grptcmon - Mathbox for Zhi Wang

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Theorem grptcmon 48198

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 2728	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2728	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2728	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		eqid 2728	. . . . 5 ⊢ (Mono‘𝐶) = (Mono‘𝐶)
5		grptcmon.c	. . . . . 6 ⊢ (𝜑 → 𝐶 = (MndToCat‘𝐺))
6		grptcmon.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Grp)
7	6	grpmndd 18917	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd)
8	5, 7	mndtccat 48196	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
9		grptcmon.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10		grptcmon.b	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
11	9, 10	eleqtrd 2831	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
12		grptcmon.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
13	12, 10	eleqtrd 2831	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
14	1, 2, 3, 4, 8, 11, 13	ismon2 17726	. . . 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 724	. . . . . . . . 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 724	. . . . . . . . 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 724	. . . . . . . . 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 1191	. . . . . . . . . 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 2832	. . . . . . . . 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 724	. . . . . . . . 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 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑌 ∈ 𝐵)
22		eqidd 2729	. . . . . . . . 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 2729	. . . . . . . . 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 48194	. . . . . . . 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 48194	. . . . . . . 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 2744	. . . . . . 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 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝐺 ∈ Grp)
28		simpr2 1192	. . . . . . . . 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 2729	. . . . . . . . . 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 48192	. . . . . . . . 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 2831	. . . . . . . 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 1193	. . . . . . . . 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 2831	. . . . . . . 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 767	. . . . . . . . 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 48192	. . . . . . . . 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 2831	. . . . . . . 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 2728	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺)
38		eqid 2728	. . . . . . . . 9 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
39	37, 38	grplcan 18971	. . . . . . . 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 1369	. . . . . . 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 278	. . . . . 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 228	. . . . 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 3201	. . . 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 47965	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑋(Mono‘𝐶)𝑌) ↔ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)))
45	44	eqrdv 2726	. 2 ⊢ (𝜑 → (𝑋(Mono‘𝐶)𝑌) = (𝑋(Hom ‘𝐶)𝑌))
46		grptcmon.m	. . 3 ⊢ (𝜑 → 𝑀 = (Mono‘𝐶))
47	46	oveqd 7443	. 2 ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋(Mono‘𝐶)𝑌))
48		grptcmon.h	. . 3 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
49	48	oveqd 7443	. 2 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
50	45, 47, 49	3eqtr4d 2778	1 ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ⟨cop 4638 ‘cfv 6553 (class class class)co 7426 Basecbs 17189 +_gcplusg 17242 Hom chom 17253 compcco 17254 Monocmon 17720 Mndcmnd 18703 Grpcgrp 18904 MndToCatcmndtc 48185

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-ot 4641 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-fz 13527 df-struct 17125 df-slot 17160 df-ndx 17172 df-base 17190 df-hom 17266 df-cco 17267 df-0g 17432 df-cat 17657 df-cid 17658 df-mon 17722 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-grp 18907 df-minusg 18908 df-mndtc 48186

This theorem is referenced by: (None)

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