Theorem grptcmon 49190

Description: All morphisms in a category converted from a group are monomorphisms. (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))
grptcmon.m	⊢ (𝜑 → 𝑀 = (Mono‘𝐶))

Assertion

Ref	Expression
grptcmon	⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌))

Proof of Theorem grptcmon

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

Step	Hyp	Ref	Expression
1		eqid 2737	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2737	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2737	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		eqid 2737	. . . . 5 ⊢ (Mono‘𝐶) = (Mono‘𝐶)
5		grptcmon.c	. . . . . 6 ⊢ (𝜑 → 𝐶 = (MndToCat‘𝐺))
6		grptcmon.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Grp)
7	6	grpmndd 18964	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd)
8	5, 7	mndtccat 49185	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
9		grptcmon.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10		grptcmon.b	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
11	9, 10	eleqtrd 2843	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
12		grptcmon.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
13	12, 10	eleqtrd 2843	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
14	1, 2, 3, 4, 8, 11, 13	ismon2 17778	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝑋(Mono‘𝐶)𝑌) ↔ (𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)((𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ))))
15	5	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝐶 = (MndToCat‘𝐺))
16	7	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝐺 ∈ Mnd)
17	10	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝐵 = (Base‘𝐶))
18		simpr1 1195	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑧 ∈ (Base‘𝐶))
19	18, 17	eleqtrrd 2844	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑧 ∈ 𝐵)
20	9	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑋 ∈ 𝐵)
21	12	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑌 ∈ 𝐵)
22		eqidd 2738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → (comp‘𝐶) = (comp‘𝐶))
23		eqidd 2738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → (⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌) = (⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌))
24	15, 16, 17, 19, 20, 21, 22, 23	mndtcco2 49183	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → (𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝑓(+g‘𝐺)𝑔))
25	15, 16, 17, 19, 20, 21, 22, 23	mndtcco2 49183	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → (𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)ℎ) = (𝑓(+g‘𝐺)ℎ))
26	24, 25	eqeq12d 2753	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → ((𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)ℎ) ↔ (𝑓(+g‘𝐺)𝑔) = (𝑓(+g‘𝐺)ℎ)))
27	6	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝐺 ∈ Grp)
28		simpr2 1196	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋))
29		eqidd 2738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → (Hom ‘𝐶) = (Hom ‘𝐶))
30	15, 16, 17, 19, 20, 29	mndtchom 49181	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → (𝑧(Hom ‘𝐶)𝑋) = (Base‘𝐺))
31	28, 30	eleqtrd 2843	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑔 ∈ (Base‘𝐺))
32		simpr3 1197	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))
33	32, 30	eleqtrd 2843	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → ℎ ∈ (Base‘𝐺))
34		simplr 769	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌))
35	15, 16, 17, 20, 21, 29	mndtchom 49181	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → (𝑋(Hom ‘𝐶)𝑌) = (Base‘𝐺))
36	34, 35	eleqtrd 2843	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑓 ∈ (Base‘𝐺))
37		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺)
38		eqid 2737	. . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝐺)
39	37, 38	grplcan 19018	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑔 ∈ (Base‘𝐺) ∧ ℎ ∈ (Base‘𝐺) ∧ 𝑓 ∈ (Base‘𝐺))) → ((𝑓(+g‘𝐺)𝑔) = (𝑓(+g‘𝐺)ℎ) ↔ 𝑔 = ℎ))
40	27, 31, 33, 36, 39	syl13anc 1374	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → ((𝑓(+g‘𝐺)𝑔) = (𝑓(+g‘𝐺)ℎ) ↔ 𝑔 = ℎ))
41	26, 40	bitrd 279	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → ((𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)ℎ) ↔ 𝑔 = ℎ))
42	41	biimpd 229	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → ((𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ))
43	42	ralrimivvva 3205	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) → ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)((𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ))
44	14, 43	mpbiran3d 48717	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑋(Mono‘𝐶)𝑌) ↔ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)))
45	44	eqrdv 2735	. 2 ⊢ (𝜑 → (𝑋(Mono‘𝐶)𝑌) = (𝑋(Hom ‘𝐶)𝑌))
46		grptcmon.m	. . 3 ⊢ (𝜑 → 𝑀 = (Mono‘𝐶))
47	46	oveqd 7448	. 2 ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋(Mono‘𝐶)𝑌))
48		grptcmon.h	. . 3 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
49	48	oveqd 7448	. 2 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
50	45, 47, 49	3eqtr4d 2787	1 ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ⟨cop 4632  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  +_gcplusg 17297  Hom chom 17308  compcco 17309  Monocmon 17772  Mndcmnd 18747  Grpcgrp 18951  MndToCatcmndtc 49174

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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-hom 17321  df-cco 17322  df-0g 17486  df-cat 17711  df-cid 17712  df-mon 17774  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-mndtc 49175

This theorem is referenced by: (None)