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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 (𝜑𝑅𝑈)
mnringnmulrdOLD.8 (𝜑𝑀𝑊)
Assertion
Ref Expression
mnringnmulrdOLD (𝜑 → (𝐸𝑉) = (𝐸𝐹))

Proof of Theorem mnringnmulrdOLD
Dummy variables 𝑎 𝑏 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mnringnmulrdOLD.2 . . . 4 𝐸 = Slot 𝑁
2 mnringnmulrdOLD.3 . . . 4 𝑁 ∈ ℕ
31, 2ndxid 16995 . . 3 𝐸 = Slot (𝐸‘ndx)
41, 2ndxarg 16994 . . . 4 (𝐸‘ndx) = 𝑁
5 mnringnmulrdOLD.4 . . . 4 𝑁 ≠ (.r‘ndx)
64, 5eqnetri 3012 . . 3 (𝐸‘ndx) ≠ (.r‘ndx)
73, 6setsnid 17007 . 2 (𝐸𝑉) = (𝐸‘(𝑉 sSet ⟨(.r‘ndx), (𝑥 ∈ (Base‘𝑉), 𝑦 ∈ (Base‘𝑉) ↦ (𝑉 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑥𝑎)(.r𝑅)(𝑦𝑏)), (0g𝑅))))))⟩))
8 mnringnmulrdOLD.1 . . . 4 𝐹 = (𝑅 MndRing 𝑀)
9 eqid 2737 . . . 4 (.r𝑅) = (.r𝑅)
10 eqid 2737 . . . 4 (0g𝑅) = (0g𝑅)
11 mnringnmulrdOLD.5 . . . 4 𝐴 = (Base‘𝑀)
12 eqid 2737 . . . 4 (+g𝑀) = (+g𝑀)
13 mnringnmulrdOLD.6 . . . 4 𝑉 = (𝑅 freeLMod 𝐴)
14 eqid 2737 . . . 4 (Base‘𝑉) = (Base‘𝑉)
15 mnringnmulrdOLD.7 . . . 4 (𝜑𝑅𝑈)
16 mnringnmulrdOLD.8 . . . 4 (𝜑𝑀𝑊)
178, 9, 10, 11, 12, 13, 14, 15, 16mnringvald 42199 . . 3 (𝜑𝐹 = (𝑉 sSet ⟨(.r‘ndx), (𝑥 ∈ (Base‘𝑉), 𝑦 ∈ (Base‘𝑉) ↦ (𝑉 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑥𝑎)(.r𝑅)(𝑦𝑏)), (0g𝑅))))))⟩))
1817fveq2d 6833 . 2 (𝜑 → (𝐸𝐹) = (𝐸‘(𝑉 sSet ⟨(.r‘ndx), (𝑥 ∈ (Base‘𝑉), 𝑦 ∈ (Base‘𝑉) ↦ (𝑉 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑥𝑎)(.r𝑅)(𝑦𝑏)), (0g𝑅))))))⟩)))
197, 18eqtr4id 2796 1 (𝜑 → (𝐸𝑉) = (𝐸𝐹))
Colors of variables: wff setvar class
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  cn 12078   sSet csts 16961  Slot cslot 16979  ndxcnx 16991  Basecbs 17009  +gcplusg 17059  .rcmulr 17060  0gc0g 17247   Σg 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
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