Theorem mnringnmulrd 44178

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 17256	. 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 2740	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
6		eqid 2740	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
7		mnringnmulrd.5	. . . 4 ⊢ 𝐴 = (Base‘𝑀)
8		eqid 2740	. . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀)
9		mnringnmulrd.6	. . . 4 ⊢ 𝑉 = (𝑅 freeLMod 𝐴)
10		eqid 2740	. . . 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 44177	. . 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 6924	. 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 2799	1 ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ifcif 4548  ⟨cop 4654   ↦ cmpt 5249  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450   sSet csts 17210  Slot cslot 17228  ndxcnx 17240  Basecbs 17258  +_gcplusg 17311  ._rcmulr 17312  0_gc0g 17499   Σ_g cgsu 17500   freeLMod cfrlm 21789   MndRing cmnring 44175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-res 5712  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-sets 17211  df-slot 17229  df-mnring 44176

This theorem is referenced by:  mnringbased  44180  mnringaddgd  44186  mnringscad  44191  mnringvscad  44193