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Theorem grptcepi 47882
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 2731 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2731 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2731 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
4 eqid 2731 . . . . 5 (Epi‘𝐶) = (Epi‘𝐶)
5 grptcmon.c . . . . . 6 (𝜑𝐶 = (MndToCat‘𝐺))
6 grptcmon.g . . . . . . 7 (𝜑𝐺 ∈ Grp)
76grpmndd 18874 . . . . . 6 (𝜑𝐺 ∈ Mnd)
85, 7mndtccat 47879 . . . . 5 (𝜑𝐶 ∈ Cat)
9 grptcmon.x . . . . . 6 (𝜑𝑋𝐵)
10 grptcmon.b . . . . . 6 (𝜑𝐵 = (Base‘𝐶))
119, 10eleqtrd 2834 . . . . 5 (𝜑𝑋 ∈ (Base‘𝐶))
12 grptcmon.y . . . . . 6 (𝜑𝑌𝐵)
1312, 10eleqtrd 2834 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐶))
141, 2, 3, 4, 8, 11, 13isepi2 17695 . . . 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 2835 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑧𝐵)
22 eqidd 2732 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → (comp‘𝐶) = (comp‘𝐶))
23 eqidd 2732 . . . . . . . . 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 47877 . . . . . . . 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 47877 . . . . . . . 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 2747 . . . . . . 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 2732 . . . . . . . . . 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 47875 . . . . . . . . 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 2834 . . . . . . . 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 2834 . . . . . . . 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 47875 . . . . . . . . 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 2834 . . . . . . . 8 (((𝜑𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (Base‘𝐺))
37 eqid 2731 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
38 eqid 2731 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
3937, 38grprcan 18901 . . . . . . . 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 279 . . . . . 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 3202 . . . 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 47647 . . 3 (𝜑 → (𝑓 ∈ (𝑋(Epi‘𝐶)𝑌) ↔ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)))
4544eqrdv 2729 . 2 (𝜑 → (𝑋(Epi‘𝐶)𝑌) = (𝑋(Hom ‘𝐶)𝑌))
46 grptcepi.e . . 3 (𝜑𝐸 = (Epi‘𝐶))
4746oveqd 7429 . 2 (𝜑 → (𝑋𝐸𝑌) = (𝑋(Epi‘𝐶)𝑌))
48 grptcmon.h . . 3 (𝜑𝐻 = (Hom ‘𝐶))
4948oveqd 7429 . 2 (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
5045, 47, 493eqtr4d 2781 1 (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060  cop 4634  cfv 6543  (class class class)co 7412  Basecbs 17151  +gcplusg 17204  Hom chom 17215  compcco 17216  Epicepi 17683  Mndcmnd 18665  Grpcgrp 18861  MndToCatcmndtc 47868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-ot 4637  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-tpos 8217  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-z 12566  df-dec 12685  df-uz 12830  df-fz 13492  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-hom 17228  df-cco 17229  df-0g 17394  df-cat 17619  df-cid 17620  df-oppc 17663  df-mon 17684  df-epi 17685  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-grp 18864  df-mndtc 47869
This theorem is referenced by: (None)
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