Theorem psdmvr 22129

Description: The partial derivative of a variable is the Kronecker delta if(𝑋 = 𝑌, 1 , 0 ). (Contributed by SN, 16-Oct-2025.)

Hypotheses

Ref	Expression
psdmvr.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
psdmvr.z	⊢ 0 = (0g‘𝑆)
psdmvr.o	⊢ 1 = (1r‘𝑆)
psdmvr.v	⊢ 𝑉 = (𝐼 mVar 𝑅)
psdmvr.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
psdmvr.r	⊢ (𝜑 → 𝑅 ∈ Ring)
psdmvr.x	⊢ (𝜑 → 𝑋 ∈ 𝐼)
psdmvr.y	⊢ (𝜑 → 𝑌 ∈ 𝐼)

Assertion

Ref	Expression
psdmvr	⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑉‘𝑌)) = if(𝑋 = 𝑌, 1 , 0 ))

Proof of Theorem psdmvr

Dummy variables ℎ 𝑘 𝑦 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psdmvr.s	. . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
2		eqid 2737	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2737	. . 3 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
4		psdmvr.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
5		psdmvr.v	. . . 4 ⊢ 𝑉 = (𝐼 mVar 𝑅)
6		psdmvr.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
7		psdmvr.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
8		psdmvr.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐼)
9	1, 5, 2, 6, 7, 8	mvrcl2 21959	. . 3 ⊢ (𝜑 → (𝑉‘𝑌) ∈ (Base‘𝑆))
10	1, 2, 3, 4, 9	psdval 22119	. 2 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑉‘𝑌)) = (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑘‘𝑋) + 1)(.g‘𝑅)((𝑉‘𝑌)‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))))
11		eqid 2737	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
12		eqid 2737	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
13	6	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑊)
14	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
15	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑌 ∈ 𝐼)
16		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
17	3	psrbagsn 22035	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝑊 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
18	6, 17	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
19	18	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
20	3	psrbagaddcl 21897	. . . . . . . 8 ⊢ ((𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
21	16, 19, 20	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
22	5, 3, 11, 12, 13, 14, 15, 21	mvrval2 21955	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑉‘𝑌)‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = if((𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0)), (1r‘𝑅), (0g‘𝑅)))
23		1red 11147	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))) → 1 ∈ ℝ)
24	3	psrbagf 21891	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑘:𝐼⟶ℕ0)
25	24	ad2antlr 728	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))) → 𝑘:𝐼⟶ℕ0)
26	4	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))) → 𝑋 ∈ 𝐼)
27	25, 26	ffvelcdmd 7041	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))) → (𝑘‘𝑋) ∈ ℕ0)
28		nn0addge2 12462	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ ∧ (𝑘‘𝑋) ∈ ℕ0) → 1 ≤ ((𝑘‘𝑋) + 1))
29	23, 27, 28	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))) → 1 ≤ ((𝑘‘𝑋) + 1))
30		fveq1 6843	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0)) → ((𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) = ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))‘𝑋))
31	30	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))) → ((𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) = ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))‘𝑋))
32	24	ffnd 6673	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑘 Fn 𝐼)
33	32	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑘 Fn 𝐼)
34		1re 11146	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℝ
35		0re 11148	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℝ
36	34, 35	ifcli 4529	. . . . . . . . . . . . . . . . . . . 20 ⊢ if(𝑦 = 𝑋, 1, 0) ∈ ℝ
37	36	elexi 3465	. . . . . . . . . . . . . . . . . . 19 ⊢ if(𝑦 = 𝑋, 1, 0) ∈ V
38		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))
39	37, 38	fnmpti 6645	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼
40	39	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) Fn 𝐼)
41		inidm 4181	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ∩ 𝐼) = 𝐼
42		eqidd 2738	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑋 ∈ 𝐼) → (𝑘‘𝑋) = (𝑘‘𝑋))
