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Theorem grptcepi 46378
Description: All morphisms in a category converted from a group are epimorphisms. (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))
grptcepi.e (𝜑𝐸 = (Epi‘𝐶))
Assertion
Ref Expression
grptcepi (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌))

Proof of Theorem grptcepi
Dummy variables 𝑓 𝑔 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2738 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2738 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2738 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
4 eqid 2738 . . . . 5 (Epi‘𝐶) = (Epi‘𝐶)
5 grptcmon.c . . . . . 6 (𝜑𝐶 = (MndToCat‘𝐺))
6 grptcmon.g . . . . . . 7 (𝜑𝐺 ∈ Grp)
76grpmndd 18589 . . . . . 6 (𝜑𝐺 ∈ Mnd)
85, 7mndtccat 46375 . . . . 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, 13isepi2 17453 . . . 4 (𝜑 → (𝑓 ∈ (𝑋(Epi‘𝐶)𝑌) ↔ (𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) → 𝑔 = ))))
155ad2antrr 723 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝐶 = (MndToCat‘𝐺))
167ad2antrr 723 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝐺 ∈ Mnd)
1710ad2antrr 723 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝐵 = (Base‘𝐶))
189ad2antrr 723 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑋𝐵)
1912ad2antrr 723 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑌𝐵)
20 simpr1 1193 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑧 ∈ (Base‘𝐶))
2120, 17eleqtrrd 2842 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑧𝐵)
22 eqidd 2739 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → (comp‘𝐶) = (comp‘𝐶))
23 eqidd 2739 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧) = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧))
2415, 16, 17, 18, 19, 21, 22, 23mndtcco2 46373 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(+g𝐺)𝑓))
2515, 16, 17, 18, 19, 21, 22, 23mndtcco2 46373 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) = ((+g𝐺)𝑓))
2624, 25eqeq12d 2754 . . . . . . 7 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) ↔ (𝑔(+g𝐺)𝑓) = ((+g𝐺)𝑓)))
276ad2antrr 723 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝐺 ∈ Grp)
28 simpr2 1194 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧))
29 eqidd 2739 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → (Hom ‘𝐶) = (Hom ‘𝐶))
3015, 16, 17, 19, 21, 29mndtchom 46371 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → (𝑌(Hom ‘𝐶)𝑧) = (Base‘𝐺))
3128, 30eleqtrd 2841 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑔 ∈ (Base‘𝐺))
32 simpr3 1195 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → ∈ (𝑌(Hom ‘𝐶)𝑧))
3332, 30eleqtrd 2841 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → ∈ (Base‘𝐺))
34 simplr 766 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌))
3515, 16, 17, 18, 19, 29mndtchom 46371 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → (𝑋(Hom ‘𝐶)𝑌) = (Base‘𝐺))
3634, 35eleqtrd 2841 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (Base‘𝐺))
37 eqid 2738 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
38 eqid 2738 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
3937, 38grprcan 18613 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑔 ∈ (Base‘𝐺) ∧ ∈ (Base‘𝐺) ∧ 𝑓 ∈ (Base‘𝐺))) → ((𝑔(+g𝐺)𝑓) = ((+g𝐺)𝑓) ↔ 𝑔 = ))
4027, 31, 33, 36, 39syl13anc 1371 . . . . . . 7 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → ((𝑔(+g𝐺)𝑓) = ((+g𝐺)𝑓) ↔ 𝑔 = ))
4126, 40bitrd 278 . . . . . 6 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) ↔ 𝑔 = ))
4241biimpd 228 . . . . 5 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) → 𝑔 = ))
4342ralrimivvva 3127 . . . 4 ((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) → ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝑓) → 𝑔 = ))
4414, 43mpbiran3d 46142 . . 3 (𝜑 → (𝑓 ∈ (𝑋(Epi‘𝐶)𝑌) ↔ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)))
4544eqrdv 2736 . 2 (𝜑 → (𝑋(Epi‘𝐶)𝑌) = (𝑋(Hom ‘𝐶)𝑌))
46 grptcepi.e . . 3 (𝜑𝐸 = (Epi‘𝐶))
4746oveqd 7292 . 2 (𝜑 → (𝑋𝐸𝑌) = (𝑋(Epi‘𝐶)𝑌))
48 grptcmon.h . . 3 (𝜑𝐻 = (Hom ‘𝐶))
4948oveqd 7292 . 2 (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
5045, 47, 493eqtr4d 2788 1 (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  cop 4567  cfv 6433  (class class class)co 7275  Basecbs 16912  +gcplusg 16962  Hom chom 16973  compcco 16974  Epicepi 17441  Mndcmnd 18385  Grpcgrp 18577  MndToCatcmndtc 46364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-ot 4570  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-tpos 8042  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-fz 13240  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-hom 16986  df-cco 16987  df-0g 17152  df-cat 17377  df-cid 17378  df-oppc 17421  df-mon 17442  df-epi 17443  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-mndtc 46365
This theorem is referenced by: (None)
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