Theorem psdmvr 22135

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 2736	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2736	. . 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 21965	. . 3 ⊢ (𝜑 → (𝑉‘𝑌) ∈ (Base‘𝑆))
10	1, 2, 3, 4, 9	psdval 22125	. 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 2736	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
12		eqid 2736	. . . . . . 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 22041	. . . . . . . . . 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 21904	. . . . . . . 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 21961	. . . . . 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 11145	. . . . . . . . . . . . . 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 21898	. . . . . . . . . . . . . . . 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 7037	. . . . . . . . . . . . . 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 12484	. . . . . . . . . . . . . 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 6839	. . . . . . . . . . . . . . 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 6669	. . . . . . . . . . . . . . . . . 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 11144	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℝ
35		0re 11146	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℝ
36	34, 35	ifcli 4514	. . . . . . . . . . . . . . . . . . . 20 ⊢ if(𝑦 = 𝑋, 1, 0) ∈ ℝ
37	36	elexi 3452	. . . . . . . . . . . . . . . . . . 19 ⊢ if(𝑦 = 𝑋, 1, 0) ∈ V
38		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))
39	37, 38	fnmpti 6641	. . . . . . . . . . . . . . . . . 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 4167	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ∩ 𝐼) = 𝐼
42		eqidd 2737	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑋 ∈ 𝐼) → (𝑘‘𝑋) = (𝑘‘𝑋))
43		iftrue 4472	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, 1, 0) = 1)
44		1ex 11140	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ V
45	43, 38, 44	fvmpt 6947	. . . . . . . . . . . . . . . . . 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 7642	. . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))
51		eqeq1 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑋 → (𝑦 = 𝑌 ↔ 𝑋 = 𝑌))
52	51	ifbid 4490	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑌, 1, 0) = if(𝑋 = 𝑌, 1, 0))
53	34, 35	ifcli 4514	. . . . . . . . . . . . . . . . 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 6971	. . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . 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 5111	. . . . . . . . . . . 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 11778	. . . . . . . . . . . . . 14 ⊢ 1 ≤ 1
60		0le1 11673	. . . . . . . . . . . . . 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 7464	. . . . . . . . . . . . 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 11262	. . . . . . . . . . . 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 2742	. . . . . . . . . 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 11107	. . . . . . . . . . 11 ⊢ 1 ≠ 0
69		iftrueb 4479	. . . . . . . . . . 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 2748	. . . . . . . . . . . . . 14 ⊢ (𝑋 = 𝑌 → (𝑦 = 𝑋 ↔ 𝑦 = 𝑌))
73	72	ifbid 4490	. . . . . . . . . . . . 13 ⊢ (𝑋 = 𝑌 → if(𝑦 = 𝑋, 1, 0) = if(𝑦 = 𝑌, 1, 0))
74	73	mpteq2dv 5179	. . . . . . . . . . . 12 ⊢ (𝑋 = 𝑌 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0)))
75	74	oveq2d 7383	. . . . . . . . . . 11 ⊢ (𝑋 = 𝑌 → (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))))
76	75	eqeq1d 2738	. . . . . . . . . 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 12453	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℕ0
79		0nn0 12452	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℕ0
80	78, 79	ifcli 4514	. . . . . . . . . . . . . 14 ⊢ if(𝑦 = 𝑌, 1, 0) ∈ ℕ0
81	80	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐼 → if(𝑦 = 𝑌, 1, 0) ∈ ℕ0)
82	50, 81	fmpti 7064	. . . . . . . . . . . 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 12447	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℂ)
85		nn0cn 12447	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℂ)
86		addcom 11332	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝑛 + 𝑚) = (𝑚 + 𝑛))
87	86	eqeq1d 2738	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → ((𝑛 + 𝑚) = 𝑚 ↔ (𝑚 + 𝑛) = 𝑚))
88		addid0 11569	. . . . . . . . . . . . . . 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 7669	. . . . . . . . . 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 4490	. . . . . 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 2771	. . . . 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 7383	. . . 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 7466	. . . . 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 6839	. . . . . . . . . . 11 ⊢ (𝑘 = (𝐼 × {0}) → (𝑘‘𝑋) = ((𝐼 × {0})‘𝑋))
102	101	oveq1d 7382	. . . . . . . . . 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 11138	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
105	104	fvconst2 7159	. . . . . . . . . . . . 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 7382	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝐼 × {0})‘𝑋) + 1) = (0 + 1))
108		0p1e1 12298	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
109	107, 108	eqtrdi 2787	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝐼 × {0})‘𝑋) + 1) = 1)
110	102, 109	sylan9eqr 2793	. . . . . . . . 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 7382	. . . . . . 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 2736	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
114	113, 12, 7	ringidcld 20247	. . . . . . . . 9 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
115		eqid 2736	. . . . . . . . . 10 ⊢ (.g‘𝑅) = (.g‘𝑅)
116	113, 115	mulg1 19057	. . . . . . . . 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 2771	. . . . . 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 20223	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Grp)
121	120	grpmndd 18922	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Mnd)
122	121	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ Mnd)
123	77, 103	ffvelcdmd 7037	. . . . . . . . 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 12502	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑘‘𝑋) + 1) ∈ ℕ0)
126	113, 115, 11	mulgnn0z 19077	. . . . . . . 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 4500	. . . . 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 2783	. . . 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 4489	. . . . . . 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 4520	. . . . . 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 2759	. . . . 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 2775	. . 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 5178	. 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 7469	. . 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 21949	. . . 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 21936	. . . . 5 ⊢ (𝜑 → 0 = ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝑅)}))
144		fconstmpt 5693	. . . . 5 ⊢ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝑅)}) = (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (0g‘𝑅))
145	143, 144	eqtrdi 2787	. . . 4 ⊢ (𝜑 → 0 = (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (0g‘𝑅)))
146	141, 145	ifeq12d 4488	. . 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 2790	. 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 2775	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 2932  {crab 3389  ifcif 4466  {csn 4567   class class class wbr 5085   ↦ cmpt 5166   × cxp 5629  ^◡ccnv 5630   “ cima 5634   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ∘_f cof 7629   ↑_m cmap 8773  Fincfn 8893  ℂcc 11036  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041   ≤ cle 11180  ℕcn 12174  ℕ₀cn0 12437  Basecbs 17179  0_gc0g 17402  Mndcmnd 18702  ._gcmg 19043  1_rcur 20162  Ringcrg 20214   mPwSer cmps 21884   mVar cmvr 21885   mPSDer cpsd 22096

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-ofr 7632  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-pm 8776  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-sup 9355  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-fzo 13609  df-seq 13964  df-hash 14293  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-mre 17548  df-mrc 17549  df-acs 17551  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-grp 18912  df-minusg 18913  df-mulg 19044  df-ghm 19188  df-cntz 19292  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-ring 20216  df-psr 21889  df-mvr 21890  df-psd 22122

This theorem is referenced by: (None)