43		iftrue 4487	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, 1, 0) = 1)
44		1ex 11142	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ V
45	43, 38, 44	fvmpt 6951	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ 𝐼 → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑋) = 1)
46	45	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑋 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑋) = 1)
47	33, 40, 13, 13, 41, 42, 46	ofval 7645	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑋 ∈ 𝐼) → ((𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) = ((𝑘‘𝑋) + 1))
48	4, 47	mpidan 690	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) = ((𝑘‘𝑋) + 1))
49	48	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))) → ((𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))‘𝑋) = ((𝑘‘𝑋) + 1))
50		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))
51		eqeq1 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑋 → (𝑦 = 𝑌 ↔ 𝑋 = 𝑌))
52	51	ifbid 4505	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑌, 1, 0) = if(𝑋 = 𝑌, 1, 0))
53	34, 35	ifcli 4529	. . . . . . . . . . . . . . . . 17 ⊢ if(𝑋 = 𝑌, 1, 0) ∈ ℝ
54	53	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → if(𝑋 = 𝑌, 1, 0) ∈ ℝ)
55	50, 52, 4, 54	fvmptd3 6975	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))‘𝑋) = if(𝑋 = 𝑌, 1, 0))
56	55	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))‘𝑋) = if(𝑋 = 𝑌, 1, 0))
57	31, 49, 56	3eqtr3d 2780	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))) → ((𝑘‘𝑋) + 1) = if(𝑋 = 𝑌, 1, 0))
58	29, 57	breqtrd 5126	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))) → 1 ≤ if(𝑋 = 𝑌, 1, 0))
59		1le1 11779	. . . . . . . . . . . . . 14 ⊢ 1 ≤ 1
60		0le1 11674	. . . . . . . . . . . . . 14 ⊢ 0 ≤ 1
61		anifp 1072	. . . . . . . . . . . . . 14 ⊢ ((1 ≤ 1 ∧ 0 ≤ 1) → if-(𝑋 = 𝑌, 1 ≤ 1, 0 ≤ 1))
62	59, 60, 61	mp2an 693	. . . . . . . . . . . . 13 ⊢ if-(𝑋 = 𝑌, 1 ≤ 1, 0 ≤ 1)
63		brif1 7467	. . . . . . . . . . . . 13 ⊢ (if(𝑋 = 𝑌, 1, 0) ≤ 1 ↔ if-(𝑋 = 𝑌, 1 ≤ 1, 0 ≤ 1))
64	62, 63	mpbir 231	. . . . . . . . . . . 12 ⊢ if(𝑋 = 𝑌, 1, 0) ≤ 1
65	34, 53	letri3i 11263	. . . . . . . . . . . 12 ⊢ (1 = if(𝑋 = 𝑌, 1, 0) ↔ (1 ≤ if(𝑋 = 𝑌, 1, 0) ∧ if(𝑋 = 𝑌, 1, 0) ≤ 1))
66	58, 64, 65	sylanblrc 591	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))) → 1 = if(𝑋 = 𝑌, 1, 0))
67	66	eqcomd 2743	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))) → if(𝑋 = 𝑌, 1, 0) = 1)
68		ax-1ne0 11109	. . . . . . . . . . 11 ⊢ 1 ≠ 0
69		iftrueb 4494	. . . . . . . . . . 11 ⊢ (1 ≠ 0 → (if(𝑋 = 𝑌, 1, 0) = 1 ↔ 𝑋 = 𝑌))
70	68, 69	ax-mp 5	. . . . . . . . . 10 ⊢ (if(𝑋 = 𝑌, 1, 0) = 1 ↔ 𝑋 = 𝑌)
71	67, 70	sylib 218	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))) → 𝑋 = 𝑌)
72		eqeq2 2749	. . . . . . . . . . . . . 14 ⊢ (𝑋 = 𝑌 → (𝑦 = 𝑋 ↔ 𝑦 = 𝑌))
73	72	ifbid 4505	. . . . . . . . . . . . 13 ⊢ (𝑋 = 𝑌 → if(𝑦 = 𝑋, 1, 0) = if(𝑦 = 𝑌, 1, 0))
74	73	mpteq2dv 5194	. . . . . . . . . . . 12 ⊢ (𝑋 = 𝑌 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0)))
75	74	oveq2d 7386	. . . . . . . . . . 11 ⊢ (𝑋 = 𝑌 → (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))))
76	75	eqeq1d 2739	. . . . . . . . . 10 ⊢ (𝑋 = 𝑌 → ((𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0)) ↔ (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))))
77	24	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑘:𝐼⟶ℕ0)
78		1nn0 12431	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℕ0
79		0nn0 12430	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℕ0
80	78, 79	ifcli 4529	. . . . . . . . . . . . . 14 ⊢ if(𝑦 = 𝑌, 1, 0) ∈ ℕ0
81	80	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐼 → if(𝑦 = 𝑌, 1, 0) ∈ ℕ0)
82	50, 81	fmpti 7068	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0)):𝐼⟶ℕ0
83	82	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0)):𝐼⟶ℕ0)
84		nn0cn 12425	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℂ)
85		nn0cn 12425	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℂ)
86		addcom 11333	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝑛 + 𝑚) = (𝑚 + 𝑛))
87	86	eqeq1d 2739	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → ((𝑛 + 𝑚) = 𝑚 ↔ (𝑚 + 𝑛) = 𝑚))
88		addid0 11570	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑚 + 𝑛) = 𝑚 ↔ 𝑛 = 0))
89	88	ancoms 458	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → ((𝑚 + 𝑛) = 𝑚 ↔ 𝑛 = 0))
