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Theorem mnringvald 41551
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 3443 . . 3 (𝜑𝑅 ∈ V)
4 mnringvald.9 . . . 4 (𝜑𝑀𝑊)
54elexd 3443 . . 3 (𝜑𝑀 ∈ V)
6 nfv 1922 . . . . 5 𝑣(𝑟 = 𝑅𝑚 = 𝑀)
7 nfcvd 2908 . . . . 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 7269 . . . . 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 767 . . . . . . . . 9 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → 𝑟 = 𝑅)
11 fveq2 6738 . . . . . . . . . . 11 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
12 mnringvald.4 . . . . . . . . . . 11 𝐴 = (Base‘𝑀)
1311, 12eqtr4di 2798 . . . . . . . . . 10 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐴)
1413ad2antlr 727 . . . . . . . . 9 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (Base‘𝑚) = 𝐴)
1510, 14oveq12d 7252 . . . . . . . 8 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑟 freeLMod (Base‘𝑚)) = (𝑅 freeLMod 𝐴))
169, 15eqtrd 2779 . . . . . . 7 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → 𝑣 = (𝑅 freeLMod 𝐴))
17 mnringvald.6 . . . . . . 7 𝑉 = (𝑅 freeLMod 𝐴)
1816, 17eqtr4di 2798 . . . . . 6 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → 𝑣 = 𝑉)
1918fveq2d 6742 . . . . . . . . 9 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (Base‘𝑣) = (Base‘𝑉))
20 mnringvald.7 . . . . . . . . 9 𝐵 = (Base‘𝑉)
2119, 20eqtr4di 2798 . . . . . . . 8 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (Base‘𝑣) = 𝐵)
22 fveq2 6738 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
23 mnringvald.5 . . . . . . . . . . . . . . . 16 + = (+g𝑀)
2422, 23eqtr4di 2798 . . . . . . . . . . . . . . 15 (𝑚 = 𝑀 → (+g𝑚) = + )
2524oveqd 7251 . . . . . . . . . . . . . 14 (𝑚 = 𝑀 → (𝑎(+g𝑚)𝑏) = (𝑎 + 𝑏))
2625ad2antlr 727 . . . . . . . . . . . . 13 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑎(+g𝑚)𝑏) = (𝑎 + 𝑏))
2726eqeq2d 2750 . . . . . . . . . . . 12 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑖 = (𝑎(+g𝑚)𝑏) ↔ 𝑖 = (𝑎 + 𝑏)))
28 fveq2 6738 . . . . . . . . . . . . . . 15 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
29 mnringvald.2 . . . . . . . . . . . . . . 15 · = (.r𝑅)
3028, 29eqtr4di 2798 . . . . . . . . . . . . . 14 (𝑟 = 𝑅 → (.r𝑟) = · )
3130oveqd 7251 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → ((𝑥𝑎)(.r𝑟)(𝑦𝑏)) = ((𝑥𝑎) · (𝑦𝑏)))
3231ad2antrr 726 . . . . . . . . . . . 12 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → ((𝑥𝑎)(.r𝑟)(𝑦𝑏)) = ((𝑥𝑎) · (𝑦𝑏)))
33 fveq2 6738 . . . . . . . . . . . . . 14 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
34 mnringvald.3 . . . . . . . . . . . . . 14 0 = (0g𝑅)
3533, 34eqtr4di 2798 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (0g𝑟) = 0 )
3635ad2antrr 726 . . . . . . . . . . . 12 (((𝑟 = 𝑅𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (0g𝑟) = 0 )
3727, 32, 36ifbieq12d 4483 . . . . . . . . . . 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 5156 . . . . . . . . . 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 7307 . . . . . . . . 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 7252 . . . . . . . 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 7307 . . . . . . 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 4807 . . . . . 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 7252 . . . . 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 3859 . . . 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 41550 . . . 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 7267 . . . 4 (𝑉 sSet ⟨(.r‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ (𝑉 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥𝑎) · (𝑦𝑏)), 0 )))))⟩) ∈ V
4744, 45, 46ovmpoa 7385 . . 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 2792 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 1543  wcel 2112  Vcvv 3423  csb 3828  ifcif 4455  cop 4563  cmpt 5151  cfv 6400  (class class class)co 7234  cmpo 7236   sSet csts 16748  ndxcnx 16776  Basecbs 16792  +gcplusg 16834  .rcmulr 16835  0gc0g 16976   Σg cgsu 16977   freeLMod cfrlm 20740   MndRing cmnring 41549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5208  ax-nul 5215  ax-pr 5338
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4456  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4836  df-br 5070  df-opab 5132  df-mpt 5152  df-id 5471  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-iota 6358  df-fun 6402  df-fv 6408  df-ov 7237  df-oprab 7238  df-mpo 7239  df-mnring 41550
This theorem is referenced by:  mnringnmulrd  41552  mnringmulrd  41561
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