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Theorem mnringnmulrd 44233
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 17246 . 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 2736 . . . 4 (.r𝑅) = (.r𝑅)
6 eqid 2736 . . . 4 (0g𝑅) = (0g𝑅)
7 mnringnmulrd.5 . . . 4 𝐴 = (Base‘𝑀)
8 eqid 2736 . . . 4 (+g𝑀) = (+g𝑀)
9 mnringnmulrd.6 . . . 4 𝑉 = (𝑅 freeLMod 𝐴)
10 eqid 2736 . . . 4 (Base‘𝑉) = (Base‘𝑉)
11 mnringnmulrd.7 . . . 4 (𝜑𝑅𝑈)
12 mnringnmulrd.8 . . . 4 (𝜑𝑀𝑊)
134, 5, 6, 7, 8, 9, 10, 11, 12mnringvald 44232 . . 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 6909 . 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 2795 1 (𝜑 → (𝐸𝑉) = (𝐸𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  wne 2939  ifcif 4524  cop 4631  cmpt 5224  cfv 6560  (class class class)co 7432  cmpo 7434   sSet csts 17201  Slot cslot 17219  ndxcnx 17231  Basecbs 17248  +gcplusg 17298  .rcmulr 17299  0gc0g 17485   Σg cgsu 17486   freeLMod cfrlm 21767   MndRing cmnring 44230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-res 5696  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-sets 17202  df-slot 17220  df-mnring 44231
This theorem is referenced by:  mnringbased  44235  mnringaddgd  44241  mnringscad  44246  mnringvscad  44248
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