90	87, 89	bitrd 279	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → ((𝑛 + 𝑚) = 𝑚 ↔ 𝑛 = 0))
91	84, 85, 90	syl2an 597	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) → ((𝑛 + 𝑚) = 𝑚 ↔ 𝑛 = 0))
92	91	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) → ((𝑛 + 𝑚) = 𝑚 ↔ 𝑛 = 0))
93	13, 77, 83, 92	caofidlcan 7672	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0)) ↔ 𝑘 = (𝐼 × {0})))
94	76, 93	sylan9bbr 510	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑋 = 𝑌) → ((𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0)) ↔ 𝑘 = (𝐼 × {0})))
95	71, 94	biadanid 823	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0)) ↔ (𝑋 = 𝑌 ∧ 𝑘 = (𝐼 × {0}))))
96	95	biancomd 463	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0)) ↔ (𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌)))
97	96	ifbid 4505	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → if((𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0)), (1r‘𝑅), (0g‘𝑅)) = if((𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌), (1r‘𝑅), (0g‘𝑅)))
98	22, 97	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑉‘𝑌)‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = if((𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌), (1r‘𝑅), (0g‘𝑅)))
99	98	oveq2d 7386	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝑘‘𝑋) + 1)(.g‘𝑅)((𝑉‘𝑌)‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) = (((𝑘‘𝑋) + 1)(.g‘𝑅)if((𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌), (1r‘𝑅), (0g‘𝑅))))
100		ovif2 7469	. . . . 5 ⊢ (((𝑘‘𝑋) + 1)(.g‘𝑅)if((𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌), (1r‘𝑅), (0g‘𝑅))) = if((𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌), (((𝑘‘𝑋) + 1)(.g‘𝑅)(1r‘𝑅)), (((𝑘‘𝑋) + 1)(.g‘𝑅)(0g‘𝑅)))
101		fveq1 6843	. . . . . . . . . . 11 ⊢ (𝑘 = (𝐼 × {0}) → (𝑘‘𝑋) = ((𝐼 × {0})‘𝑋))
102	101	oveq1d 7385	. . . . . . . . . 10 ⊢ (𝑘 = (𝐼 × {0}) → ((𝑘‘𝑋) + 1) = (((𝐼 × {0})‘𝑋) + 1))
103	4	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑋 ∈ 𝐼)
104		c0ex 11140	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
105	104	fvconst2 7162	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝐼 → ((𝐼 × {0})‘𝑋) = 0)
106	103, 105	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝐼 × {0})‘𝑋) = 0)
107	106	oveq1d 7385	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝐼 × {0})‘𝑋) + 1) = (0 + 1))
108		0p1e1 12276	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
109	107, 108	eqtrdi 2788	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝐼 × {0})‘𝑋) + 1) = 1)
110	102, 109	sylan9eqr 2794	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 = (𝐼 × {0})) → ((𝑘‘𝑋) + 1) = 1)
111	110	adantrr 718	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌)) → ((𝑘‘𝑋) + 1) = 1)
112	111	oveq1d 7385	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌)) → (((𝑘‘𝑋) + 1)(.g‘𝑅)(1r‘𝑅)) = (1(.g‘𝑅)(1r‘𝑅)))
113		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
114	113, 12, 7	ringidcld 20218	. . . . . . . . 9 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
115		eqid 2737	. . . . . . . . . 10 ⊢ (.g‘𝑅) = (.g‘𝑅)
116	113, 115	mulg1 19028	. . . . . . . . 9 ⊢ ((1r‘𝑅) ∈ (Base‘𝑅) → (1(.g‘𝑅)(1r‘𝑅)) = (1r‘𝑅))
117	114, 116	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1(.g‘𝑅)(1r‘𝑅)) = (1r‘𝑅))
118	117	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌)) → (1(.g‘𝑅)(1r‘𝑅)) = (1r‘𝑅))
119	112, 118	eqtrd 2772	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌)) → (((𝑘‘𝑋) + 1)(.g‘𝑅)(1r‘𝑅)) = (1r‘𝑅))
120	7	ringgrpd 20194	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Grp)
121	120	grpmndd 18893	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Mnd)
122	121	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ Mnd)
123	77, 103	ffvelcdmd 7041	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑘‘𝑋) ∈ ℕ0)
124	78	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 1 ∈ ℕ0)
125	123, 124	nn0addcld 12480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑘‘𝑋) + 1) ∈ ℕ0)
126	113, 115, 11	mulgnn0z 19048	. . . . . . . 8 ⊢ ((𝑅 ∈ Mnd ∧ ((𝑘‘𝑋) + 1) ∈ ℕ0) → (((𝑘‘𝑋) + 1)(.g‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
127	122, 125, 126	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝑘‘𝑋) + 1)(.g‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
