Theorem psdmvr 22301

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 2769	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2769	. . 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 22105	. . 3 ⊢ (𝜑 → (𝑉‘𝑌) ∈ (Base‘𝑆))
10	1, 2, 3, 4, 9	psdval 22291	. 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 2769	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
12		eqid 2769	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
13	6	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑊)
14	7	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
15	8	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑌 ∈ 𝐼)
16		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
17	3	psrbagsn 22183	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝑊 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
18	6, 17	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
19	18	adantr 485	. . . . . . . 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 22043	. . . . . . . 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 595	. . . . . . 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 22101	. . . . . 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 11209	. . . . . . . . . . . . . 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 22037	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑘:𝐼⟶ℕ0)
25	24	ad2antlr 739	. . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . 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 7081	. . . . . . . . . . . . . 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 12551	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ ∧ (𝑘‘𝑋) ∈ ℕ0) → 1 ≤ ((𝑘‘𝑋) + 1))
29	23, 27, 28	syl2anc 595	. . . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 6707	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑘 Fn 𝐼)
33	32	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑘 Fn 𝐼)
34		1re 11208	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℝ
35		0re 11210	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℝ
36	34, 35	ifcli 4538	. . . . . . . . . . . . . . . . . . . 20 ⊢ if(𝑦 = 𝑋, 1, 0) ∈ ℝ
37	36	elexi 3483	. . . . . . . . . . . . . . . . . . 19 ⊢ if(𝑦 = 𝑋, 1, 0) ∈ V
38		eqid 2769	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))
39	37, 38	fnmpti 6679	. . . . . . . . . . . . . . . . . 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 4185	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ∩ 𝐼) = 𝐼
42		eqidd 2770	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑋 ∈ 𝐼) → (𝑘‘𝑋) = (𝑘‘𝑋))
43		iftrue 4496	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, 1, 0) = 1)
44		1ex 11203	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ V
45	43, 38, 44	fvmpt 6990	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ 𝐼 → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑋) = 1)
46	45	adantl 486	. . . . . . . . . . . . . . . . 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 7686	. . . . . . . . . . . . . . . 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 701	. . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))
51		eqeq1 2773	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑋 → (𝑦 = 𝑌 ↔ 𝑋 = 𝑌))
52	51	ifbid 4514	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑌, 1, 0) = if(𝑋 = 𝑌, 1, 0))
53	34, 35	ifcli 4538	. . . . . . . . . . . . . . . . 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 7014	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))‘𝑋) = if(𝑋 = 𝑌, 1, 0))
56	55	ad2antrr 738	. . . . . . . . . . . . . 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 2812	. . . . . . . . . . . . 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 5139	. . . . . . . . . . . 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 11842	. . . . . . . . . . . . . 14 ⊢ 1 ≤ 1
60		0le1 11737	. . . . . . . . . . . . . 14 ⊢ 0 ≤ 1
61		anifp 1086	. . . . . . . . . . . . . 14 ⊢ ((1 ≤ 1 ∧ 0 ≤ 1) → if-(𝑋 = 𝑌, 1 ≤ 1, 0 ≤ 1))
62	59, 60, 61	mp2an 704	. . . . . . . . . . . . 13 ⊢ if-(𝑋 = 𝑌, 1 ≤ 1, 0 ≤ 1)
63		brif1 7508	. . . . . . . . . . . . 13 ⊢ (if(𝑋 = 𝑌, 1, 0) ≤ 1 ↔ if-(𝑋 = 𝑌, 1 ≤ 1, 0 ≤ 1))
64	62, 63	mpbir 234	. . . . . . . . . . . 12 ⊢ if(𝑋 = 𝑌, 1, 0) ≤ 1
