Theorem esplyfvaln 33715

Description: The last elementary symmetric polynomial is the product of all variables. (Contributed by Thierry Arnoux, 16-Mar-2026.)

Hypotheses

Ref	Expression
esplyfval1.w	⊢ 𝑊 = (𝐼 mPoly 𝑅)
esplyfval1.v	⊢ 𝑉 = (𝐼 mVar 𝑅)
esplyfval1.e	⊢ 𝐸 = (𝐼eSymPoly𝑅)
esplyfval1.i	⊢ (𝜑 → 𝐼 ∈ Fin)
esplyfvaln.r	⊢ (𝜑 → 𝑅 ∈ CRing)
esplyfvaln.n	⊢ 𝑁 = (♯‘𝐼)
esplyfvaln.m	⊢ 𝑀 = (mulGrp‘𝑊)

Assertion

Ref	Expression
esplyfvaln	⊢ (𝜑 → (𝐸‘𝑁) = (𝑀 Σg 𝑉))

Proof of Theorem esplyfvaln

Dummy variables 𝑓 ℎ 𝑖 𝑗 𝑡 𝑢 𝑘 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		esplyfval1.e	. . 3 ⊢ 𝐸 = (𝐼eSymPoly𝑅)
2	1	fveq1i 6839	. 2 ⊢ (𝐸‘𝑁) = ((𝐼eSymPoly𝑅)‘𝑁)
3		eqid 2737	. . . 4 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
4		esplyfval1.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ Fin)
5		esplyfvaln.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing)
6	5	crngringd 20224	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
7		esplyfvaln.n	. . . . 5 ⊢ 𝑁 = (♯‘𝐼)
8		hashcl 14315	. . . . . 6 ⊢ (𝐼 ∈ Fin → (♯‘𝐼) ∈ ℕ0)
9	4, 8	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘𝐼) ∈ ℕ0)
10	7, 9	eqeltrid 2841	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
11		eqid 2737	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
12		eqid 2737	. . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅)
13	3, 4, 6, 10, 11, 12	esplyfval3 33713	. . 3 ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝑁) = (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝑁), (1r‘𝑅), (0g‘𝑅))))
14		esplyfval1.w	. . . . 5 ⊢ 𝑊 = (𝐼 mPoly 𝑅)
15		eqid 2737	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
16		breq1 5089	. . . . . . 7 ⊢ (ℎ = ((𝟭‘𝐼)‘{𝑖}) → (ℎ finSupp 0 ↔ ((𝟭‘𝐼)‘{𝑖}) finSupp 0))
17		nn0ex 12440	. . . . . . . . 9 ⊢ ℕ0 ∈ V
18	17	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ℕ0 ∈ V)
19	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ Fin)
20		snssi 4730	. . . . . . . . . 10 ⊢ (𝑖 ∈ 𝐼 → {𝑖} ⊆ 𝐼)
21		indf 12162	. . . . . . . . . 10 ⊢ ((𝐼 ∈ Fin ∧ {𝑖} ⊆ 𝐼) → ((𝟭‘𝐼)‘{𝑖}):𝐼⟶{0, 1})
22	4, 20, 21	syl2an 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}):𝐼⟶{0, 1})
23		0nn0 12449	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ0
24	23	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 0 ∈ ℕ0)
25		1nn0 12450	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ0
26	25	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 1 ∈ ℕ0)
27	24, 26	prssd 4766	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → {0, 1} ⊆ ℕ0)
28	22, 27	fssd 6683	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}):𝐼⟶ℕ0)
29	18, 19, 28	elmapdd 8785	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}) ∈ (ℕ0 ↑m 𝐼))
30	22, 19, 24	fidmfisupp 9282	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}) finSupp 0)
31	16, 29, 30	elrabd 3637	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
32	31	fmpttd 7065	. . . . 5 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖})):𝐼⟶{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
33		esplyfvaln.m	. . . . 5 ⊢ 𝑀 = (mulGrp‘𝑊)
34		eqeq2 2749	. . . . . . . . 9 ⊢ (𝑡 = 𝑦 → (𝑢 = 𝑡 ↔ 𝑢 = 𝑦))
35	34	ifbid 4491	. . . . . . . 8 ⊢ (𝑡 = 𝑦 → if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅)) = if(𝑢 = 𝑦, (1r‘𝑅), (0g‘𝑅)))
36	35	mpteq2dv 5180	. . . . . . 7 ⊢ (𝑡 = 𝑦 → (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅))) = (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = 𝑦, (1r‘𝑅), (0g‘𝑅))))
37		eqeq1 2741	. . . . . . . . 9 ⊢ (𝑢 = 𝑧 → (𝑢 = 𝑦 ↔ 𝑧 = 𝑦))
38	37	ifbid 4491	. . . . . . . 8 ⊢ (𝑢 = 𝑧 → if(𝑢 = 𝑦, (1r‘𝑅), (0g‘𝑅)) = if(𝑧 = 𝑦, (1r‘𝑅), (0g‘𝑅)))
39	38	cbvmptv 5190	. . . . . . 7 ⊢ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = 𝑦, (1r‘𝑅), (0g‘𝑅))) = (𝑧 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑧 = 𝑦, (1r‘𝑅), (0g‘𝑅)))
40	36, 39	eqtrdi 2788	. . . . . 6 ⊢ (𝑡 = 𝑦 → (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅))) = (𝑧 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑧 = 𝑦, (1r‘𝑅), (0g‘𝑅))))
41	40	cbvmptv 5190	. . . . 5 ⊢ (𝑡 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅)))) = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑧 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑧 = 𝑦, (1r‘𝑅), (0g‘𝑅))))
42	14, 15, 5, 4, 3, 4, 32, 12, 11, 33, 41	mplmonprod 33695	. . . 4 ⊢ (𝜑 → (𝑀 Σg ((𝑡 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅)))) ∘ (𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖})))) = ((𝑡 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅))))‘(𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))))))
