Theorem psdmvr 22490

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 2760	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2760	. . 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 22294	. . 3 ⊢ (𝜑 → (𝑉‘𝑌) ∈ (Base‘𝑆))
10	1, 2, 3, 4, 9	psdval 22480	. 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 2760	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
12		eqid 2760	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
13	6	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑊)
14	7	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
15	8	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑌 ∈ 𝐼)
16		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
17	3	psrbagsn 22372	. . . . . . . . . 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 486	. . . . . . . 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 22232	. . . . . . . 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 596	. . . . . . 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 22290	. . . . . 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 11309	. . . . . . . . . . . . . 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 22226	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑘:𝐼⟶ℕ0)
25	24	ad2antlr 740	. . . . . . . . . . . . . . 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 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))) → 𝑋 ∈ 𝐼)
27	25, 26	ffvelcdmd 7085	. . . . . . . . . . . . . 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 12653	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ ∧ (𝑘‘𝑋) ∈ ℕ0) → 1 ≤ ((𝑘‘𝑋) + 1))
29	23, 27, 28	syl2anc 596	. . . . . . . . . . . . 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 6884	. . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . 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 6710	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑘 Fn 𝐼)
33	32	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑘 Fn 𝐼)
34		1re 11308	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℝ
35		0re 11310	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℝ
36	34, 35	ifcli 4529	. . . . . . . . . . . . . . . . . . . 20 ⊢ if(𝑦 = 𝑋, 1, 0) ∈ ℝ
37	36	elexi 3472	. . . . . . . . . . . . . . . . . . 19 ⊢ if(𝑦 = 𝑋, 1, 0) ∈ V
38		eqid 2760	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))
39	37, 38	fnmpti 6682	. . . . . . . . . . . . . . . . . 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 4123	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ∩ 𝐼) = 𝐼
42		eqidd 2761	. . . . . . . . . . . . . . . . 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 11303	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ V
45	43, 38, 44	fvmpt 6993	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ 𝐼 → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))‘𝑋) = 1)
46	45	adantl 487	. . . . . . . . . . . . . . . . 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 7704	. . . . . . . . . . . . . . . 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 702	. . . . . . . . . . . . . . 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 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) = ((𝑘‘𝑋) + 1))
50		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))
51		eqeq1 2764	. . . . . . . . . . . . . . . . 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 7017	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))‘𝑋) = if(𝑋 = 𝑌, 1, 0))
56	55	ad2antrr 739	. . . . . . . . . . . . . 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 2803	. . . . . . . . . . . . 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 5130	. . . . . . . . . . . 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 11944	. . . . . . . . . . . . . 14 ⊢ 1 ≤ 1
60		0le1 11839	. . . . . . . . . . . . . 14 ⊢ 0 ≤ 1
61		anifp 1088	. . . . . . . . . . . . . 14 ⊢ ((1 ≤ 1 ∧ 0 ≤ 1) → if-(𝑋 = 𝑌, 1 ≤ 1, 0 ≤ 1))
62	59, 60, 61	mp2an 705	. . . . . . . . . . . . 13 ⊢ if-(𝑋 = 𝑌, 1 ≤ 1, 0 ≤ 1)
63		brif1 7517	. . . . . . . . . . . . 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 11426	. . . . . . . . . . . 12 ⊢ (1 = if(𝑋 = 𝑌, 1, 0) ↔ (1 ≤ if(𝑋 = 𝑌, 1, 0) ∧ if(𝑋 = 𝑌, 1, 0) ≤ 1))
66	58, 64, 65	sylanblrc 602	. . . . . . . . . . 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 2766	. . . . . . . . . 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 11269	. . . . . . . . . . 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 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 2772	. . . . . . . . . . . . . 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 5198	. . . . . . . . . . . 12 ⊢ (𝑋 = 𝑌 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0)))
75	74	oveq2d 7436	. . . . . . . . . . 11 ⊢ (𝑋 = 𝑌 → (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑌, 1, 0))))
76	75	eqeq1d 2762	. . . . . . . . . 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 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑘:𝐼⟶ℕ0)
78		1nn0 12622	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℕ0
79		0nn0 12621	. . . . . . . . . . . . . . 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 7112	. . . . . . . . . . . 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 12616	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℂ)
85		nn0cn 12616	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℂ)
86		addcom 11496	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝑛 + 𝑚) = (𝑚 + 𝑛))
87	86	eqeq1d 2762	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → ((𝑛 + 𝑚) = 𝑚 ↔ (𝑚 + 𝑛) = 𝑚))
