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Theorem grptcmon 47928
Description: All morphisms in a category converted from a group are monomorphisms. (Contributed by Zhi Wang, 23-Sep-2024.)
Hypotheses
Ref Expression
grptcmon.c (𝜑𝐶 = (MndToCat‘𝐺))
grptcmon.g (𝜑𝐺 ∈ Grp)
grptcmon.b (𝜑𝐵 = (Base‘𝐶))
grptcmon.x (𝜑𝑋𝐵)
grptcmon.y (𝜑𝑌𝐵)
grptcmon.h (𝜑𝐻 = (Hom ‘𝐶))
grptcmon.m (𝜑𝑀 = (Mono‘𝐶))
Assertion
Ref Expression
grptcmon (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌))

Proof of Theorem grptcmon
Dummy variables 𝑓 𝑔 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2724 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2724 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2724 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
4 eqid 2724 . . . . 5 (Mono‘𝐶) = (Mono‘𝐶)
5 grptcmon.c . . . . . 6 (𝜑𝐶 = (MndToCat‘𝐺))
6 grptcmon.g . . . . . . 7 (𝜑𝐺 ∈ Grp)
76grpmndd 18868 . . . . . 6 (𝜑𝐺 ∈ Mnd)
85, 7mndtccat 47926 . . . . 5 (𝜑𝐶 ∈ Cat)
9 grptcmon.x . . . . . 6 (𝜑𝑋𝐵)
10 grptcmon.b . . . . . 6 (𝜑𝐵 = (Base‘𝐶))
119, 10eleqtrd 2827 . . . . 5 (𝜑𝑋 ∈ (Base‘𝐶))
12 grptcmon.y . . . . . 6 (𝜑𝑌𝐵)
1312, 10eleqtrd 2827 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐶))
141, 2, 3, 4, 8, 11, 13ismon2 17682 . . . 4 (𝜑 → (𝑓 ∈ (𝑋(Mono‘𝐶)𝑌) ↔ (𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)∀ ∈ (𝑧(Hom ‘𝐶)𝑋)((𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)) → 𝑔 = ))))
155ad2antrr 723 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝐶 = (MndToCat‘𝐺))
167ad2antrr 723 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝐺 ∈ Mnd)
1710ad2antrr 723 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝐵 = (Base‘𝐶))
18 simpr1 1191 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑧 ∈ (Base‘𝐶))
1918, 17eleqtrrd 2828 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑧𝐵)
209ad2antrr 723 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑋𝐵)
2112ad2antrr 723 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑌𝐵)
22 eqidd 2725 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → (comp‘𝐶) = (comp‘𝐶))
23 eqidd 2725 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → (⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌) = (⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌))
2415, 16, 17, 19, 20, 21, 22, 23mndtcco2 47924 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → (𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝑓(+g𝐺)𝑔))
2515, 16, 17, 19, 20, 21, 22, 23mndtcco2 47924 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → (𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)) = (𝑓(+g𝐺)))
2624, 25eqeq12d 2740 . . . . . . 7 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → ((𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)) ↔ (𝑓(+g𝐺)𝑔) = (𝑓(+g𝐺))))
276ad2antrr 723 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝐺 ∈ Grp)
28 simpr2 1192 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋))
29 eqidd 2725 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → (Hom ‘𝐶) = (Hom ‘𝐶))
3015, 16, 17, 19, 20, 29mndtchom 47922 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → (𝑧(Hom ‘𝐶)𝑋) = (Base‘𝐺))
3128, 30eleqtrd 2827 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑔 ∈ (Base‘𝐺))
32 simpr3 1193 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → ∈ (𝑧(Hom ‘𝐶)𝑋))
3332, 30eleqtrd 2827 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → ∈ (Base‘𝐺))
34 simplr 766 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌))
3515, 16, 17, 20, 21, 29mndtchom 47922 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → (𝑋(Hom ‘𝐶)𝑌) = (Base‘𝐺))
3634, 35eleqtrd 2827 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑓 ∈ (Base‘𝐺))
37 eqid 2724 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
38 eqid 2724 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
3937, 38grplcan 18922 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑔 ∈ (Base‘𝐺) ∧ ∈ (Base‘𝐺) ∧ 𝑓 ∈ (Base‘𝐺))) → ((𝑓(+g𝐺)𝑔) = (𝑓(+g𝐺)) ↔ 𝑔 = ))
4027, 31, 33, 36, 39syl13anc 1369 . . . . . . 7 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → ((𝑓(+g𝐺)𝑔) = (𝑓(+g𝐺)) ↔ 𝑔 = ))
4126, 40bitrd 279 . . . . . 6 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → ((𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)) ↔ 𝑔 = ))
4241biimpd 228 . . . . 5 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → ((𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)) → 𝑔 = ))
4342ralrimivvva 3195 . . . 4 ((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) → ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)∀ ∈ (𝑧(Hom ‘𝐶)𝑋)((𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)) → 𝑔 = ))
4414, 43mpbiran3d 47695 . . 3 (𝜑 → (𝑓 ∈ (𝑋(Mono‘𝐶)𝑌) ↔ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)))
4544eqrdv 2722 . 2 (𝜑 → (𝑋(Mono‘𝐶)𝑌) = (𝑋(Hom ‘𝐶)𝑌))
46 grptcmon.m . . 3 (𝜑𝑀 = (Mono‘𝐶))
4746oveqd 7419 . 2 (𝜑 → (𝑋𝑀𝑌) = (𝑋(Mono‘𝐶)𝑌))
48 grptcmon.h . . 3 (𝜑𝐻 = (Hom ‘𝐶))
4948oveqd 7419 . 2 (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
5045, 47, 493eqtr4d 2774 1 (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3053  cop 4627  cfv 6534  (class class class)co 7402  Basecbs 17145  +gcplusg 17198  Hom chom 17209  compcco 17210  Monocmon 17676  Mndcmnd 18659  Grpcgrp 18855  MndToCatcmndtc 47915
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 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-tp 4626  df-op 4628  df-ot 4630  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11444  df-neg 11445  df-nn 12211  df-2 12273  df-3 12274  df-4 12275  df-5 12276  df-6 12277  df-7 12278  df-8 12279  df-9 12280  df-n0 12471  df-z 12557  df-dec 12676  df-uz 12821  df-fz 13483  df-struct 17081  df-slot 17116  df-ndx 17128  df-base 17146  df-hom 17222  df-cco 17223  df-0g 17388  df-cat 17613  df-cid 17614  df-mon 17678  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-minusg 18859  df-mndtc 47916
This theorem is referenced by: (None)
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