43		eqid 2737	. . . . 5 ⊢ (𝑡 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅)))) = (𝑡 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅))))
44		eqeq2 2749	. . . . . . . 8 ⊢ (𝑡 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) → (𝑢 = 𝑡 ↔ 𝑢 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))))))
45	44	ifbid 4491	. . . . . . 7 ⊢ (𝑡 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) → if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅)) = if(𝑢 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))), (1r‘𝑅), (0g‘𝑅)))
46	45	mpteq2dv 5180	. . . . . 6 ⊢ (𝑡 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) → (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅))) = (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))), (1r‘𝑅), (0g‘𝑅))))
47		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑢 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))))) → 𝑢 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))))
48	47	rneqd 5891	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑢 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))))) → ran 𝑢 = ran (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))))
49		nfv 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
50		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))))
51		eqid 2737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖})) = (𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))
52		sneq 4578	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑘 → {𝑖} = {𝑘})
53	52	fveq2d 6842	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑘 → ((𝟭‘𝐼)‘{𝑖}) = ((𝟭‘𝐼)‘{𝑘}))
54		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼)
55		fvexd 6853	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑘}) ∈ V)
56	51, 53, 54, 55	fvmptd3 6969	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → ((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘) = ((𝟭‘𝐼)‘{𝑘}))
57	56	fveq1d 6840	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗) = (((𝟭‘𝐼)‘{𝑘})‘𝑗))
58	4	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → 𝐼 ∈ Fin)
59	54	snssd 4731	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → {𝑘} ⊆ 𝐼)
60		simplr 769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → 𝑗 ∈ 𝐼)
61		indfval 12163	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼 ∈ Fin ∧ {𝑘} ⊆ 𝐼 ∧ 𝑗 ∈ 𝐼) → (((𝟭‘𝐼)‘{𝑘})‘𝑗) = if(𝑗 ∈ {𝑘}, 1, 0))
62	58, 59, 60, 61	syl3anc 1374	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (((𝟭‘𝐼)‘{𝑘})‘𝑗) = if(𝑗 ∈ {𝑘}, 1, 0))
63		velsn 4584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ {𝑘} ↔ 𝑗 = 𝑘)
64		equcom 2020	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑘 ↔ 𝑘 = 𝑗)
65	63, 64	bitri 275	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ {𝑘} ↔ 𝑘 = 𝑗)
66	65	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑗 ∈ {𝑘} ↔ 𝑘 = 𝑗))
67	66	ifbid 4491	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → if(𝑗 ∈ {𝑘}, 1, 0) = if(𝑘 = 𝑗, 1, 0))
68	57, 62, 67	3eqtrd 2776	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗) = if(𝑘 = 𝑗, 1, 0))
69	68	mpteq2dva 5179	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1, 0)))
70	69	oveq2d 7380	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))) = (ℂfld Σg (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1, 0))))
71		cnfld0 21373	. . . . . . . . . . . . . . . . . 18 ⊢ 0 = (0g‘ℂfld)
72		cnfldfld 33399	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℂfld ∈ Field
73		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℂfld ∈ Field → ℂfld ∈ Field)
74	73	fldcrngd 20716	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℂfld ∈ Field → ℂfld ∈ CRing)
75		crngring 20223	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℂfld ∈ CRing → ℂfld ∈ Ring)
76		ringcmn 20260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℂfld ∈ Ring → ℂfld ∈ CMnd)
77	74, 75, 76	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℂfld ∈ Field → ℂfld ∈ CMnd)
78	72, 77	mp1i 13	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → ℂfld ∈ CMnd)
79	78	cmnmndd 19776	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → ℂfld ∈ Mnd)
80	4	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → 𝐼 ∈ Fin)
81		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → 𝑗 ∈ 𝐼)
82		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1, 0))
83		ax-1cn 11093	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℂ
84		cnfldbas 21353	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℂ = (Base‘ℂfld)
