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Theorem mndtcval 45867
Description: Value of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
Hypotheses
Ref Expression
mndtcbas.c (𝜑𝐶 = (MndToCat‘𝑀))
mndtcbas.m (𝜑𝑀 ∈ Mnd)
Assertion
Ref Expression
mndtcval (𝜑𝐶 = {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g𝑀)⟩}⟩})

Proof of Theorem mndtcval
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 mndtcbas.c . 2 (𝜑𝐶 = (MndToCat‘𝑀))
2 mndtcbas.m . . 3 (𝜑𝑀 ∈ Mnd)
3 sneq 4536 . . . . . 6 (𝑚 = 𝑀 → {𝑚} = {𝑀})
43opeq2d 4778 . . . . 5 (𝑚 = 𝑀 → ⟨(Base‘ndx), {𝑚}⟩ = ⟨(Base‘ndx), {𝑀}⟩)
5 id 22 . . . . . . . 8 (𝑚 = 𝑀𝑚 = 𝑀)
6 fveq2 6686 . . . . . . . 8 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
75, 5, 6oteq123d 4786 . . . . . . 7 (𝑚 = 𝑀 → ⟨𝑚, 𝑚, (Base‘𝑚)⟩ = ⟨𝑀, 𝑀, (Base‘𝑀)⟩)
87sneqd 4538 . . . . . 6 (𝑚 = 𝑀 → {⟨𝑚, 𝑚, (Base‘𝑚)⟩} = {⟨𝑀, 𝑀, (Base‘𝑀)⟩})
98opeq2d 4778 . . . . 5 (𝑚 = 𝑀 → ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩ = ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩)
105, 5, 5oteq123d 4786 . . . . . . . 8 (𝑚 = 𝑀 → ⟨𝑚, 𝑚, 𝑚⟩ = ⟨𝑀, 𝑀, 𝑀⟩)
11 fveq2 6686 . . . . . . . 8 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
1210, 11opeq12d 4779 . . . . . . 7 (𝑚 = 𝑀 → ⟨⟨𝑚, 𝑚, 𝑚⟩, (+g𝑚)⟩ = ⟨⟨𝑀, 𝑀, 𝑀⟩, (+g𝑀)⟩)
1312sneqd 4538 . . . . . 6 (𝑚 = 𝑀 → {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g𝑚)⟩} = {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g𝑀)⟩})
1413opeq2d 4778 . . . . 5 (𝑚 = 𝑀 → ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g𝑚)⟩}⟩ = ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g𝑀)⟩}⟩)
154, 9, 14tpeq123d 4649 . . . 4 (𝑚 = 𝑀 → {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g𝑚)⟩}⟩} = {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g𝑀)⟩}⟩})
16 df-mndtc 45866 . . . 4 MndToCat = (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g𝑚)⟩}⟩})
17 tpex 7500 . . . 4 {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g𝑀)⟩}⟩} ∈ V
1815, 16, 17fvmpt 6787 . . 3 (𝑀 ∈ Mnd → (MndToCat‘𝑀) = {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g𝑀)⟩}⟩})
192, 18syl 17 . 2 (𝜑 → (MndToCat‘𝑀) = {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g𝑀)⟩}⟩})
201, 19eqtrd 2774 1 (𝜑𝐶 = {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g𝑀)⟩}⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114  {csn 4526  {ctp 4530  cop 4532  cotp 4534  cfv 6349  ndxcnx 16595  Basecbs 16598  +gcplusg 16680  Hom chom 16691  compcco 16692  Mndcmnd 18039  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-sep 5177  ax-nul 5184  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  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-ral 3059  df-rex 3060  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-ot 4535  df-uni 4807  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-iota 6307  df-fun 6351  df-fv 6357  df-mndtc 45866
This theorem is referenced by:  mndtcbasval  45868  mndtchom  45871  mndtcco  45872
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