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Theorem mnringnmulrdOLD 42480
Description: Obsolete version of mnringnmulrd 42479 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 17069 . . 3 𝐸 = Slot (𝐸‘ndx)
41, 2ndxarg 17068 . . . 4 (𝐸‘ndx) = 𝑁
5 mnringnmulrdOLD.4 . . . 4 𝑁 ≠ (.r‘ndx)
64, 5eqnetri 3014 . . 3 (𝐸‘ndx) ≠ (.r‘ndx)
73, 6setsnid 17081 . 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 2736 . . . 4 (.r𝑅) = (.r𝑅)
10 eqid 2736 . . . 4 (0g𝑅) = (0g𝑅)
11 mnringnmulrdOLD.5 . . . 4 𝐴 = (Base‘𝑀)
12 eqid 2736 . . . 4 (+g𝑀) = (+g𝑀)
13 mnringnmulrdOLD.6 . . . 4 𝑉 = (𝑅 freeLMod 𝐴)
14 eqid 2736 . . . 4 (Base‘𝑉) = (Base‘𝑉)
15 mnringnmulrdOLD.7 . . . 4 (𝜑𝑅𝑈)
16 mnringnmulrdOLD.8 . . . 4 (𝜑𝑀𝑊)
178, 9, 10, 11, 12, 13, 14, 15, 16mnringvald 42478 . . 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 6846 . 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 2795 1 (𝜑 → (𝐸𝑉) = (𝐸𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  wne 2943  ifcif 4486  cop 4592  cmpt 5188  cfv 6496  (class class class)co 7357  cmpo 7359  cn 12153   sSet csts 17035  Slot cslot 17053  ndxcnx 17065  Basecbs 17083  +gcplusg 17133  .rcmulr 17134  0gc0g 17321   Σg cgsu 17322   freeLMod cfrlm 21152   MndRing cmnring 42476
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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-1cn 11109  ax-addcl 11111
This theorem depends on definitions:  df-bi 206  df-an 397  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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-nn 12154  df-sets 17036  df-slot 17054  df-ndx 17066  df-mnring 42477
This theorem is referenced by:  mnringbasedOLD  42482  mnringaddgdOLD  42488  mnringscadOLD  42493  mnringvscadOLD  42495
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