Theorem mndtcval 48752

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 4658	. . . . . 6 ⊢ (𝑚 = 𝑀 → {𝑚} = {𝑀})
4	3	opeq2d 4904	. . . . 5 ⊢ (𝑚 = 𝑀 → ⟨(Base‘ndx), {𝑚}⟩ = ⟨(Base‘ndx), {𝑀}⟩)
5		id 22	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → 𝑚 = 𝑀)
6		fveq2 6920	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
7	5, 5, 6	oteq123d 4912	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ⟨𝑚, 𝑚, (Base‘𝑚)⟩ = ⟨𝑀, 𝑀, (Base‘𝑀)⟩)
8	7	sneqd 4660	. . . . . 6 ⊢ (𝑚 = 𝑀 → {⟨𝑚, 𝑚, (Base‘𝑚)⟩} = {⟨𝑀, 𝑀, (Base‘𝑀)⟩})
9	8	opeq2d 4904	. . . . 5 ⊢ (𝑚 = 𝑀 → ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩ = ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩)
10	5, 5, 5	oteq123d 4912	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → ⟨𝑚, 𝑚, 𝑚⟩ = ⟨𝑀, 𝑀, 𝑀⟩)
11		fveq2 6920	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
12	10, 11	opeq12d 4905	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩ = ⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩)
13	12	sneqd 4660	. . . . . 6 ⊢ (𝑚 = 𝑀 → {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩} = {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩})
14	13	opeq2d 4904	. . . . 5 ⊢ (𝑚 = 𝑀 → ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩ = ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩)
15	4, 9, 14	tpeq123d 4773	. . . 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 48751	. . . 4 ⊢ MndToCat = (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩})
17		tpex 7781	. . . 4 ⊢ {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩} ∈ V
18	15, 16, 17	fvmpt 7029	. . 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 2780	1 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩})

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  {csn 4648  {ctp 4652  ⟨cop 4654  ⟨cotp 4656  ‘cfv 6573  ndxcnx 17240  Basecbs 17258  +_gcplusg 17311  Hom chom 17322  compcco 17323  Mndcmnd 18772  MndToCatcmndtc 48750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-mndtc 48751

This theorem is referenced by:  mndtcbasval  48753  mndtchom  48757  mndtcco  48758