Theorem mndtcval 50656

Description: Value of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)

Hypotheses

Ref	Expression
mndtcbaseu.c	⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))
mndtcbaseu.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		mndtcbaseu.c	. 2 ⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))
2		mndtcbaseu.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ Mnd)
3		sneq 4594	. . . . . 6 ⊢ (𝑚 = 𝑀 → {𝑚} = {𝑀})
4	3	opeq2d 4840	. . . . 5 ⊢ (𝑚 = 𝑀 → ⟨(Base‘ndx), {𝑚}⟩ = ⟨(Base‘ndx), {𝑀}⟩)
5		id 23	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → 𝑚 = 𝑀)
6		fveq2 6883	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
7	5, 5, 6	oteq123d 4848	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ⟨𝑚, 𝑚, (Base‘𝑚)⟩ = ⟨𝑀, 𝑀, (Base‘𝑀)⟩)
8	7	sneqd 4596	. . . . . 6 ⊢ (𝑚 = 𝑀 → {⟨𝑚, 𝑚, (Base‘𝑚)⟩} = {⟨𝑀, 𝑀, (Base‘𝑀)⟩})
9	8	opeq2d 4840	. . . . 5 ⊢ (𝑚 = 𝑀 → ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩ = ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩)
10	5, 5, 5	oteq123d 4848	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → ⟨𝑚, 𝑚, 𝑚⟩ = ⟨𝑀, 𝑀, 𝑀⟩)
11		fveq2 6883	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
12	10, 11	opeq12d 4841	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩ = ⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩)
13	12	sneqd 4596	. . . . . 6 ⊢ (𝑚 = 𝑀 → {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩} = {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩})
14	13	opeq2d 4840	. . . . 5 ⊢ (𝑚 = 𝑀 → ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩ = ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩)
15	4, 9, 14	tpeq123d 4709	. . . 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 50655	. . . 4 ⊢ MndToCat = (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩})
17		tpex 7760	. . . 4 ⊢ {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩} ∈ V
18	15, 16, 17	fvmpt 6991	. . 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 18	. 2 ⊢ (𝜑 → (MndToCat‘𝑀) = {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩})
20	1, 19	eqtrd 2796	1 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {csn 4584  {ctp 4588  ⟨cop 4590  ⟨cotp 4592  ‘cfv 6537  ndxcnx 17364  Basecbs 17380  +_gcplusg 17421  Hom chom 17432  compcco 17433  Mndcmnd 18916  MndToCatcmndtc 50654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6493  df-fun 6539  df-fv 6545  df-mndtc 50655

This theorem is used by:  mndtcbasval  50657  mndtchom  50661  mndtcco  50662