Theorem mnringnmulrd 44886

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 17267	. 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 2761	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
6		eqid 2761	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
7		mnringnmulrd.5	. . . 4 ⊢ 𝐴 = (Base‘𝑀)
8		eqid 2761	. . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀)
9		mnringnmulrd.6	. . . 4 ⊢ 𝑉 = (𝑅 freeLMod 𝐴)
10		eqid 2761	. . . 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 44885	. . 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 6885	. 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 2815	1 ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹))

Syntax hints:   → wi 4   = wceq 1568   ∈ wcel 2141   ≠ wne 2956  ifcif 4486  ⟨cop 4594   ↦ cmpt 5191  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412   sSet csts 17222  Slot cslot 17240  ndxcnx 17252  Basecbs 17268  +_gcplusg 17309  ._rcmulr 17310  0_gc0g 17491   Σ_g cgsu 17492   freeLMod cfrlm 21875   MndRing cmnring 44883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-res 5673  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-sets 17223  df-slot 17241  df-mnring 44884

This theorem is referenced by:  mnringbased  44887  mnringaddgd  44892  mnringscad  44896  mnringvscad  44897