Theorem mnringnmulrdOLD 44206

Description: Obsolete version of mnringnmulrd 44205 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 ∈ ℕ
mnringnmulrdOLD.4	⊢ 𝑁 ≠ (.r‘ndx)
mnringnmulrdOLD.5	⊢ 𝐴 = (Base‘𝑀)
mnringnmulrdOLD.6	⊢ 𝑉 = (𝑅 freeLMod 𝐴)
mnringnmulrdOLD.7	⊢ (𝜑 → 𝑅 ∈ 𝑈)
mnringnmulrdOLD.8	⊢ (𝜑 → 𝑀 ∈ 𝑊)

Assertion

Ref	Expression
mnringnmulrdOLD	⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹))

Proof of Theorem mnringnmulrdOLD

Dummy variables 𝑎 𝑏 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mnringnmulrdOLD.2	. . . 4 ⊢ 𝐸 = Slot 𝑁
2		mnringnmulrdOLD.3	. . . 4 ⊢ 𝑁 ∈ ℕ
3	1, 2	ndxid 17231	. . 3 ⊢ 𝐸 = Slot (𝐸‘ndx)
4	1, 2	ndxarg 17230	. . . 4 ⊢ (𝐸‘ndx) = 𝑁
5		mnringnmulrdOLD.4	. . . 4 ⊢ 𝑁 ≠ (.r‘ndx)
6	4, 5	eqnetri 3009	. . 3 ⊢ (𝐸‘ndx) ≠ (.r‘ndx)
7	3, 6	setsnid 17243	. 2 ⊢ (𝐸‘𝑉) = (𝐸‘(𝑉 sSet ⟨(.r‘ndx), (𝑥 ∈ (Base‘𝑉), 𝑦 ∈ (Base‘𝑉) ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑥‘𝑎)(.r‘𝑅)(𝑦‘𝑏)), (0g‘𝑅))))))⟩))
8		mnringnmulrdOLD.1	. . . 4 ⊢ 𝐹 = (𝑅 MndRing 𝑀)
9		eqid 2735	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
10		eqid 2735	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
11		mnringnmulrdOLD.5	. . . 4 ⊢ 𝐴 = (Base‘𝑀)
12		eqid 2735	. . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀)
13		mnringnmulrdOLD.6	. . . 4 ⊢ 𝑉 = (𝑅 freeLMod 𝐴)
14		eqid 2735	. . . 4 ⊢ (Base‘𝑉) = (Base‘𝑉)
15		mnringnmulrdOLD.7	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑈)
16		mnringnmulrdOLD.8	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑊)
17	8, 9, 10, 11, 12, 13, 14, 15, 16	mnringvald 44204	. . 3 ⊢ (𝜑 → 𝐹 = (𝑉 sSet ⟨(.r‘ndx), (𝑥 ∈ (Base‘𝑉), 𝑦 ∈ (Base‘𝑉) ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑥‘𝑎)(.r‘𝑅)(𝑦‘𝑏)), (0g‘𝑅))))))⟩))
18	17	fveq2d 6911	. 2 ⊢ (𝜑 → (𝐸‘𝐹) = (𝐸‘(𝑉 sSet ⟨(.r‘ndx), (𝑥 ∈ (Base‘𝑉), 𝑦 ∈ (Base‘𝑉) ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑥‘𝑎)(.r‘𝑅)(𝑦‘𝑏)), (0g‘𝑅))))))⟩)))
19	7, 18	eqtr4id 2794	1 ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ifcif 4531  ⟨cop 4637   ↦ cmpt 5231  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  ℕcn 12264   sSet csts 17197  Slot cslot 17215  ndxcnx 17227  Basecbs 17245  +_gcplusg 17298  ._rcmulr 17299  0_gc0g 17486   Σ_g cgsu 17487   freeLMod cfrlm 21784   MndRing cmnring 44202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-nn 12265  df-sets 17198  df-slot 17216  df-ndx 17228  df-mnring 44203

This theorem is referenced by:  mnringbasedOLD  44208  mnringaddgdOLD  44214  mnringscadOLD  44219  mnringvscadOLD  44221