	. . . 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 738	. . . . . . . . 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 738	. . . . . . . . 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 738	. . . . . . . . 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 738	. . . . . . . . 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 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑌 ∈ 𝐵)
20		simpr1 1211	. . . . . . . . . 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 2864	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑧 ∈ 𝐵)
22		eqidd 2762	. . . . . . . . 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 2762	. . . . . . . . 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 50331	. . . . . . . 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 50331	. . . . . . . 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 2777	. . . . . . 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 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝐺 ∈ Grp)
28		simpr2 1212	. . . . . . . . 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 2762	. . . . . . . . . 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 50329	. . . . . . . . 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 2863	. . . . . . . 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 1213	. . . . . . . . 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 2863	. . . . . . . 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 780	. . . . . . . . 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 50329	. . . . . . . . 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 2863	. . . . . . . 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 2761	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺)
38		eqid 2761	. . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝐺)
39	37, 38	grprcan 19039	. . . . . . . 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 1397	. . . . . . 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 282	. . . . . 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 232	. . . . 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 3209	. . . 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 49542	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑋(Epi‘𝐶)𝑌) ↔ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)))
45	44	eqrdv 2759	. 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 2806	1 ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141  ∀wral 3077  ⟨cop 4594  ‘cfv 6536  (class class class)co 7410  Basecbs 17268  +_gcplusg 17309  Hom chom 17320  compcco 17321  Epicepi 17785  Mndcmnd 18791  Grpcgrp 18999  MndToCatcmndtc 50322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-ot 4597  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  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 7862  df-1st 7985  df-2nd 7986  df-tpos 8221  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-9 12309  df-n0 12504  df-z 12591  df-dec 12711  df-uz 12862  df-fz 13535  df-struct 17206  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-hom 17333  df-cco 17334  df-0g 17493  df-cat 17723  df-cid 17724  df-oppc 17767  df-mon 17786  df-epi 17787  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-grp 19002  df-mndtc 50323

This theorem is referenced by: (None)