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Theorem mnringnmulrd 42839
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 (𝜑𝑅𝑈)
mnringnmulrd.8 (𝜑𝑀𝑊)
Assertion
Ref Expression
mnringnmulrd (𝜑 → (𝐸𝑉) = (𝐸𝐹))

Proof of Theorem mnringnmulrd
Dummy variables 𝑎 𝑏 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mnringnmulrd.2 . . 3 𝐸 = Slot (𝐸‘ndx)
2 mnringnmulrd.4 . . 3 (𝐸‘ndx) ≠ (.r‘ndx)
31, 2setsnid 17129 . 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))))))⟩))
4 mnringnmulrd.1 . . . 4 𝐹 = (𝑅 MndRing 𝑀)
5 eqid 2733 . . . 4 (.r𝑅) = (.r𝑅)
6 eqid 2733 . . . 4 (0g𝑅) = (0g𝑅)
7 mnringnmulrd.5 . . . 4 𝐴 = (Base‘𝑀)
8 eqid 2733 . . . 4 (+g𝑀) = (+g𝑀)
9 mnringnmulrd.6 . . . 4 𝑉 = (𝑅 freeLMod 𝐴)
10 eqid 2733 . . . 4 (Base‘𝑉) = (Base‘𝑉)
11 mnringnmulrd.7 . . . 4 (𝜑𝑅𝑈)
12 mnringnmulrd.8 . . . 4 (𝜑𝑀𝑊)
134, 5, 6, 7, 8, 9, 10, 11, 12mnringvald 42838 . . 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))))))⟩))
1413fveq2d 6885 . 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))))))⟩)))
153, 14eqtr4id 2792 1 (𝜑 → (𝐸𝑉) = (𝐸𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wne 2941  ifcif 4524  cop 4630  cmpt 5227  cfv 6535  (class class class)co 7396  cmpo 7398   sSet csts 17083  Slot cslot 17101  ndxcnx 17113  Basecbs 17131  +gcplusg 17184  .rcmulr 17185  0gc0g 17372   Σg cgsu 17373   freeLMod cfrlm 21274   MndRing cmnring 42836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-res 5684  df-iota 6487  df-fun 6537  df-fv 6543  df-ov 7399  df-oprab 7400  df-mpo 7401  df-sets 17084  df-slot 17102  df-mnring 42837
This theorem is referenced by:  mnringbased  42841  mnringaddgd  42847  mnringscad  42852  mnringvscad  42854
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