Theorem mnringnmulrdOLD 42969

Description: Obsolete version of mnringnmulrd 42968 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 17130	. . 3 ⊢ 𝐸 = Slot (𝐸‘ndx)
4	1, 2	ndxarg 17129	. . . 4 ⊢ (𝐸‘ndx) = 𝑁
5		mnringnmulrdOLD.4	. . . 4 ⊢ 𝑁 ≠ (.r‘ndx)
6	4, 5	eqnetri 3012	. . 3 ⊢ (𝐸‘ndx) ≠ (.r‘ndx)
7	3, 6	setsnid 17142	. 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 2733	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
10		eqid 2733	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
11		mnringnmulrdOLD.5	. . . 4 ⊢ 𝐴 = (Base‘𝑀)
12		eqid 2733	. . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀)
13		mnringnmulrdOLD.6	. . . 4 ⊢ 𝑉 = (𝑅 freeLMod 𝐴)
14		eqid 2733	. . . 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 42967	. . 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 6896	. 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 2792	1 ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ifcif 4529  ⟨cop 4635   ↦ cmpt 5232  ‘cfv 6544  (class class class)co 7409   ∈ cmpo 7411  ℕcn 12212   sSet csts 17096  Slot cslot 17114  ndxcnx 17126  Basecbs 17144  +_gcplusg 17197  ._rcmulr 17198  0_gc0g 17385   Σ_g cgsu 17386   freeLMod cfrlm 21301   MndRing cmnring 42965

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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-1cn 11168  ax-addcl 11170

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 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-nn 12213  df-sets 17097  df-slot 17115  df-ndx 17127  df-mnring 42966

This theorem is referenced by:  mnringbasedOLD  42971  mnringaddgdOLD  42977  mnringscadOLD  42982  mnringvscadOLD  42984