88		addid0 11735	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑚 + 𝑛) = 𝑚 ↔ 𝑛 = 0))
89	88	ancoms 464	. . . . . . . . . . . . . 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 608	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) → ((𝑛 + 𝑚) = 𝑚 ↔ 𝑛 = 0))
92	91	adantl 487	. . . . . . . . . . 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 7731	. . . . . . . . . 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 520	. . . . . . . . 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 835	. . . . . . . 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 469	. . . . . . 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 2795	. . . . 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 7436	. . . 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 7519	. . . . 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 6884	. . . . . . . . . . 11 ⊢ (𝑘 = (𝐼 × {0}) → (𝑘‘𝑋) = ((𝐼 × {0})‘𝑋))
102	101	oveq1d 7435	. . . . . . . . . 10 ⊢ (𝑘 = (𝐼 × {0}) → ((𝑘‘𝑋) + 1) = (((𝐼 × {0})‘𝑋) + 1))
103	4	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑋 ∈ 𝐼)
104		c0ex 11300	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
105	104	fvconst2 7210	. . . . . . . . . . . . 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 7435	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝐼 × {0})‘𝑋) + 1) = (0 + 1))
108		0p1e1 12463	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
109	107, 108	eqtrdi 2811	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝐼 × {0})‘𝑋) + 1) = 1)
110	102, 109	sylan9eqr 2817	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 = (𝐼 × {0})) → ((𝑘‘𝑋) + 1) = 1)
111	110	adantrr 730	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ (𝑘 = (𝐼 × {0}) ∧ 𝑋 = 𝑌)) → ((𝑘‘𝑋) + 1) = 1)
112	111	oveq1d 7435	. . . . . . 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 2760	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
114	113, 12, 7	ringidcld 20495	. . . . . . . . 9 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
115		eqid 2760	. . . . . . . . . 10 ⊢ (.g‘𝑅) = (.g‘𝑅)
116	113, 115	mulg1 19291	. . . . . . . . 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 739	. . . . . . 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 2795	. . . . . 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 20469	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Grp)
121	120	grpmndd 19157	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Mnd)
122	121	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ Mnd)
123	77, 103	ffvelcdmd 7085	. . . . . . . . 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 12671	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑘‘𝑋) + 1) ∈ ℕ0)
126	113, 115, 11	mulgnn0z 19311	. . . . . . . 8 ⊢ ((𝑅 ∈ Mnd ∧ ((𝑘‘𝑋) + 1) ∈ ℕ0) → (((𝑘‘𝑋) + 1)(.g‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
127	122, 125, 126	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝑘‘𝑋) + 1)(.g‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
128	127	adantr 486	. . . . . 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 2807	. . . 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 466	. . . . . . 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 2783	. . . . 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 2799	. . 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 5197	. 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 7522	. . 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 22278	. . . 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 22265	. . . . 5 ⊢ (𝜑 → 0 = ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝑅)}))
144		fconstmpt 5712	. . . . 5 ⊢ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝑅)}) = (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (0g‘𝑅))
145	143, 144	eqtrdi 2811	. . . 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 2814	. 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 2799	1 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝑉‘𝑌)) = if(𝑋 = 𝑌, 1 , 0 ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  if-wif 1078   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  {crab 3412  ifcif 4481  {csn 4583   class class class wbr 5102   ↦ cmpt 5185   × cxp 5648  ^◡ccnv 5649   “ cima 5653   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∘_f cof 7691   ↑_m cmap 8847  Fincfn 8973  ℂcc 11198  ℝcr 11199  0cc0 11200  1c1 11201   + caddc 11203   ≤ cle 11344  ℕcn 12335  ℕ₀cn0 12606  Basecbs 17387  0_gc0g 17610  Mndcmnd 18923  ._gcmg 19277  1_rcur 20407  Ringcrg 20459   mPwSer cmps 22212   mVar cmvr 22213   mPSDer cpsd 22455

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-ofr 7694  df-om 7878  df-1st 8001  df-2nd 8002  df-supp 8178  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-map 8849  df-pm 8850  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-fsupp 9354  df-sup 9434  df-oi 9504  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-fzo 13789  df-seq 14145  df-hash 14475  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-sca 17444  df-vsca 17445  df-ip 17446  df-tset 17447  df-ple 17448  df-ds 17450  df-hom 17452  df-cco 17453  df-0g 17612  df-gsum 17613  df-prds 17618  df-pws 17620  df-mre 17756  df-mrc 17757  df-acs 17759  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-mhm 18978  df-submnd 18979  df-grp 19147  df-minusg 19148  df-mulg 19278  df-ghm 19428  df-cntz 19531  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-ur 20408  df-ring 20461  df-psr 22217  df-mvr 22218  df-psd 22477

This theorem is used by: (None)