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Theorem mndtcval 47783
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 (πœ‘ β†’ 𝐢 = (MndToCatβ€˜π‘€))
mndtcbas.m (πœ‘ β†’ 𝑀 ∈ Mnd)
Assertion
Ref Expression
mndtcval (πœ‘ β†’ 𝐢 = {⟨(Baseβ€˜ndx), {𝑀}⟩, ⟨(Hom β€˜ndx), {βŸ¨π‘€, 𝑀, (Baseβ€˜π‘€)⟩}⟩, ⟨(compβ€˜ndx), {βŸ¨βŸ¨π‘€, 𝑀, π‘€βŸ©, (+gβ€˜π‘€)⟩}⟩})

Proof of Theorem mndtcval
Dummy variable π‘š is distinct from all other variables.
StepHypRef Expression
1 mndtcbas.c . 2 (πœ‘ β†’ 𝐢 = (MndToCatβ€˜π‘€))
2 mndtcbas.m . . 3 (πœ‘ β†’ 𝑀 ∈ Mnd)
3 sneq 4638 . . . . . 6 (π‘š = 𝑀 β†’ {π‘š} = {𝑀})
43opeq2d 4880 . . . . 5 (π‘š = 𝑀 β†’ ⟨(Baseβ€˜ndx), {π‘š}⟩ = ⟨(Baseβ€˜ndx), {𝑀}⟩)
5 id 22 . . . . . . . 8 (π‘š = 𝑀 β†’ π‘š = 𝑀)
6 fveq2 6891 . . . . . . . 8 (π‘š = 𝑀 β†’ (Baseβ€˜π‘š) = (Baseβ€˜π‘€))
75, 5, 6oteq123d 4888 . . . . . . 7 (π‘š = 𝑀 β†’ βŸ¨π‘š, π‘š, (Baseβ€˜π‘š)⟩ = βŸ¨π‘€, 𝑀, (Baseβ€˜π‘€)⟩)
87sneqd 4640 . . . . . 6 (π‘š = 𝑀 β†’ {βŸ¨π‘š, π‘š, (Baseβ€˜π‘š)⟩} = {βŸ¨π‘€, 𝑀, (Baseβ€˜π‘€)⟩})
98opeq2d 4880 . . . . 5 (π‘š = 𝑀 β†’ ⟨(Hom β€˜ndx), {βŸ¨π‘š, π‘š, (Baseβ€˜π‘š)⟩}⟩ = ⟨(Hom β€˜ndx), {βŸ¨π‘€, 𝑀, (Baseβ€˜π‘€)⟩}⟩)
105, 5, 5oteq123d 4888 . . . . . . . 8 (π‘š = 𝑀 β†’ βŸ¨π‘š, π‘š, π‘šβŸ© = βŸ¨π‘€, 𝑀, π‘€βŸ©)
11 fveq2 6891 . . . . . . . 8 (π‘š = 𝑀 β†’ (+gβ€˜π‘š) = (+gβ€˜π‘€))
1210, 11opeq12d 4881 . . . . . . 7 (π‘š = 𝑀 β†’ βŸ¨βŸ¨π‘š, π‘š, π‘šβŸ©, (+gβ€˜π‘š)⟩ = βŸ¨βŸ¨π‘€, 𝑀, π‘€βŸ©, (+gβ€˜π‘€)⟩)
1312sneqd 4640 . . . . . 6 (π‘š = 𝑀 β†’ {βŸ¨βŸ¨π‘š, π‘š, π‘šβŸ©, (+gβ€˜π‘š)⟩} = {βŸ¨βŸ¨π‘€, 𝑀, π‘€βŸ©, (+gβ€˜π‘€)⟩})
1413opeq2d 4880 . . . . 5 (π‘š = 𝑀 β†’ ⟨(compβ€˜ndx), {βŸ¨βŸ¨π‘š, π‘š, π‘šβŸ©, (+gβ€˜π‘š)⟩}⟩ = ⟨(compβ€˜ndx), {βŸ¨βŸ¨π‘€, 𝑀, π‘€βŸ©, (+gβ€˜π‘€)⟩}⟩)
