Theorem mnringvald 44204

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	⊢ · = (.r‘𝑅)
mnringvald.3	⊢ 0 = (0g‘𝑅)
mnringvald.4	⊢ 𝐴 = (Base‘𝑀)
mnringvald.5	⊢ + = (+g‘𝑀)
mnringvald.6	⊢ 𝑉 = (𝑅 freeLMod 𝐴)
mnringvald.7	⊢ 𝐵 = (Base‘𝑉)
mnringvald.8	⊢ (𝜑 → 𝑅 ∈ 𝑈)
mnringvald.9	⊢ (𝜑 → 𝑀 ∈ 𝑊)

Assertion

Ref	Expression
mnringvald	⊢ (𝜑 → 𝐹 = (𝑉 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 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.

Step	Hyp	Ref	Expression
1		mnringvald.1	. 2 ⊢ 𝐹 = (𝑅 MndRing 𝑀)
2		mnringvald.8	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑈)
3	2	elexd 3488	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
4		mnringvald.9	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑊)
5	4	elexd 3488	. . 3 ⊢ (𝜑 → 𝑀 ∈ V)
6		nfv 1914	. . . . 5 ⊢ Ⅎ𝑣(𝑟 = 𝑅 ∧ 𝑚 = 𝑀)
7		nfcvd 2900	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → Ⅎ𝑣(𝑉 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))))⟩))
8		ovexd 7445	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → (𝑟 freeLMod (Base‘𝑚)) ∈ V)
9		simpr 484	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → 𝑣 = (𝑟 freeLMod (Base‘𝑚)))
10		simpll 766	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → 𝑟 = 𝑅)
11		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
12		mnringvald.4	. . . . . . . . . . 11 ⊢ 𝐴 = (Base‘𝑀)
13	11, 12	eqtr4di 2789	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐴)
14	13	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (Base‘𝑚) = 𝐴)
15	10, 14	oveq12d 7428	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑟 freeLMod (Base‘𝑚)) = (𝑅 freeLMod 𝐴))
16	9, 15	eqtrd 2771	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → 𝑣 = (𝑅 freeLMod 𝐴))
17		mnringvald.6	. . . . . . 7 ⊢ 𝑉 = (𝑅 freeLMod 𝐴)
18	16, 17	eqtr4di 2789	. . . . . 6 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → 𝑣 = 𝑉)
19	18	fveq2d 6885	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (Base‘𝑣) = (Base‘𝑉))
20		mnringvald.7	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑉)
21	19, 20	eqtr4di 2789	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (Base‘𝑣) = 𝐵)
22		fveq2 6881	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
23		mnringvald.5	. . . . . . . . . . . . . . . 16 ⊢ + = (+g‘𝑀)
24	22, 23	eqtr4di 2789	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = + )
25	24	oveqd 7427	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑀 → (𝑎(+g‘𝑚)𝑏) = (𝑎 + 𝑏))
26	25	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑎(+g‘𝑚)𝑏) = (𝑎 + 𝑏))
27	26	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑖 = (𝑎(+g‘𝑚)𝑏) ↔ 𝑖 = (𝑎 + 𝑏)))
28		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
29		mnringvald.2	. . . . . . . . . . . . . . 15 ⊢ · = (.r‘𝑅)
30	28, 29	eqtr4di 2789	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
31	30	oveqd 7427	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → ((𝑥‘𝑎)(.r‘𝑟)(𝑦‘𝑏)) = ((𝑥‘𝑎) · (𝑦‘𝑏)))
32	31	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → ((𝑥‘𝑎)(.r‘𝑟)(𝑦‘𝑏)) = ((𝑥‘𝑎) · (𝑦‘𝑏)))
33		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
34		mnringvald.3	. . . . . . . . . . . . . 14 ⊢ 0 = (0g‘𝑅)
35	33, 34	eqtr4di 2789	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
36	35	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (0g‘𝑟) = 0 )
37	27, 32, 36	ifbieq12d 4534	. . . . . . . . . . 11 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → if(𝑖 = (𝑎(+g‘𝑚)𝑏), ((𝑥‘𝑎)(.r‘𝑟)(𝑦‘𝑏)), (0g‘𝑟)) = if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 ))
38	14, 37	mpteq12dv 5212	. . . . . . . . . 10 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+g‘𝑚)𝑏), ((𝑥‘𝑎)(.r‘𝑟)(𝑦‘𝑏)), (0g‘𝑟))) = (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))
39	14, 14, 38	mpoeq123dv 7487	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑎 ∈ (Base‘𝑚), 𝑏 ∈ (Base‘𝑚) ↦ (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+g‘𝑚)𝑏), ((𝑥‘𝑎)(.r‘𝑟)(𝑦‘𝑏)), (0g‘𝑟)))) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 ))))
40	18, 39	oveq12d 7428	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑣 Σg (𝑎 ∈ (Base‘𝑚), 𝑏 ∈ (Base‘𝑚) ↦ (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+g‘𝑚)𝑏), ((𝑥‘𝑎)(.r‘𝑟)(𝑦‘𝑏)), (0g‘𝑟))))) = (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))))
41	21, 21, 40	mpoeq123dv 7487	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 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(+g‘𝑚)𝑏), ((𝑥‘𝑎)(.r‘𝑟)(𝑦‘𝑏)), (0g‘𝑟)))))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 ))))))
42	41	opeq2d 4861	. . . . . 6 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 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(+g‘𝑚)𝑏), ((𝑥‘𝑎)(.r‘𝑟)(𝑦‘𝑏)), (0g‘𝑟))))))⟩ = ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))))⟩)
43	18, 42	oveq12d 7428	. . . . 5 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 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(+g‘𝑚)𝑏), ((𝑥‘𝑎)(.r‘𝑟)(𝑦‘𝑏)), (0g‘𝑟))))))⟩) = (𝑉 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))))⟩))
44	6, 7, 8, 43	csbiedf 3909	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 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(+g‘𝑚)𝑏), ((𝑥‘𝑎)(.r‘𝑟)(𝑦‘𝑏)), (0g‘𝑟))))))⟩) = (𝑉 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))))⟩))
45		df-mnring 44203	. . . 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(+g‘𝑚)𝑏), ((𝑥‘𝑎)(.r‘𝑟)(𝑦‘𝑏)), (0g‘𝑟))))))⟩))
46		ovex 7443	. . . 4 ⊢ (𝑉 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))))⟩) ∈ V
47	44, 45, 46	ovmpoa 7567	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑀 ∈ V) → (𝑅 MndRing 𝑀) = (𝑉 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))))⟩))
48	3, 5, 47	syl2anc 584	. 2 ⊢ (𝜑 → (𝑅 MndRing 𝑀) = (𝑉 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))))⟩))
49	1, 48	eqtrid 2783	1 ⊢ (𝜑 → 𝐹 = (𝑉 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑉 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))))⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3464  ⦋csb 3879  ifcif 4505  ⟨cop 4612   ↦ cmpt 5206  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412   sSet csts 17187  ndxcnx 17217  Basecbs 17233  +_gcplusg 17276  ._rcmulr 17277  0_gc0g 17458   Σ_g cgsu 17459   freeLMod cfrlm 21711   MndRing cmnring 44202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-mnring 44203

This theorem is referenced by:  mnringnmulrd  44205  mnringmulrd  44214