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Theorem grptcmon 48902
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 2735 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2735 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2735 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
4 eqid 2735 . . . . 5 (Mono‘𝐶) = (Mono‘𝐶)
5 grptcmon.c . . . . . 6 (𝜑𝐶 = (MndToCat‘𝐺))
6 grptcmon.g . . . . . . 7 (𝜑𝐺 ∈ Grp)
76grpmndd 18977 . . . . . 6 (𝜑𝐺 ∈ Mnd)
85, 7mndtccat 48897 . . . . 5 (𝜑𝐶 ∈ Cat)
9 grptcmon.x . . . . . 6 (𝜑𝑋𝐵)
10 grptcmon.b . . . . . 6 (𝜑𝐵 = (Base‘𝐶))
119, 10eleqtrd 2841 . . . . 5 (𝜑𝑋 ∈ (Base‘𝐶))
12 grptcmon.y . . . . . 6 (𝜑𝑌𝐵)
1312, 10eleqtrd 2841 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐶))
141, 2, 3, 4, 8, 11, 13ismon2 17782 . . . 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 726 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝐶 = (MndToCat‘𝐺))
167ad2antrr 726 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝐺 ∈ Mnd)
1710ad2antrr 726 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝐵 = (Base‘𝐶))
18 simpr1 1193 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑧 ∈ (Base‘𝐶))
1918, 17eleqtrrd 2842 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑧𝐵)
209ad2antrr 726 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑋𝐵)
2112ad2antrr 726 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑌𝐵)
22 eqidd 2736 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → (comp‘𝐶) = (comp‘𝐶))
23 eqidd 2736 . . . . . . . . 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 48895 . . . . . . . 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 48895 . . . . . . . 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 2751 . . . . . . 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 726 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝐺 ∈ Grp)
28 simpr2 1194 . . . . . . . . 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 2736 . . . . . . . . . 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 48893 . . . . . . . . 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 2841 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑔 ∈ (Base‘𝐺))
32 simpr3 1195 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → ∈ (𝑧(Hom ‘𝐶)𝑋))
3332, 30eleqtrd 2841 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → ∈ (Base‘𝐺))
34 simplr 769 . . . . . . . . 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 48893 . . . . . . . . 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 2841 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → 𝑓 ∈ (Base‘𝐺))
37 eqid 2735 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
38 eqid 2735 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
3937, 38grplcan 19031 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑔 ∈ (Base‘𝐺) ∧ ∈ (Base‘𝐺) ∧ 𝑓 ∈ (Base‘𝐺))) → ((𝑓(+g𝐺)𝑔) = (𝑓(+g𝐺)) ↔ 𝑔 = ))
4027, 31, 33, 36, 39syl13anc 1371 . . . . . . 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 229 . . . . 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 3203 . . . 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 48646 . . 3 (𝜑 → (𝑓 ∈ (𝑋(Mono‘𝐶)𝑌) ↔ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)))
4544eqrdv 2733 . 2 (𝜑 → (𝑋(Mono‘𝐶)𝑌) = (𝑋(Hom ‘𝐶)𝑌))
46 grptcmon.m . . 3 (𝜑𝑀 = (Mono‘𝐶))
4746oveqd 7448 . 2 (𝜑 → (𝑋𝑀𝑌) = (𝑋(Mono‘𝐶)𝑌))
48 grptcmon.h . . 3 (𝜑𝐻 = (Hom ‘𝐶))
4948oveqd 7448 . 2 (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
5045, 47, 493eqtr4d 2785 1 (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  cop 4637  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Hom chom 17309  compcco 17310  Monocmon 17776  Mndcmnd 18760  Grpcgrp 18964  MndToCatcmndtc 48886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-hom 17322  df-cco 17323  df-0g 17488  df-cat 17713  df-cid 17714  df-mon 17778  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-mndtc 48887
This theorem is referenced by: (None)
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