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Theorem grptcepi 45878
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 2739 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2739 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2739 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
4 eqid 2739 . . . . 5 (Epi‘𝐶) = (Epi‘𝐶)
5 grptcmon.c . . . . . 6 (𝜑𝐶 = (MndToCat‘𝐺))
6 grptcmon.g . . . . . . 7 (𝜑𝐺 ∈ Grp)
76grpmndd 18243 . . . . . 6 (𝜑𝐺 ∈ Mnd)
85, 7mndtccat 45875 . . . . 5 (𝜑𝐶 ∈ Cat)
9 grptcmon.x . . . . . 6 (𝜑𝑋𝐵)
10 grptcmon.b . . . . . 6 (𝜑𝐵 = (Base‘𝐶))
119, 10eleqtrd 2836 . . . . 5 (𝜑𝑋 ∈ (Base‘𝐶))
12 grptcmon.y . . . . . 6 (𝜑𝑌𝐵)
1312, 10eleqtrd 2836 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐶))
141, 2, 3, 4, 8, 11, 13isepi2 17128 . . . 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 726 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝐶 = (MndToCat‘𝐺))
167ad2antrr 726 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝐺 ∈ Mnd)
1710ad2antrr 726 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝐵 = (Base‘𝐶))
189ad2antrr 726 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑋𝐵)
1912ad2antrr 726 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑌𝐵)
20 simpr1 1195 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑧 ∈ (Base‘𝐶))
2120, 17eleqtrrd 2837 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑧𝐵)
22 eqidd 2740 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → (comp‘𝐶) = (comp‘𝐶))
23 eqidd 2740 . . . . . . . . 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 45873 . . . . . . . 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 45873 . . . . . . . 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 2755 . . . . . . 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 726 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝐺 ∈ Grp)
28 simpr2 1196 . . . . . . . . 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 2740 . . . . . . . . . 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 45871 . . . . . . . . 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 2836 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑔 ∈ (Base‘𝐺))
32 simpr3 1197 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → ∈ (𝑌(Hom ‘𝐶)𝑧))
3332, 30eleqtrd 2836 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → ∈ (Base‘𝐺))
34 simplr 769 . . . . . . . . 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 45871 . . . . . . . . 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 2836 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (Base‘𝐺))
37 eqid 2739 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
38 eqid 2739 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
3937, 38grprcan 18267 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑔 ∈ (Base‘𝐺) ∧ ∈ (Base‘𝐺) ∧ 𝑓 ∈ (Base‘𝐺))) → ((𝑔(+g𝐺)𝑓) = ((+g𝐺)𝑓) ↔ 𝑔 = ))
4027, 31, 33, 36, 39syl13anc 1373 . . . . . . 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 282 . . . . . 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 232 . . . . 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 3105 . . . 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, 43monepilem 45723 . . 3 (𝜑 → (𝑓 ∈ (𝑋(Epi‘𝐶)𝑌) ↔ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)))
4544eqrdv 2737 . 2 (𝜑 → (𝑋(Epi‘𝐶)𝑌) = (𝑋(Hom ‘𝐶)𝑌))
46 grptcepi.e . . 3 (𝜑𝐸 = (Epi‘𝐶))
4746oveqd 7199 . 2 (𝜑 → (𝑋𝐸𝑌) = (𝑋(Epi‘𝐶)𝑌))
48 grptcmon.h . . 3 (𝜑𝐻 = (Hom ‘𝐶))
4948oveqd 7199 . 2 (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
5045, 47, 493eqtr4d 2784 1 (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3054  cop 4532  cfv 6349  (class class class)co 7182  Basecbs 16598  +gcplusg 16680  Hom chom 16691  compcco 16692  Epicepi 17116  Mndcmnd 18039  Grpcgrp 18231  MndToCatcmndtc 45865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491  ax-cnex 10683  ax-resscn 10684  ax-1cn 10685  ax-icn 10686  ax-addcl 10687  ax-addrcl 10688  ax-mulcl 10689  ax-mulrcl 10690  ax-mulcom 10691  ax-addass 10692  ax-mulass 10693  ax-distr 10694  ax-i2m1 10695  ax-1ne0 10696  ax-1rid 10697  ax-rnegex 10698  ax-rrecex 10699  ax-cnre 10700  ax-pre-lttri 10701  ax-pre-lttrn 10702  ax-pre-ltadd 10703  ax-pre-mulgt0 10704
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-ot 4535  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-pred 6139  df-ord 6185  df-on 6186  df-lim 6187  df-suc 6188  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7139  df-ov 7185  df-oprab 7186  df-mpo 7187  df-om 7612  df-1st 7726  df-2nd 7727  df-tpos 7933  df-wrecs 7988  df-recs 8049  df-rdg 8087  df-1o 8143  df-er 8332  df-en 8568  df-dom 8569  df-sdom 8570  df-fin 8571  df-pnf 10767  df-mnf 10768  df-xr 10769  df-ltxr 10770  df-le 10771  df-sub 10962  df-neg 10963  df-nn 11729  df-2 11791  df-3 11792  df-4 11793  df-5 11794  df-6 11795  df-7 11796  df-8 11797  df-9 11798  df-n0 11989  df-z 12075  df-dec 12192  df-uz 12337  df-fz 12994  df-struct 16600  df-ndx 16601  df-slot 16602  df-base 16604  df-sets 16605  df-hom 16704  df-cco 16705  df-0g 16830  df-cat 17054  df-cid 17055  df-oppc 17098  df-mon 17117  df-epi 17118  df-mgm 17980  df-sgrp 18029  df-mnd 18040  df-grp 18234  df-mndtc 45866
This theorem is referenced by: (None)
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