Theorem grptcmon 48763

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 2740	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2740	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2740	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		eqid 2740	. . . . 5 ⊢ (Mono‘𝐶) = (Mono‘𝐶)
5		grptcmon.c	. . . . . 6 ⊢ (𝜑 → 𝐶 = (MndToCat‘𝐺))
6		grptcmon.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Grp)
7	6	grpmndd 18986	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd)
8	5, 7	mndtccat 48761	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
9		grptcmon.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10		grptcmon.b	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
11	9, 10	eleqtrd 2846	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
12		grptcmon.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
13	12, 10	eleqtrd 2846	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
14	1, 2, 3, 4, 8, 11, 13	ismon2 17795	. . . 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 725	. . . . . . . . 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 725	. . . . . . . . 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 725	. . . . . . . . 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 1194	. . . . . . . . . 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 2847	. . . . . . . . 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 725	. . . . . . . . 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 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑌 ∈ 𝐵)
22		eqidd 2741	. . . . . . . . 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 2741	. . . . . . . . 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 48759	. . . . . . . 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 48759	. . . . . . . 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 2756	. . . . . . 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 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝐺 ∈ Grp)
28		simpr2 1195	. . . . . . . . 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 2741	. . . . . . . . . 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 48757	. . . . . . . . 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 2846	. . . . . . . 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 1196	. . . . . . . . 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 2846	. . . . . . . 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 768	. . . . . . . . 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 48757	. . . . . . . . 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 2846	. . . . . . . 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 2740	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺)
38		eqid 2740	. . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝐺)
39	37, 38	grplcan 19040	. . . . . . . 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 1372	. . . . . . 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 3211	. . . 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 48530	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑋(Mono‘𝐶)𝑌) ↔ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)))
45	44	eqrdv 2738	. 2 ⊢ (𝜑 → (𝑋(Mono‘𝐶)𝑌) = (𝑋(Hom ‘𝐶)𝑌))
46		grptcmon.m	. . 3 ⊢ (𝜑 → 𝑀 = (Mono‘𝐶))
47	46	oveqd 7465	. 2 ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋(Mono‘𝐶)𝑌))
48		grptcmon.h	. . 3 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
49	48	oveqd 7465	. 2 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
50	45, 47, 49	3eqtr4d 2790	1 ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ⟨cop 4654  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  Hom chom 17322  compcco 17323  Monocmon 17789  Mndcmnd 18772  Grpcgrp 18973  MndToCatcmndtc 48750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-struct 17194  df-slot 17229  df-ndx 17241  df-base 17259  df-hom 17335  df-cco 17336  df-0g 17501  df-cat 17726  df-cid 17727  df-mon 17791  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977  df-mndtc 48751

This theorem is referenced by: (None)