65	34, 53	letri3i 11326	. . . . . . . . . . . 12 ⊢ (1 = if(𝑋 = 𝑌, 1, 0) ↔ (1 ≤ if(𝑋 = 𝑌, 1, 0) ∧ if(𝑋 = 𝑌, 1, 0) ≤ 1))
66	58, 64, 65	sylanblrc 601	. . . . . . . . . . 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 2775	. . . . . . . . . 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 11169	. . . . . . . . . . 11 ⊢ 1 ≠ 0
69		iftrueb 4503	. . . . . . . . . . 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 221	. . . . . . . . 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 2781	. . . . . . . . . . . . . 14 ⊢ (𝑋 = 𝑌 → (𝑦 = 𝑋 ↔ 𝑦 = 𝑌))
73	72	ifbid 4514	. . . . . . . . . . . . 13 ⊢ (𝑋 = 𝑌 → if(𝑦 = 𝑋, 1, 0) = if(𝑦 = 𝑌, 1, 0))
74	73	mpteq2dv 5207	. . . . . . . . . . . 12 ⊢ (𝑋 = 𝑌 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0)))
75	74	oveq2d 7427	. . . . . . . . . . 11 ⊢ (𝑋 = 𝑌 → (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))))
76	75	eqeq1d 2771	. . . . . . . . . 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 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑘:𝐼⟶ℕ0)
78		1nn0 12520	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℕ0
79		0nn0 12519	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℕ0
80	78, 79	ifcli 4538	. . . . . . . . . . . . . 14 ⊢ if(𝑦 = 𝑌, 1, 0) ∈ ℕ0
81	80	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐼 → if(𝑦 = 𝑌, 1, 0) ∈ ℕ0)
82	50, 81	fmpti 7108	. . . . . . . . . . . 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 12514	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℂ)
85		nn0cn 12514	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℂ)
86		addcom 11396	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝑛 + 𝑚) = (𝑚 + 𝑛))
87	86	eqeq1d 2771	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → ((𝑛 + 𝑚) = 𝑚 ↔ (𝑚 + 𝑛) = 𝑚))
88		addid0 11633	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑚 + 𝑛) = 𝑚 ↔ 𝑛 = 0))
89	88	ancoms 463	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → ((𝑚 + 𝑛) = 𝑚 ↔ 𝑛 = 0))
90	87, 89	bitrd 282	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → ((𝑛 + 𝑚) = 𝑚 ↔ 𝑛 = 0))
91	84, 85, 90	syl2an 607	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) → ((𝑛 + 𝑚) = 𝑚 ↔ 𝑛 = 0))
92	91	adantl 486	. . . . . . . . . . 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 7713	. . . . . . . . . 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 519	. . . . . . . . 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 834	. . . . . . . 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 468	. . . . . . 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 4514	. . . . . 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 2804	. . . . 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 7427	. . . 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 7510	. . . . 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 6881	. . . . . . . . . . 11 ⊢ (𝑘 = (𝐼 × {0}) → (𝑘‘𝑋) = ((𝐼 × {0})‘𝑋))
102	101	oveq1d 7426	. . . . . . . . . 10 ⊢ (𝑘 = (𝐼 × {0}) → ((𝑘‘𝑋) + 1) = (((𝐼 × {0})‘𝑋) + 1))
103	4	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑋 ∈ 𝐼)
104		c0ex 11200	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
105	104	fvconst2 7203	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝐼 → ((𝐼 × {0})‘𝑋) = 0)
106	103, 105	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝐼 × {0})‘𝑋) = 0)
107	106	oveq1d 7426	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝐼 × {0})‘𝑋) + 1) = (0 + 1))
108		0p1e1 12361	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
109	107, 108	eqtrdi 2820	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝐼 × {0})‘𝑋) + 1) = 1)
110	102, 109	sylan9eqr 2826	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 = (𝐼 × {0})) → ((𝑘‘𝑋) + 1) = 1)
111	110	adantrr 729	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌)) → ((𝑘‘𝑋) + 1) = 1)
112	111	oveq1d 7426	. . . . . . 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 2769	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
114	113, 12, 7	ringidcld 20349	. . . . . . . . 9 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
115		eqid 2769	. . . . . . . . . 10 ⊢ (.g‘𝑅) = (.g‘𝑅)
