Theorem mnringnmulrd 44665

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 17173	. 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 2737	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
6		eqid 2737	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
7		mnringnmulrd.5	. . . 4 ⊢ 𝐴 = (Base‘𝑀)
8		eqid 2737	. . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀)
9		mnringnmulrd.6	. . . 4 ⊢ 𝑉 = (𝑅 freeLMod 𝐴)
10		eqid 2737	. . . 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 44664	. . 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 6840	. 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 2791	1 ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ifcif 4467  ⟨cop 4574   ↦ cmpt 5167  ‘cfv 6494  (class class class)co 7362   ∈ cmpo 7364   sSet csts 17128  Slot cslot 17146  ndxcnx 17158  Basecbs 17174  +_gcplusg 17215  ._rcmulr 17216  0_gc0g 17397   Σ_g cgsu 17398   freeLMod cfrlm 21740   MndRing cmnring 44662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-res 5638  df-iota 6450  df-fun 6496  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-sets 17129  df-slot 17147  df-mnring 44663

This theorem is referenced by:  mnringbased  44666  mnringaddgd  44671  mnringscad  44675  mnringvscad  44676