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Theorem mnringnmulrdOLD 42582
Description: Obsolete version of mnringnmulrd 42581 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 ∈ β„•
mnringnmulrdOLD.4 𝑁 β‰  (.rβ€˜ndx)
mnringnmulrdOLD.5 𝐴 = (Baseβ€˜π‘€)
mnringnmulrdOLD.6 𝑉 = (𝑅 freeLMod 𝐴)
mnringnmulrdOLD.7 (πœ‘ β†’ 𝑅 ∈ π‘ˆ)
mnringnmulrdOLD.8 (πœ‘ β†’ 𝑀 ∈ π‘Š)
Assertion
Ref Expression
mnringnmulrdOLD (πœ‘ β†’ (πΈβ€˜π‘‰) = (πΈβ€˜πΉ))

Proof of Theorem mnringnmulrdOLD
Dummy variables π‘Ž 𝑏 𝑖 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mnringnmulrdOLD.2 . . . 4 𝐸 = Slot 𝑁
2 mnringnmulrdOLD.3 . . . 4 𝑁 ∈ β„•
31, 2ndxid 17077 . . 3 𝐸 = Slot (πΈβ€˜ndx)
41, 2ndxarg 17076 . . . 4 (πΈβ€˜ndx) = 𝑁
5 mnringnmulrdOLD.4 . . . 4 𝑁 β‰  (.rβ€˜ndx)
64, 5eqnetri 3011 . . 3 (πΈβ€˜ndx) β‰  (.rβ€˜ndx)
73, 6setsnid 17089 . 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‘…))))))⟩))
8 mnringnmulrdOLD.1 . . . 4 𝐹 = (𝑅 MndRing 𝑀)
9 eqid 2733 . . . 4 (.rβ€˜π‘…) = (.rβ€˜π‘…)
10 eqid 2733 . . . 4 (0gβ€˜π‘…) = (0gβ€˜π‘…)
11 mnringnmulrdOLD.5 . . . 4 𝐴 = (Baseβ€˜π‘€)
12 eqid 2733 . . . 4 (+gβ€˜π‘€) = (+gβ€˜π‘€)
13 mnringnmulrdOLD.6 . . . 4 𝑉 = (𝑅 freeLMod 𝐴)
14 eqid 2733 . . . 4 (Baseβ€˜π‘‰) = (Baseβ€˜π‘‰)
15 mnringnmulrdOLD.7 . . . 4 (πœ‘ β†’ 𝑅 ∈ π‘ˆ)
16 mnringnmulrdOLD.8 . . . 4 (πœ‘ β†’ 𝑀 ∈ π‘Š)
178, 9, 10, 11, 12, 13, 14, 15, 16mnringvald 42580 . . 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‘…))))))⟩))
1817fveq2d 6850 . 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‘…))))))⟩)))
197, 18eqtr4id 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 2940  ifcif 4490  βŸ¨cop 4596   ↦ cmpt 5192  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  β„•cn 12161   sSet csts 17043  Slot cslot 17061  ndxcnx 17073  Basecbs 17091  +gcplusg 17141  .rcmulr 17142  0gc0g 17329   Ξ£g cgsu 17330   freeLMod cfrlm 21175   MndRing cmnring 42578
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 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-1cn 11117  ax-addcl 11119
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-nn 12162  df-sets 17044  df-slot 17062  df-ndx 17074  df-mnring 42579
This theorem is referenced by:  mnringbasedOLD  42584  mnringaddgdOLD  42590  mnringscadOLD  42595  mnringvscadOLD  42597
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