mnringvald - Mathbox for Rohan Ridenour

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Theorem mnringvald 42724

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 = (0_g‘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 Σ_g (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 3490	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
4		mnringvald.9	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑊)
5	4	elexd 3490	. . 3 ⊢ (𝜑 → 𝑀 ∈ V)
6		nfv 1917	. . . . 5 ⊢ Ⅎ𝑣(𝑟 = 𝑅 ∧ 𝑚 = 𝑀)
7		nfcvd 2903	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → Ⅎ𝑣(𝑉 sSet ⟨(._r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑉 Σ_g (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))))⟩))
8		ovexd 7425	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → (𝑟 freeLMod (Base‘𝑚)) ∈ V)
9		simpr 485	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → 𝑣 = (𝑟 freeLMod (Base‘𝑚)))
10		simpll 765	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → 𝑟 = 𝑅)
11		fveq2 6875	. . . . . . . . . . 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 725	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (Base‘𝑚) = 𝐴)
15	10, 14	oveq12d 7408	. . . . . . . 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 6879	. . . . . . . . 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 6875	. . . . . . . . . . . . . . . 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 7407	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑀 → (𝑎(+_g‘𝑚)𝑏) = (𝑎 + 𝑏))
26	25	ad2antlr 725	. . . . . . . . . . . . 13 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑎(+_g‘𝑚)𝑏) = (𝑎 + 𝑏))
27	26	eqeq2d 2742	. . . . . . . . . . . 12 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑖 = (𝑎(+_g‘𝑚)𝑏) ↔ 𝑖 = (𝑎 + 𝑏)))
28		fveq2 6875	. . . . . . . . . . . . . . 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 7407	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → ((𝑥‘𝑎)(._r‘𝑟)(𝑦‘𝑏)) = ((𝑥‘𝑎) · (𝑦‘𝑏)))
32	31	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → ((𝑥‘𝑎)(._r‘𝑟)(𝑦‘𝑏)) = ((𝑥‘𝑎) · (𝑦‘𝑏)))
33		fveq2 6875	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑅 → (0_g‘𝑟) = (0_g‘𝑅))
34		mnringvald.3	. . . . . . . . . . . . . 14 ⊢ 0 = (0_g‘𝑅)
35	33, 34	eqtr4di 2789	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (0_g‘𝑟) = 0 )
36	35	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (0_g‘𝑟) = 0 )
37	27, 32, 36	ifbieq12d 4547	. . . . . . . . . . 11 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → if(𝑖 = (𝑎(+_g‘𝑚)𝑏), ((𝑥‘𝑎)(._r‘𝑟)(𝑦‘𝑏)), (0_g‘𝑟)) = if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 ))
38	14, 37	mpteq12dv 5229	. . . . . . . . . 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)), (0_g‘𝑟))) = (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))
39	14, 14, 38	mpoeq123dv 7465	. . . . . . . . 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)), (0_g‘𝑟)))) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 ))))
40	18, 39	oveq12d 7408	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑣 Σ_g (𝑎 ∈ (Base‘𝑚), 𝑏 ∈ (Base‘𝑚) ↦ (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+_g‘𝑚)𝑏), ((𝑥‘𝑎)(._r‘𝑟)(𝑦‘𝑏)), (0_g‘𝑟))))) = (𝑉 Σ_g (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))))
