Theorem mndtcval 47705

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	⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))
mndtcbas.m	⊢ (𝜑 → 𝑀 ∈ Mnd)

Assertion

Ref	Expression
mndtcval	⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩})

Proof of Theorem mndtcval

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mndtcbas.c	. 2 ⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))
2		mndtcbas.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ Mnd)
3		sneq 4639	. . . . . 6 ⊢ (𝑚 = 𝑀 → {𝑚} = {𝑀})
4	3	opeq2d 4881	. . . . 5 ⊢ (𝑚 = 𝑀 → ⟨(Base‘ndx), {𝑚}⟩ = ⟨(Base‘ndx), {𝑀}⟩)
5		id 22	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → 𝑚 = 𝑀)
6		fveq2 6892	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
7	5, 5, 6	oteq123d 4889	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ⟨𝑚, 𝑚, (Base‘𝑚)⟩ = ⟨𝑀, 𝑀, (Base‘𝑀)⟩)
8	7	sneqd 4641	. . . . . 6 ⊢ (𝑚 = 𝑀 → {⟨𝑚, 𝑚, (Base‘𝑚)⟩} = {⟨𝑀, 𝑀, (Base‘𝑀)⟩})
9	8	opeq2d 4881	. . . . 5 ⊢ (𝑚 = 𝑀 → ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩ = ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩)
10	5, 5, 5	oteq123d 4889	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → ⟨𝑚, 𝑚, 𝑚⟩ = ⟨𝑀, 𝑀, 𝑀⟩)
11		fveq2 6892	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
12	10, 11	opeq12d 4882	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩ = ⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩)
13	12	sneqd 4641	. . . . . 6 ⊢ (𝑚 = 𝑀 → {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩} = {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩})
14	13	opeq2d 4881	. . . . 5 ⊢ (𝑚 = 𝑀 → ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩ = ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩)
15	4, 9, 14	tpeq123d 4753	. . . 4 ⊢ (𝑚 = 𝑀 → {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩} = {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩})
16		df-mndtc 47704	. . . 4 ⊢ MndToCat = (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩})
17		tpex 7734	. . . 4 ⊢ {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩} ∈ V
18	15, 16, 17	fvmpt 6999	. . 3 ⊢ (𝑀 ∈ Mnd → (MndToCat‘𝑀) = {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩})
19	2, 18	syl 17	. 2 ⊢ (𝜑 → (MndToCat‘𝑀) = {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩})
20	1, 19	eqtrd 2773	1 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩})

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  {csn 4629  {ctp 4633  ⟨cop 4635  ⟨cotp 4637  ‘cfv 6544  ndxcnx 17126  Basecbs 17144  +_gcplusg 17197  Hom chom 17208  compcco 17209  Mndcmnd 18625  MndToCatcmndtc 47703

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-ot 4638  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-mndtc 47704

This theorem is referenced by:  mndtcbasval  47706  mndtchom  47710  mndtcco  47711