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Theorem mnringvald 40845
 Description: Value of the monoid ring function. (Contributed by Rohan Ridenour, 14-May-2024.)
Hypotheses
Ref Expression
mnringvald.1 𝐹 = (𝑅 MndRing 𝑀)
mnringvald.2 · = (.r𝑅)
mnringvald.3 0 = (0g𝑅)
mnringvald.4 𝐴 = (Base‘𝑀)
mnringvald.5 + = (+g𝑀)
mnringvald.6 𝑉 = (𝑅 freeLMod 𝐴)
mnringvald.7 𝐵 = (Base‘𝑉)
mnringvald.8 (𝜑𝑅𝑈)
mnringvald.9 (𝜑𝑀𝑊)
Assertion
Ref Expression
mnringvald (𝜑𝐹 = (𝑉 sSet ⟨(.r‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑉 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥𝑎) · (𝑦𝑏)), 0 )))))⟩))
Distinct variable groups:   𝑅,𝑎,𝑏,𝑖,𝑥,𝑦   𝑀,𝑎,𝑏,𝑖,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑖,𝑎,𝑏)   𝐴(𝑥,𝑦,𝑖,𝑎,𝑏)   𝐵(𝑥,𝑦,𝑖,𝑎,𝑏)   + (𝑥,𝑦,𝑖,𝑎,𝑏)   · (𝑥,𝑦,𝑖,𝑎,𝑏)   𝑈(𝑥,𝑦,𝑖,𝑎,𝑏)   𝐹(𝑥,𝑦,𝑖,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑖,𝑎,𝑏)   𝑊(𝑥,𝑦,𝑖,𝑎,𝑏)   0 (𝑥,𝑦,𝑖,𝑎,𝑏)

Proof of Theorem mnringvald
Dummy variables 𝑚 𝑟 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mnringvald.1 . 2 𝐹 = (𝑅 MndRing 𝑀)
2 mnringvald.8 . . . 4 (𝜑𝑅𝑈)
32elexd 3500 . . 3 (𝜑𝑅 ∈ V)
4 mnringvald.9 . . . 4 (𝜑𝑀𝑊)
54elexd 3500 . . 3 (𝜑𝑀 ∈ V)
6 nfv 1916 . . . . 5 𝑣(𝑟 = 𝑅𝑚 = 𝑀)
7 nfcvd 2983 . . . . 5 ((𝑟 = 𝑅𝑚 = 𝑀) → 𝑣(𝑉 sSet ⟨(.r‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑉 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥𝑎) · (𝑦𝑏)), 0 )))))⟩))
8 ovexd 7184 . . . . 5 ((𝑟 = 𝑅𝑚 = 𝑀) → (𝑟 freeLMod (Base‘𝑚)) ∈ V)
9 simpr 488 . . . . . . . 8 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → 𝑣 = (𝑟 freeLMod (Base‘𝑚)))
10 simpll 766 . . . . . . . . 9 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → 𝑟 = 𝑅)
11 fveq2 6661 . . . . . . . . . . 11 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
12 mnringvald.4 . . . . . . . . . . 11 𝐴 = (Base‘𝑀)
1311, 12syl6eqr 2877 . . . . . . . . . 10 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐴)
1413ad2antlr 726 . . . . . . . . 9 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (Base‘𝑚) = 𝐴)
1510, 14oveq12d 7167 . . . . . . . 8 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑟 freeLMod (Base‘𝑚)) = (𝑅 freeLMod 𝐴))
169, 15eqtrd 2859 . . . . . . 7 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → 𝑣 = (𝑅 freeLMod 𝐴))
17 mnringvald.6 . . . . . . 7 𝑉 = (𝑅 freeLMod 𝐴)
1816, 17syl6eqr 2877 . . . . . 6 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → 𝑣 = 𝑉)
1918fveq2d 6665 . . . . . . . . 9 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (Base‘𝑣) = (Base‘𝑉))
20 mnringvald.7 . . . . . . . . 9 𝐵 = (Base‘𝑉)
2119, 20syl6eqr 2877 . . . . . . . 8 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (Base‘𝑣) = 𝐵)
22 fveq2 6661 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