116	113, 115	mulg1 19147	. . . . . . . . 9 ⊢ ((1r‘𝑅) ∈ (Base‘𝑅) → (1(.g‘𝑅)(1r‘𝑅)) = (1r‘𝑅))
117	114, 116	syl 18	. . . . . . . 8 ⊢ (𝜑 → (1(.g‘𝑅)(1r‘𝑅)) = (1r‘𝑅))
118	117	ad2antrr 738	. . . . . . 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 2804	. . . . . 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 20324	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Grp)
121	120	grpmndd 19013	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Mnd)
122	121	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ Mnd)
123	77, 103	ffvelcdmd 7081	. . . . . . . . 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 12569	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑘‘𝑋) + 1) ∈ ℕ0)
126	113, 115, 11	mulgnn0z 19167	. . . . . . . 8 ⊢ ((𝑅 ∈ Mnd ∧ ((𝑘‘𝑋) + 1) ∈ ℕ0) → (((𝑘‘𝑋) + 1)(.g‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
127	122, 125, 126	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝑘‘𝑋) + 1)(.g‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
128	127	adantr 485	. . . . . 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 4524	. . . . 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 2816	. . . 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 465	. . . . . . 7 ⊢ ((𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌) ↔ (𝑋 = 𝑌 ∧ 𝑘 = (𝐼 × {0})))
132		ifbi 4513	. . . . . . 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 4544	. . . . . 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 2792	. . . . 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 2808	. . 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 5206	. 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 7513	. . 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 22089	. . . 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 22076	. . . . 5 ⊢ (𝜑 → 0 = ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝑅)}))
144		fconstmpt 5724	. . . . 5 ⊢ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝑅)}) = (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (0g‘𝑅))
145	143, 144	eqtrdi 2820	. . . 4 ⊢ (𝜑 → 0 = (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (0g‘𝑅)))
146	141, 145	ifeq12d 4512	. . 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 2823	. 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 2808	1 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑉‘𝑌)) = if(𝑋 = 𝑌, 1 , 0 ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  if-wif 1076   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  {crab 3422  ifcif 4490  {csn 4592   class class class wbr 5111   ↦ cmpt 5194   × cxp 5660  ^◡ccnv 5661   “ cima 5665   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ∘_f cof 7673   ↑_m cmap 8824  Fincfn 8943  ℂcc 11098  ℝcr 11099  0cc0 11100  1c1 11101   + caddc 11103   ≤ cle 11244  ℕcn 12233  ℕ₀cn0 12504  Basecbs 17269  0_gc0g 17492  Mndcmnd 18792  ._gcmg 19133  1_rcur 20263  Ringcrg 20315   mPwSer cmps 22023   mVar cmvr 22024   mPSDer cpsd 22266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1077  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7675  df-ofr 7676  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8157  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-er 8694  df-map 8826  df-pm 8827  df-ixp 8896  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fsupp 9322  df-sup 9402  df-oi 9472  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12505  df-z 12592  df-dec 12712  df-uz 12863  df-fz 13536  df-fzo 13683  df-seq 14038  df-hash 14367  df-struct 17207  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-ress 17291  df-plusg 17323  df-mulr 17324  df-sca 17326  df-vsca 17327  df-ip 17328  df-tset 17329  df-ple 17330  df-ds 17332  df-hom 17334  df-cco 17335  df-0g 17494  df-gsum 17495  df-prds 17500  df-pws 17502  df-mre 17638  df-mrc 17639  df-acs 17641  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-mhm 18841  df-submnd 18842  df-grp 19003  df-minusg 19004  df-mulg 19134  df-ghm 19284  df-cntz 19387  df-cmn 19852  df-abl 19853  df-mgp 20217  df-rng 20231  df-ur 20264  df-ring 20317  df-psr 22028  df-mvr 22029  df-psd 22288

This theorem is referenced by: (None)