Theorem mnringnmulrd 41716

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 16838	. 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 2738	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
6		eqid 2738	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
7		mnringnmulrd.5	. . . 4 ⊢ 𝐴 = (Base‘𝑀)
8		eqid 2738	. . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀)
9		mnringnmulrd.6	. . . 4 ⊢ 𝑉 = (𝑅 freeLMod 𝐴)
10		eqid 2738	. . . 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 41715	. . 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 6760	. 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 2798	1 ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ifcif 4456  ⟨cop 4564   ↦ cmpt 5153  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257   sSet csts 16792  Slot cslot 16810  ndxcnx 16822  Basecbs 16840  +_gcplusg 16888  ._rcmulr 16889  0_gc0g 17067   Σ_g cgsu 17068   freeLMod cfrlm 20863   MndRing cmnring 41713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-res 5592  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-sets 16793  df-slot 16811  df-mnring 41714

This theorem is referenced by:  mnringbased  41718  mnringaddgd  41724  mnringscad  41729  mnringvscad  41731