23 mnringvald.5 . . . . . . . . . . . . . . . 16 + = (+g𝑀)
2422, 23syl6eqr 2877 . . . . . . . . . . . . . . 15 (𝑚 = 𝑀 → (+g𝑚) = + )
2524oveqd 7166 . . . . . . . . . . . . . 14 (𝑚 = 𝑀 → (𝑎(+g𝑚)𝑏) = (𝑎 + 𝑏))
2625ad2antlr 726 . . . . . . . . . . . . 13 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑎(+g𝑚)𝑏) = (𝑎 + 𝑏))
2726eqeq2d 2835 . . . . . . . . . . . 12 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑖 = (𝑎(+g𝑚)𝑏) ↔ 𝑖 = (𝑎 + 𝑏)))
28 fveq2 6661 . . . . . . . . . . . . . . 15 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
29 mnringvald.2 . . . . . . . . . . . . . . 15 · = (.r𝑅)
3028, 29syl6eqr 2877 . . . . . . . . . . . . . 14 (𝑟 = 𝑅 → (.r𝑟) = · )
3130oveqd 7166 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → ((𝑥𝑎)(.r𝑟)(𝑦𝑏)) = ((𝑥𝑎) · (𝑦𝑏)))
3231ad2antrr 725 . . . . . . . . . . . 12 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → ((𝑥𝑎)(.r𝑟)(𝑦𝑏)) = ((𝑥𝑎) · (𝑦𝑏)))
33 fveq2 6661 . . . . . . . . . . . . . 14 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
34 mnringvald.3 . . . . . . . . . . . . . 14 0 = (0g𝑅)
3533, 34syl6eqr 2877 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (0g𝑟) = 0 )
3635ad2antrr 725 . . . . . . . . . . . 12 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (0g𝑟) = 0 )
3727, 32, 36ifbieq12d 4477 . . . . . . . . . . 11 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → if(𝑖 = (𝑎(+g𝑚)𝑏), ((𝑥𝑎)(.r𝑟)(𝑦𝑏)), (0g𝑟)) = if(𝑖 = (𝑎 + 𝑏), ((𝑥𝑎) · (𝑦𝑏)), 0 ))
3814, 37mpteq12dv 5137 . . . . . . . . . 10 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+g𝑚)𝑏), ((𝑥𝑎)(.r𝑟)(𝑦𝑏)), (0g𝑟))) = (𝑖𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥𝑎) · (𝑦𝑏)), 0 )))
3914, 14, 38mpoeq123dv 7222 . . . . . . . . 9 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑎 ∈ (Base‘𝑚), 𝑏 ∈ (Base‘𝑚) ↦ (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+g𝑚)𝑏), ((𝑥𝑎)(.r𝑟)(𝑦𝑏)), (0g𝑟)))) = (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥𝑎) · (𝑦𝑏)), 0 ))))
4018, 39oveq12d 7167 . . . . . . . 8 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑣 Σg (𝑎 ∈ (Base‘𝑚), 𝑏 ∈ (Base‘𝑚) ↦ (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+g𝑚)𝑏), ((𝑥𝑎)(.r𝑟)(𝑦𝑏)), (0g𝑟))))) = (𝑉 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥𝑎) · (𝑦𝑏)), 0 )))))
4121, 21, 40mpoeq123dv 7222 . . . . . . 7 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑥 ∈ (Base‘𝑣), 𝑦 ∈ (Base‘𝑣) ↦ (𝑣 Σg (𝑎 ∈ (Base‘𝑚), 𝑏 ∈ (Base‘𝑚) ↦ (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+g𝑚)𝑏), ((𝑥𝑎)(.r𝑟)(𝑦𝑏)), (0g𝑟)))))) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑉 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥𝑎) · (𝑦𝑏)), 0 ))))))
