Theorem mnringnmulrdOLD 42201

Description: Obsolete version of mnringnmulrd 42200 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 16995	. . 3 ⊢ 𝐸 = Slot (𝐸‘ndx)
4	1, 2	ndxarg 16994	. . . 4 ⊢ (𝐸‘ndx) = 𝑁
5		mnringnmulrdOLD.4	. . . 4 ⊢ 𝑁 ≠ (.r‘ndx)
6	4, 5	eqnetri 3012	. . 3 ⊢ (𝐸‘ndx) ≠ (.r‘ndx)
7	3, 6	setsnid 17007	. 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 42199	. . 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 6833	. 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 1541   ∈ wcel 2106   ≠ wne 2941  ifcif 4477  ⟨cop 4583   ↦ cmpt 5179  ‘cfv 6483  (class class class)co 7341   ∈ cmpo 7343  ℕcn 12078   sSet csts 16961  Slot cslot 16979  ndxcnx 16991  Basecbs 17009  +_gcplusg 17059  ._rcmulr 17060  0_gc0g 17247   Σ_g cgsu 17248   freeLMod cfrlm 21058   MndRing cmnring 42197

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5247  ax-nul 5254  ax-pow 5312  ax-pr 5376  ax-un 7654  ax-cnex 11032  ax-1cn 11034  ax-addcl 11036

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3920  df-nul 4274  df-if 4478  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-iun 4947  df-br 5097  df-opab 5159  df-mpt 5180  df-tr 5214  df-id 5522  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5579  df-we 5581  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6242  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6435  df-fun 6485  df-fn 6486  df-f 6487  df-f1 6488  df-fo 6489  df-f1o 6490  df-fv 6491  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7785  df-2nd 7904  df-frecs 8171  df-wrecs 8202  df-recs 8276  df-rdg 8315  df-nn 12079  df-sets 16962  df-slot 16980  df-ndx 16992  df-mnring 42198

This theorem is referenced by:  mnringbasedOLD  42203  mnringaddgdOLD  42209  mnringscadOLD  42214  mnringvscadOLD  42216