128	127	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ ¬ (𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌)) → (((𝑘‘𝑋) + 1)(.g‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
129	119, 128	ifeq12da 4515	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → if((𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌), (((𝑘‘𝑋) + 1)(.g‘𝑅)(1r‘𝑅)), (((𝑘‘𝑋) + 1)(.g‘𝑅)(0g‘𝑅))) = if((𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌), (1r‘𝑅), (0g‘𝑅)))
130	100, 129	eqtrid 2784	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝑘‘𝑋) + 1)(.g‘𝑅)if((𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌), (1r‘𝑅), (0g‘𝑅))) = if((𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌), (1r‘𝑅), (0g‘𝑅)))
131		ancom 460	. . . . . . 7 ⊢ ((𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌) ↔ (𝑋 = 𝑌 ∧ 𝑘 = (𝐼 × {0})))
132		ifbi 4504	. . . . . . 7 ⊢ (((𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌) ↔ (𝑋 = 𝑌 ∧ 𝑘 = (𝐼 × {0}))) → if((𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌), (1r‘𝑅), (0g‘𝑅)) = if((𝑋 = 𝑌 ∧ 𝑘 = (𝐼 × {0})), (1r‘𝑅), (0g‘𝑅)))
133	131, 132	ax-mp 5	. . . . . 6 ⊢ if((𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌), (1r‘𝑅), (0g‘𝑅)) = if((𝑋 = 𝑌 ∧ 𝑘 = (𝐼 × {0})), (1r‘𝑅), (0g‘𝑅))
134		ifan 4535	. . . . . 6 ⊢ if((𝑋 = 𝑌 ∧ 𝑘 = (𝐼 × {0})), (1r‘𝑅), (0g‘𝑅)) = if(𝑋 = 𝑌, if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)), (0g‘𝑅))
135	133, 134	eqtri 2760	. . . . 5 ⊢ if((𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌), (1r‘𝑅), (0g‘𝑅)) = if(𝑋 = 𝑌, if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)), (0g‘𝑅))
136	135	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → if((𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌), (1r‘𝑅), (0g‘𝑅)) = if(𝑋 = 𝑌, if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)), (0g‘𝑅)))
137	99, 130, 136	3eqtrd 2776	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝑘‘𝑋) + 1)(.g‘𝑅)((𝑉‘𝑌)‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) = if(𝑋 = 𝑌, if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)), (0g‘𝑅)))
138	137	mpteq2dva 5193	. 2 ⊢ (𝜑 → (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑘‘𝑋) + 1)(.g‘𝑅)((𝑉‘𝑌)‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))) = (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑋 = 𝑌, if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)), (0g‘𝑅))))
139		ifmpt2v 7472	. . 3 ⊢ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑋 = 𝑌, if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)), (0g‘𝑅))) = if(𝑋 = 𝑌, (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))), (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (0g‘𝑅)))
140		psdmvr.o	. . . . 5 ⊢ 1 = (1r‘𝑆)
141	1, 6, 7, 3, 11, 12, 140	psr1 21943	. . . 4 ⊢ (𝜑 → 1 = (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))))
142		psdmvr.z	. . . . . 6 ⊢ 0 = (0g‘𝑆)
143	1, 6, 120, 3, 11, 142	psr0 21930	. . . . 5 ⊢ (𝜑 → 0 = ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝑅)}))
144		fconstmpt 5696	. . . . 5 ⊢ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝑅)}) = (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (0g‘𝑅))
145	143, 144	eqtrdi 2788	. . . 4 ⊢ (𝜑 → 0 = (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (0g‘𝑅)))
146	141, 145	ifeq12d 4503	. . 3 ⊢ (𝜑 → if(𝑋 = 𝑌, 1 , 0 ) = if(𝑋 = 𝑌, (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))), (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (0g‘𝑅))))
147	139, 146	eqtr4id 2791	. 2 ⊢ (𝜑 → (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑋 = 𝑌, if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)), (0g‘𝑅))) = if(𝑋 = 𝑌, 1 , 0 ))
148	10, 138, 147	3eqtrd 2776	1 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑉‘𝑌)) = if(𝑋 = 𝑌, 1 , 0 ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  if-wif 1063   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  {crab 3401  ifcif 4481  {csn 4582   class class class wbr 5100   ↦ cmpt 5181   × cxp 5632  ^◡ccnv 5633   “ cima 5637   Fn wfn 6497  ⟶wf 6498  ‘cfv 6502  (class class class)co 7370   ∘_f cof 7632   ↑_m cmap 8777  Fincfn 8897  ℂcc 11038  ℝcr 11039  0cc0 11040  1c1 11041   + caddc 11043   ≤ cle 11181  ℕcn 12159  ℕ₀cn0 12415  Basecbs 17150  0_gc0g 17373  Mndcmnd 18673  ._gcmg 19014  1_rcur 20133  Ringcrg 20185   mPwSer cmps 21877   mVar cmvr 21878   mPSDer cpsd 22090