4241opeq2d 4796 . . . . . 6 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → ⟨(.r‘ndx), (𝑥 ∈ (Base‘𝑣), 𝑦 ∈ (Base‘𝑣) ↦ (𝑣 Σg (𝑎 ∈ (Base‘𝑚), 𝑏 ∈ (Base‘𝑚) ↦ (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+g𝑚)𝑏), ((𝑥𝑎)(.r𝑟)(𝑦𝑏)), (0g𝑟))))))⟩ = ⟨(.r‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑉 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥𝑎) · (𝑦𝑏)), 0 )))))⟩)
4318, 42oveq12d 7167 . . . . 5 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑣 sSet ⟨(.r‘ndx), (𝑥 ∈ (Base‘𝑣), 𝑦 ∈ (Base‘𝑣) ↦ (𝑣 Σg (𝑎 ∈ (Base‘𝑚), 𝑏 ∈ (Base‘𝑚) ↦ (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+g𝑚)𝑏), ((𝑥𝑎)(.r𝑟)(𝑦𝑏)), (0g𝑟))))))⟩) = (𝑉 sSet ⟨(.r‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑉 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥𝑎) · (𝑦𝑏)), 0 )))))⟩))
446, 7, 8, 43csbiedf 3896 . . . 4 ((𝑟 = 𝑅𝑚 = 𝑀) → (𝑟 freeLMod (Base‘𝑚)) / 𝑣(𝑣 sSet ⟨(.r‘ndx), (𝑥 ∈ (Base‘𝑣), 𝑦 ∈ (Base‘𝑣) ↦ (𝑣 Σg (𝑎 ∈ (Base‘𝑚), 𝑏 ∈ (Base‘𝑚) ↦ (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+g𝑚)𝑏), ((𝑥𝑎)(.r𝑟)(𝑦𝑏)), (0g𝑟))))))⟩) = (𝑉 sSet ⟨(.r‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑉 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥𝑎) · (𝑦𝑏)), 0 )))))⟩))
45 df-mnring 40844 . . . 4 MndRing = (𝑟 ∈ V, 𝑚 ∈ V ↦ (𝑟 freeLMod (Base‘𝑚)) / 𝑣(𝑣 sSet ⟨(.r‘ndx), (𝑥 ∈ (Base‘𝑣), 𝑦 ∈ (Base‘𝑣) ↦ (𝑣 Σg (𝑎 ∈ (Base‘𝑚), 𝑏 ∈ (Base‘𝑚) ↦ (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+g𝑚)𝑏), ((𝑥𝑎)(.r𝑟)(𝑦𝑏)), (0g𝑟))))))⟩))
46 ovex 7182 . . . 4 (𝑉 sSet ⟨(.r‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑉 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥𝑎) · (𝑦𝑏)), 0 )))))⟩) ∈ V
4744, 45, 46ovmpoa 7298 . . 3 ((𝑅 ∈ V ∧ 𝑀 ∈ V) → (𝑅 MndRing 𝑀) = (𝑉 sSet ⟨(.r‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑉 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥𝑎) · (𝑦𝑏)), 0 )))))⟩))
483, 5, 47syl2anc 587 . 2 (𝜑 → (𝑅 MndRing 𝑀) = (𝑉 sSet ⟨(.r‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑉 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥𝑎) · (𝑦𝑏)), 0 )))))⟩))
491, 48syl5eq 2871 1 (𝜑𝐹 = (𝑉 sSet ⟨(.r‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑉 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥𝑎) · (𝑦𝑏)), 0 )))))⟩))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  Vcvv 3480  ⦋csb 3866  ifcif 4450  ⟨cop 4556   ↦ cmpt 5132  ‘cfv 6343  (class class class)co 7149   ∈ cmpo 7151  ndxcnx 16480   sSet csts 16481  Basecbs 16483  +gcplusg 16565  .rcmulr 16566  0gc0g 16713   Σg cgsu 16714   freeLMod cfrlm 20492   MndRing cmnring 40843 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-mnring 40844 This theorem is referenced by:  mnringnmulrd  40846  mnringmulrd  40855
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