85	83, 84	eleqtri 2835	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ (Base‘ℂfld)
86	85	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → 1 ∈ (Base‘ℂfld))
87	71, 79, 80, 81, 82, 86	gsummptif1n0 19938	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1, 0))) = 1)
88	70, 87	eqtrd 2772	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))) = 1)
89		1ex 11137	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ V
90	89	prid2 4708	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ {0, 1}
91	88, 90	eqeltrdi 2845	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))) ∈ {0, 1})
92	91	adantlr 716	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑗 ∈ 𝐼) → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))) ∈ {0, 1})
93	49, 50, 92	rnmptssd 7074	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ran (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) ⊆ {0, 1})
94	93	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑢 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))))) → ran (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) ⊆ {0, 1})
95	48, 94	eqsstrd 3957	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑢 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))))) → ran 𝑢 ⊆ {0, 1})
96	47	oveq1d 7379	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑢 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))))) → (𝑢 supp 0) = ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) supp 0))
97		suppssdm 8124	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) supp 0) ⊆ dom (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))))
98		nn0subm 21399	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℕ0 ∈ (SubMnd‘ℂfld)
99	98	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → ℕ0 ∈ (SubMnd‘ℂfld))
100	23	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → 0 ∈ ℕ0)
101	25	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → 1 ∈ ℕ0)
102	100, 101	prssd 4766	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → {0, 1} ⊆ ℕ0)
103		indf 12162	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐼 ∈ Fin ∧ {𝑘} ⊆ 𝐼) → ((𝟭‘𝐼)‘{𝑘}):𝐼⟶{0, 1})
104	58, 59, 103	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑘}):𝐼⟶{0, 1})
105	104, 60	ffvelcdmd 7035	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (((𝟭‘𝐼)‘{𝑘})‘𝑗) ∈ {0, 1})
106	102, 105	sseldd 3923	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (((𝟭‘𝐼)‘{𝑘})‘𝑗) ∈ ℕ0)
107	57, 106	eqeltrd 2837	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗) ∈ ℕ0)
108	107	fmpttd 7065	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)):𝐼⟶ℕ0)
109	23	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → 0 ∈ ℕ0)
110	108, 80, 109	fdmfifsupp 9285	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)) finSupp 0)
111	71, 78, 80, 99, 108, 110	gsumsubmcl 19891	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))) ∈ ℕ0)
112	50, 111	dmmptd 6641	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) = 𝐼)
113	97, 112	sseqtrid 3965	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) supp 0) ⊆ 𝐼)
114		nfv 1916	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑖 ∈ 𝐼)
115		ovexd 7399	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗))) ∈ V)
116		eqid 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗)))) = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗))))
117	114, 115, 116	fnmptd 6637	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗)))) Fn 𝐼)
118		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
119		fveq2 6838	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑖 → (((𝟭‘𝐼)‘{𝑘})‘𝑗) = (((𝟭‘𝐼)‘{𝑘})‘𝑖))
120	119	mpteq2dv 5180	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 𝑖 → (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗)) = (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑖)))
121	120	oveq2d 7380	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑖 → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗))) = (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑖))))
122		ovexd 7399	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑖))) ∈ V)
123	116, 121, 118, 122	fvmptd3 6969	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗))))‘𝑖) = (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑖))))
124	4	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → 𝐼 ∈ Fin)
125		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼)
126	125	snssd 4731	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → {𝑘} ⊆ 𝐼)