154, 9, 14tpeq123d 4752 . . . 4 (π‘š = 𝑀 β†’ {⟨(Baseβ€˜ndx), {π‘š}⟩, ⟨(Hom β€˜ndx), {βŸ¨π‘š, π‘š, (Baseβ€˜π‘š)⟩}⟩, ⟨(compβ€˜ndx), {βŸ¨βŸ¨π‘š, π‘š, π‘šβŸ©, (+gβ€˜π‘š)⟩}⟩} = {⟨(Baseβ€˜ndx), {𝑀}⟩, ⟨(Hom β€˜ndx), {βŸ¨π‘€, 𝑀, (Baseβ€˜π‘€)⟩}⟩, ⟨(compβ€˜ndx), {βŸ¨βŸ¨π‘€, 𝑀, π‘€βŸ©, (+gβ€˜π‘€)⟩}⟩})
16 df-mndtc 47782 . . . 4 MndToCat = (π‘š ∈ Mnd ↦ {⟨(Baseβ€˜ndx), {π‘š}⟩, ⟨(Hom β€˜ndx), {βŸ¨π‘š, π‘š, (Baseβ€˜π‘š)⟩}⟩, ⟨(compβ€˜ndx), {βŸ¨βŸ¨π‘š, π‘š, π‘šβŸ©, (+gβ€˜π‘š)⟩}⟩})
17 tpex 7736 . . . 4 {⟨(Baseβ€˜ndx), {𝑀}⟩, ⟨(Hom β€˜ndx), {βŸ¨π‘€, 𝑀, (Baseβ€˜π‘€)⟩}⟩, ⟨(compβ€˜ndx), {βŸ¨βŸ¨π‘€, 𝑀, π‘€βŸ©, (+gβ€˜π‘€)⟩}⟩} ∈ V
1815, 16, 17fvmpt 6998 . . 3 (𝑀 ∈ Mnd β†’ (MndToCatβ€˜π‘€) = {⟨(Baseβ€˜ndx), {𝑀}⟩, ⟨(Hom β€˜ndx), {βŸ¨π‘€, 𝑀, (Baseβ€˜π‘€)⟩}⟩, ⟨(compβ€˜ndx), {βŸ¨βŸ¨π‘€, 𝑀, π‘€βŸ©, (+gβ€˜π‘€)⟩}⟩})
192, 18syl 17 . 2 (πœ‘ β†’ (MndToCatβ€˜π‘€) = {⟨(Baseβ€˜ndx), {𝑀}⟩, ⟨(Hom β€˜ndx), {βŸ¨π‘€, 𝑀, (Baseβ€˜π‘€)⟩}⟩, ⟨(compβ€˜ndx), {βŸ¨βŸ¨π‘€, 𝑀, π‘€βŸ©, (+gβ€˜π‘€)⟩}⟩})
201, 19eqtrd 2772 1 (πœ‘ β†’ 𝐢 = {⟨(Baseβ€˜ndx), {𝑀}⟩, ⟨(Hom β€˜ndx), {βŸ¨π‘€, 𝑀, (Baseβ€˜π‘€)⟩}⟩, ⟨(compβ€˜ndx), {βŸ¨βŸ¨π‘€, 𝑀, π‘€βŸ©, (+gβ€˜π‘€)⟩}⟩})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  {csn 4628  {ctp 4632  βŸ¨cop 4634  βŸ¨cotp 4636  β€˜cfv 6543  ndxcnx 17128  Basecbs 17146  +gcplusg 17199  Hom chom 17210  compcco 17211  Mndcmnd 18627  MndToCatcmndtc 47781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-ot 4637  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-mndtc 47782
This theorem is referenced by:  mndtcbasval  47784  mndtchom  47788  mndtcco  47789
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