Theorem mnringnmulrd 45039

Description: Components of a monoid ring other than its ring product match its underlying free module. (Contributed by Rohan Ridenour, 14-May-2024.) (Revised by AV, 1-Nov-2024.)

Hypotheses

Ref	Expression
mnringnmulrd.1	⊢ 𝐹 = (𝑅 MndRing 𝑀)
mnringnmulrd.2	⊢ 𝐸 = Slot (𝐸‘ndx)
mnringnmulrd.4	⊢ (𝐸‘ndx) ≠ (.r‘ndx)
mnringnmulrd.5	⊢ 𝐴 = (Base‘𝑀)
mnringnmulrd.6	⊢ 𝑉 = (𝑅 freeLMod 𝐴)
mnringnmulrd.7	⊢ (𝜑 → 𝑅 ∈ 𝑈)
mnringnmulrd.8	⊢ (𝜑 → 𝑀 ∈ 𝑊)

Assertion

Ref	Expression
mnringnmulrd	⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹))

Proof of Theorem mnringnmulrd

Dummy variables 𝑎 𝑏 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mnringnmulrd.2	. . 3 ⊢ 𝐸 = Slot (𝐸‘ndx)
2		mnringnmulrd.4	. . 3 ⊢ (𝐸‘ndx) ≠ (.r‘ndx)
3	1, 2	setsnid 17304	. 2 ⊢ (𝐸‘𝑉) = (𝐸‘(𝑉 sSet ⟨(.r‘ndx), (𝑥 ∈ (Base‘𝑉), 𝑦 ∈ (Base‘𝑉) ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑥‘𝑎)(.r‘𝑅)(𝑦‘𝑏)), (0g‘𝑅))))))⟩))
4		mnringnmulrd.1	. . . 4 ⊢ 𝐹 = (𝑅 MndRing 𝑀)
5		eqid 2762	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
6		eqid 2762	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
7		mnringnmulrd.5	. . . 4 ⊢ 𝐴 = (Base‘𝑀)
8		eqid 2762	. . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀)
9		mnringnmulrd.6	. . . 4 ⊢ 𝑉 = (𝑅 freeLMod 𝐴)
10		eqid 2762	. . . 4 ⊢ (Base‘𝑉) = (Base‘𝑉)
11		mnringnmulrd.7	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑈)
12		mnringnmulrd.8	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑊)
13	4, 5, 6, 7, 8, 9, 10, 11, 12	mnringvald 45038	. . 3 ⊢ (𝜑 → 𝐹 = (𝑉 sSet ⟨(.r‘ndx), (𝑥 ∈ (Base‘𝑉), 𝑦 ∈ (Base‘𝑉) ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑥‘𝑎)(.r‘𝑅)(𝑦‘𝑏)), (0g‘𝑅))))))⟩))
14	13	fveq2d 6886	. 2 ⊢ (𝜑 → (𝐸‘𝐹) = (𝐸‘(𝑉 sSet ⟨(.r‘ndx), (𝑥 ∈ (Base‘𝑉), 𝑦 ∈ (Base‘𝑉) ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑥‘𝑎)(.r‘𝑅)(𝑦‘𝑏)), (0g‘𝑅))))))⟩)))
15	3, 14	eqtr4id 2816	1 ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ifcif 4485  ⟨cop 4593   ↦ cmpt 5190  ‘cfv 6537  (class class class)co 7416   ∈ cmpo 7418   sSet csts 17259  Slot cslot 17277  ndxcnx 17289  Basecbs 17305  +_gcplusg 17346  ._rcmulr 17347  0_gc0g 17528   Σ_g cgsu 17529   freeLMod cfrlm 21960   MndRing cmnring 45036

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 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-res 5671  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-sets 17260  df-slot 17278  df-mnring 45037

This theorem is used by:  mnringbased  45040  mnringaddgd  45045  mnringscad  45049  mnringvscad  45050