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Theorem mnringnmulrd 42968
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 (πΈβ€˜ndx)
mnringnmulrd.4 (πΈβ€˜ndx) β‰  (.rβ€˜ndx)
mnringnmulrd.5 𝐴 = (Baseβ€˜π‘€)
mnringnmulrd.6 𝑉 = (𝑅 freeLMod 𝐴)
mnringnmulrd.7 (πœ‘ β†’ 𝑅 ∈ π‘ˆ)
mnringnmulrd.8 (πœ‘ β†’ 𝑀 ∈ π‘Š)
Assertion
Ref Expression
mnringnmulrd (πœ‘ β†’ (πΈβ€˜π‘‰) = (πΈβ€˜πΉ))

Proof of Theorem mnringnmulrd
Dummy variables π‘Ž 𝑏 𝑖 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mnringnmulrd.2 . . 3 𝐸 = Slot (πΈβ€˜ndx)
2 mnringnmulrd.4 . . 3 (πΈβ€˜ndx) β‰  (.rβ€˜ndx)
31, 2setsnid 17142 . 2 (πΈβ€˜π‘‰) = (πΈβ€˜(𝑉 sSet ⟨(.rβ€˜ndx), (π‘₯ ∈ (Baseβ€˜π‘‰), 𝑦 ∈ (Baseβ€˜π‘‰) ↦ (𝑉 Ξ£g (π‘Ž ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (π‘Ž(+gβ€˜π‘€)𝑏), ((π‘₯β€˜π‘Ž)(.rβ€˜π‘…)(π‘¦β€˜π‘)), (0gβ€˜π‘…))))))⟩))
4 mnringnmulrd.1 . . . 4 𝐹 = (𝑅 MndRing 𝑀)
5 eqid 2733 . . . 4 (.rβ€˜π‘…) = (.rβ€˜π‘…)
6 eqid 2733 . . . 4 (0gβ€˜π‘…) = (0gβ€˜π‘…)
7 mnringnmulrd.5 . . . 4 𝐴 = (Baseβ€˜π‘€)
8 eqid 2733 . . . 4 (+gβ€˜π‘€) = (+gβ€˜π‘€)
9 mnringnmulrd.6 . . . 4 𝑉 = (𝑅 freeLMod 𝐴)
10 eqid 2733 . . . 4 (Baseβ€˜π‘‰) = (Baseβ€˜π‘‰)
11 mnringnmulrd.7 . . . 4 (πœ‘ β†’ 𝑅 ∈ π‘ˆ)
12 mnringnmulrd.8 . . . 4 (πœ‘ β†’ 𝑀 ∈ π‘Š)
134, 5, 6, 7, 8, 9, 10, 11, 12mnringvald 42967 . . 3 (πœ‘ β†’ 𝐹 = (𝑉 sSet ⟨(.rβ€˜ndx), (π‘₯ ∈ (Baseβ€˜π‘‰), 𝑦 ∈ (Baseβ€˜π‘‰) ↦ (𝑉 Ξ£g (π‘Ž ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (π‘Ž(+gβ€˜π‘€)𝑏), ((π‘₯β€˜π‘Ž)(.rβ€˜π‘…)(π‘¦β€˜π‘)), (0gβ€˜π‘…))))))⟩))
1413fveq2d 6896 . 2 (πœ‘ β†’ (πΈβ€˜πΉ) = (πΈβ€˜(𝑉 sSet ⟨(.rβ€˜ndx), (π‘₯ ∈ (Baseβ€˜π‘‰), 𝑦 ∈ (Baseβ€˜π‘‰) ↦ (𝑉 Ξ£g (π‘Ž ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (π‘Ž(+gβ€˜π‘€)𝑏), ((π‘₯β€˜π‘Ž)(.rβ€˜π‘…)(π‘¦β€˜π‘)), (0gβ€˜π‘…))))))⟩)))
153, 14eqtr4id 2792 1 (πœ‘ β†’ (πΈβ€˜π‘‰) = (πΈβ€˜πΉ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  ifcif 4529  βŸ¨cop 4635   ↦ cmpt 5232  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411   sSet csts 17096  Slot cslot 17114  ndxcnx 17126  Basecbs 17144  +gcplusg 17197  .rcmulr 17198  0gc0g 17385   Ξ£g cgsu 17386   freeLMod cfrlm 21301   MndRing cmnring 42965
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 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-sets 17097  df-slot 17115  df-mnring 42966
This theorem is referenced by:  mnringbased  42970  mnringaddgd  42976  mnringscad  42981  mnringvscad  42983
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