Theorem mnringvald 45196

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 3474	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
4		mnringvald.9	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑊)
5	4	elexd 3474	. . 3 ⊢ (𝜑 → 𝑀 ∈ V)
6		nfv 1947	. . . . 5 ⊢ Ⅎ𝑣(𝑟 = 𝑅 ∧ 𝑚 = 𝑀)
7		nfcvd 2924	. . . . 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 7453	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → (𝑟 freeLMod (Base‘𝑚)) ∈ V)
9		simpr 490	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → 𝑣 = (𝑟 freeLMod (Base‘𝑚)))
10		simpll 779	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → 𝑟 = 𝑅)
11		fveq2 6883	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
12		mnringvald.4	. . . . . . . . . . 11 ⊢ 𝐴 = (Base‘𝑀)
13	11, 12	eqtr4di 2814	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐴)
14	13	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (Base‘𝑚) = 𝐴)
15	10, 14	oveq12d 7436	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑟 freeLMod (Base‘𝑚)) = (𝑅 freeLMod 𝐴))
16	9, 15	eqtrd 2796	. . . . . . 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 2814	. . . . . 6 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → 𝑣 = 𝑉)
19	18	fveq2d 6887	. . . . . . . . 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 2814	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (Base‘𝑣) = 𝐵)
22		fveq2 6883	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
23		mnringvald.5	. . . . . . . . . . . . . . . 16 ⊢ + = (+g‘𝑀)
24	22, 23	eqtr4di 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = + )
25	24	oveqd 7435	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑀 → (𝑎(+g‘𝑚)𝑏) = (𝑎 + 𝑏))
26	25	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑎(+g‘𝑚)𝑏) = (𝑎 + 𝑏))
27	26	eqeq2d 2772	. . . . . . . . . . . 12 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑖 = (𝑎(+g‘𝑚)𝑏) ↔ 𝑖 = (𝑎 + 𝑏)))
28		fveq2 6883	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
29		mnringvald.2	. . . . . . . . . . . . . . 15 ⊢ · = (.r‘𝑅)
30	28, 29	eqtr4di 2814	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
31	30	oveqd 7435	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → ((𝑥‘𝑎)(.r‘𝑟)(𝑦‘𝑏)) = ((𝑥‘𝑎) · (𝑦‘𝑏)))
32	31	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → ((𝑥‘𝑎)(.r‘𝑟)(𝑦‘𝑏)) = ((𝑥‘𝑎) · (𝑦‘𝑏)))
33		fveq2 6883	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
34		mnringvald.3	. . . . . . . . . . . . . 14 ⊢ 0 = (0g‘𝑅)
35	33, 34	eqtr4di 2814	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
36	35	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (0g‘𝑟) = 0 )
37	27, 32, 36	ifbieq12d 4511	. . . . . . . . . . 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 5192	. . . . . . . . . 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 7493	. . . . . . . . 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 7436	. . . . . . . 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 7493	. . . . . . 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 4840	. . . . . 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 7436	. . . . 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 3877	. . . 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 45195	. . . 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 7451	. . . 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 7573	. . 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 596	. 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 2808	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 )))))⟩))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ⦋csb 3847  ifcif 4482  ⟨cop 4590   ↦ cmpt 5186  ‘cfv 6537  (class class class)co 7418   ∈ cmpo 7420   sSet csts 17334  ndxcnx 17364  Basecbs 17380  +_gcplusg 17421  ._rcmulr 17422  0_gc0g 17603   Σ_g cgsu 17604   freeLMod cfrlm 22045   MndRing cmnring 45194

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-mnring 45195

This theorem is used by:  mnringnmulrd  45197  mnringmulrd  45206