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	⊢ (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 4536	. . . . . 6 ⊢ (𝑚 = 𝑀 → {𝑚} = {𝑀})
4	3	opeq2d 4778	. . . . 5 ⊢ (𝑚 = 𝑀 → ⟨(Base‘ndx), {𝑚}⟩ = ⟨(Base‘ndx), {𝑀}⟩)
5		id 22	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → 𝑚 = 𝑀)
6		fveq2 6686	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
7	5, 5, 6	oteq123d 4786	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ⟨𝑚, 𝑚, (Base‘𝑚)⟩ = ⟨𝑀, 𝑀, (Base‘𝑀)⟩)
8	7	sneqd 4538	. . . . . 6 ⊢ (𝑚 = 𝑀 → {⟨𝑚, 𝑚, (Base‘𝑚)⟩} = {⟨𝑀, 𝑀, (Base‘𝑀)⟩})
9	8	opeq2d 4778	. . . . 5 ⊢ (𝑚 = 𝑀 → ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩ = ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩)
10	5, 5, 5	oteq123d 4786	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → ⟨𝑚, 𝑚, 𝑚⟩ = ⟨𝑀, 𝑀, 𝑀⟩)
11		fveq2 6686	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
12	10, 11	opeq12d 4779	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩ = ⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩)
13	12	sneqd 4538	. . . . . 6 ⊢ (𝑚 = 𝑀 → {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩} = {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩})
14	13	opeq2d 4778	. . . . 5 ⊢ (𝑚 = 𝑀 → ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩ = ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩)
15	4, 9, 14	tpeq123d 4649	. . . 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 45866	. . . 4 ⊢ MndToCat = (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩})
17		tpex 7500	. . . 4 ⊢ {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩} ∈ V
18	15, 16, 17	fvmpt 6787	. . 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 2774	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 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