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-se 5588  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-isom 6511  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-of 7634  df-ofr 7635  df-om 7821  df-1st 7945  df-2nd 7946  df-supp 8115  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-2o 8410  df-er 8647  df-map 8779  df-pm 8780  df-ixp 8850  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-fsupp 9279  df-sup 9359  df-oi 9429  df-card 9865  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-2 12222  df-3 12223  df-4 12224  df-5 12225  df-6 12226  df-7 12227  df-8 12228  df-9 12229  df-n0 12416  df-z 12503  df-dec 12622  df-uz 12766  df-fz 13438  df-fzo 13585  df-seq 13939  df-hash 14268  df-struct 17088  df-sets 17105  df-slot 17123  df-ndx 17135  df-base 17151  df-ress 17172  df-plusg 17204  df-mulr 17205  df-sca 17207  df-vsca 17208  df-ip 17209  df-tset 17210  df-ple 17211  df-ds 17213  df-hom 17215  df-cco 17216  df-0g 17375  df-gsum 17376  df-prds 17381  df-pws 17383  df-mre 17519  df-mrc 17520  df-acs 17522  df-mgm 18579  df-sgrp 18658  df-mnd 18674  df-mhm 18722  df-submnd 18723  df-grp 18883  df-minusg 18884  df-mulg 19015  df-ghm 19159  df-cntz 19263  df-cmn 19728  df-abl 19729  df-mgp 20093  df-rng 20105  df-ur 20134  df-ring 20187  df-psr 21882  df-mvr 21883  df-psd 22116

This theorem is referenced by: (None)