Theorem mnringnmulrdOLD 42480

Description: Obsolete version of mnringnmulrd 42479 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 17069	. . 3 ⊢ 𝐸 = Slot (𝐸‘ndx)
4	1, 2	ndxarg 17068	. . . 4 ⊢ (𝐸‘ndx) = 𝑁
5		mnringnmulrdOLD.4	. . . 4 ⊢ 𝑁 ≠ (.r‘ndx)
6	4, 5	eqnetri 3014	. . 3 ⊢ (𝐸‘ndx) ≠ (.r‘ndx)
7	3, 6	setsnid 17081	. 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 2736	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
10		eqid 2736	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
11		mnringnmulrdOLD.5	. . . 4 ⊢ 𝐴 = (Base‘𝑀)
12		eqid 2736	. . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀)
13		mnringnmulrdOLD.6	. . . 4 ⊢ 𝑉 = (𝑅 freeLMod 𝐴)
14		eqid 2736	. . . 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 42478	. . 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 6846	. 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 2795	1 ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ifcif 4486  ⟨cop 4592   ↦ cmpt 5188  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  ℕcn 12153   sSet csts 17035  Slot cslot 17053  ndxcnx 17065  Basecbs 17083  +_gcplusg 17133  ._rcmulr 17134  0_gc0g 17321   Σ_g cgsu 17322   freeLMod cfrlm 21152   MndRing cmnring 42476

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-1cn 11109  ax-addcl 11111

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-nn 12154  df-sets 17036  df-slot 17054  df-ndx 17066  df-mnring 42477

This theorem is referenced by:  mnringbasedOLD  42482  mnringaddgdOLD  42488  mnringscadOLD  42493  mnringvscadOLD  42495