41	21, 21, 40	mpoeq123dv 7465	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑥 ∈ (Base‘𝑣), 𝑦 ∈ (Base‘𝑣) ↦ (𝑣 Σ_g (𝑎 ∈ (Base‘𝑚), 𝑏 ∈ (Base‘𝑚) ↦ (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+_g‘𝑚)𝑏), ((𝑥‘𝑎)(._r‘𝑟)(𝑦‘𝑏)), (0_g‘𝑟)))))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑉 Σ_g (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 ))))))
42	41	opeq2d 4870	. . . . . 6 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → ⟨(._r‘ndx), (𝑥 ∈ (Base‘𝑣), 𝑦 ∈ (Base‘𝑣) ↦ (𝑣 Σ_g (𝑎 ∈ (Base‘𝑚), 𝑏 ∈ (Base‘𝑚) ↦ (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+_g‘𝑚)𝑏), ((𝑥‘𝑎)(._r‘𝑟)(𝑦‘𝑏)), (0_g‘𝑟))))))⟩ = ⟨(._r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑉 Σ_g (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))))⟩)
43	18, 42	oveq12d 7408	. . . . 5 ⊢ (((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) ∧ 𝑣 = (𝑟 freeLMod (Base‘𝑚))) → (𝑣 sSet ⟨(._r‘ndx), (𝑥 ∈ (Base‘𝑣), 𝑦 ∈ (Base‘𝑣) ↦ (𝑣 Σ_g (𝑎 ∈ (Base‘𝑚), 𝑏 ∈ (Base‘𝑚) ↦ (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+_g‘𝑚)𝑏), ((𝑥‘𝑎)(._r‘𝑟)(𝑦‘𝑏)), (0_g‘𝑟))))))⟩) = (𝑉 sSet ⟨(._r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑉 Σ_g (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))))⟩))
44	6, 7, 8, 43	csbiedf 3917	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → ⦋(𝑟 freeLMod (Base‘𝑚)) / 𝑣⦌(𝑣 sSet ⟨(._r‘ndx), (𝑥 ∈ (Base‘𝑣), 𝑦 ∈ (Base‘𝑣) ↦ (𝑣 Σ_g (𝑎 ∈ (Base‘𝑚), 𝑏 ∈ (Base‘𝑚) ↦ (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+_g‘𝑚)𝑏), ((𝑥‘𝑎)(._r‘𝑟)(𝑦‘𝑏)), (0_g‘𝑟))))))⟩) = (𝑉 sSet ⟨(._r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑉 Σ_g (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))))⟩))
45		df-mnring 42723	. . . 4 ⊢ MndRing = (𝑟 ∈ V, 𝑚 ∈ V ↦ ⦋(𝑟 freeLMod (Base‘𝑚)) / 𝑣⦌(𝑣 sSet ⟨(._r‘ndx), (𝑥 ∈ (Base‘𝑣), 𝑦 ∈ (Base‘𝑣) ↦ (𝑣 Σ_g (𝑎 ∈ (Base‘𝑚), 𝑏 ∈ (Base‘𝑚) ↦ (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+_g‘𝑚)𝑏), ((𝑥‘𝑎)(._r‘𝑟)(𝑦‘𝑏)), (0_g‘𝑟))))))⟩))
46		ovex 7423	. . . 4 ⊢ (𝑉 sSet ⟨(._r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑉 Σ_g (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))))⟩) ∈ V
47	44, 45, 46	ovmpoa 7543	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑀 ∈ V) → (𝑅 MndRing 𝑀) = (𝑉 sSet ⟨(._r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑉 Σ_g (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 Σ_g (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 Σ_g (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎 + 𝑏), ((𝑥‘𝑎) · (𝑦‘𝑏)), 0 )))))⟩))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3470 ⦋csb 3886 ifcif 4519 ⟨cop 4625 ↦ cmpt 5221 ‘cfv 6529 (class class class)co 7390 ∈ cmpo 7392 sSet csts 17075 ndxcnx 17105 Basecbs 17123 +_gcplusg 17176 ._rcmulr 17177 0_gc0g 17364 Σ_g cgsu 17365 freeLMod cfrlm 21229 MndRing cmnring 42722

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 2702 ax-sep 5289 ax-nul 5296 ax-pr 5417

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4520 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-iota 6481 df-fun 6531 df-fv 6537 df-ov 7393 df-oprab 7394 df-mpo 7395 df-mnring 42723

This theorem is referenced by: mnringnmulrd 42725 mnringnmulrdOLD 42726 mnringmulrd 42737

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