Theorem mnringnmulrdOLD 44229

Description: Obsolete version of mnringnmulrd 44228 as of 1-Nov-2024. Components of a monoid ring other than its ring product match its underlying free module. (Contributed by Rohan Ridenour, 14-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
mnringnmulrdOLD.1	⊢ 𝐹 = (𝑅 MndRing 𝑀)
mnringnmulrdOLD.2	⊢ 𝐸 = Slot 𝑁
mnringnmulrdOLD.3	⊢ 𝑁 ∈ ℕ
mnringnmulrdOLD.4	⊢ 𝑁 ≠ (.r‘ndx)
mnringnmulrdOLD.5	⊢ 𝐴 = (Base‘𝑀)
mnringnmulrdOLD.6	⊢ 𝑉 = (𝑅 freeLMod 𝐴)
mnringnmulrdOLD.7	⊢ (𝜑 → 𝑅 ∈ 𝑈)
mnringnmulrdOLD.8	⊢ (𝜑 → 𝑀 ∈ 𝑊)

Assertion

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

Proof of Theorem mnringnmulrdOLD

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

Step	Hyp	Ref	Expression
1		mnringnmulrdOLD.2	. . . 4 ⊢ 𝐸 = Slot 𝑁
2		mnringnmulrdOLD.3	. . . 4 ⊢ 𝑁 ∈ ℕ
3	1, 2	ndxid 17234	. . 3 ⊢ 𝐸 = Slot (𝐸‘ndx)
4	1, 2	ndxarg 17233	. . . 4 ⊢ (𝐸‘ndx) = 𝑁
5		mnringnmulrdOLD.4	. . . 4 ⊢ 𝑁 ≠ (.r‘ndx)
6	4, 5	eqnetri 3011	. . 3 ⊢ (𝐸‘ndx) ≠ (.r‘ndx)
7	3, 6	setsnid 17245	. 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))))))⟩))
8		mnringnmulrdOLD.1	. . . 4 ⊢ 𝐹 = (𝑅 MndRing 𝑀)
9		eqid 2737	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
10		eqid 2737	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
11		mnringnmulrdOLD.5	. . . 4 ⊢ 𝐴 = (Base‘𝑀)
12		eqid 2737	. . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀)
13		mnringnmulrdOLD.6	. . . 4 ⊢ 𝑉 = (𝑅 freeLMod 𝐴)
14		eqid 2737	. . . 4 ⊢ (Base‘𝑉) = (Base‘𝑉)
15		mnringnmulrdOLD.7	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑈)
16		mnringnmulrdOLD.8	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑊)
17	8, 9, 10, 11, 12, 13, 14, 15, 16	mnringvald 44227	. . 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))))))⟩))
18	17	fveq2d 6910	. 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))))))⟩)))
19	7, 18	eqtr4id 2796	1 ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ifcif 4525  ⟨cop 4632   ↦ cmpt 5225  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  ℕcn 12266   sSet csts 17200  Slot cslot 17218  ndxcnx 17230  Basecbs 17247  +_gcplusg 17297  ._rcmulr 17298  0_gc0g 17484   Σ_g cgsu 17485   freeLMod cfrlm 21766   MndRing cmnring 44225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-addcl 11215

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-nn 12267  df-sets 17201  df-slot 17219  df-ndx 17231  df-mnring 44226

This theorem is referenced by:  mnringbasedOLD  44231  mnringaddgdOLD  44237  mnringscadOLD  44242  mnringvscadOLD  44244