127		simplr 769	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → 𝑖 ∈ 𝐼)
128		indfval 12163	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐼 ∈ Fin ∧ {𝑘} ⊆ 𝐼 ∧ 𝑖 ∈ 𝐼) → (((𝟭‘𝐼)‘{𝑘})‘𝑖) = if(𝑖 ∈ {𝑘}, 1, 0))
129	124, 126, 127, 128	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (((𝟭‘𝐼)‘{𝑘})‘𝑖) = if(𝑖 ∈ {𝑘}, 1, 0))
130		velsn 4584	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 ∈ {𝑘} ↔ 𝑖 = 𝑘)
131		equcom 2020	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 = 𝑘 ↔ 𝑘 = 𝑖)
132	130, 131	bitri 275	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ {𝑘} ↔ 𝑘 = 𝑖)
133	132	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑖 ∈ {𝑘} ↔ 𝑘 = 𝑖))
134	133	ifbid 4491	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → if(𝑖 ∈ {𝑘}, 1, 0) = if(𝑘 = 𝑖, 1, 0))
135	129, 134	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (((𝟭‘𝐼)‘{𝑘})‘𝑖) = if(𝑘 = 𝑖, 1, 0))
136	135	mpteq2dva 5179	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑖)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑖, 1, 0)))
137	136	oveq2d 7380	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑖))) = (ℂfld Σg (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑖, 1, 0))))
138	72, 77	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ℂfld ∈ CMnd)
139	138	cmnmndd 19776	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ℂfld ∈ Mnd)
140		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑖, 1, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑖, 1, 0))
141	85	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 1 ∈ (Base‘ℂfld))
142	71, 139, 19, 118, 140, 141	gsummptif1n0 19938	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑖, 1, 0))) = 1)
143	123, 137, 142	3eqtrd 2776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗))))‘𝑖) = 1)
144		ax-1ne0 11104	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ≠ 0
145	144	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 1 ≠ 0)
146	143, 145	eqnetrd 3000	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗))))‘𝑖) ≠ 0)
147	117, 19, 24, 118, 146	elsuppfnd 32752	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗)))) supp 0))
148	147	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑖 ∈ 𝐼 → 𝑖 ∈ ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗)))) supp 0)))
149	148	ssrdv 3928	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐼 ⊆ ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗)))) supp 0))
150	57	mpteq2dva 5179	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)) = (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗)))
151	150	oveq2d 7380	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))) = (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗))))
152	151	mpteq2dva 5179	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗)))))
153	152	oveq1d 7379	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) supp 0) = ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗)))) supp 0))
154	149, 153	sseqtrrd 3960	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ⊆ ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) supp 0))
155	113, 154	eqssd 3940	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) supp 0) = 𝐼)
156	155	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑢 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))))) → ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) supp 0) = 𝐼)
157	96, 156	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑢 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))))) → (𝑢 supp 0) = 𝐼)
158	157	fveq2d 6842	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑢 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))))) → (♯‘(𝑢 supp 0)) = (♯‘𝐼))
159	158, 7	eqtr4di 2790	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑢 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))))) → (♯‘(𝑢 supp 0)) = 𝑁)
160	95, 159	jca 511	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑢 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))))) → (ran 𝑢 ⊆ {0, 1} ∧ (♯‘(𝑢 supp 0)) = 𝑁))
161		simpllr 776	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → ran 𝑢 ⊆ {0, 1})
162	4	ad3antrrr 731	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) → 𝐼 ∈ Fin)
163	17	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) → ℕ0 ∈ V)
164		ssrab2 4021	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐼)
165	164	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐼))
166	165	sselda 3922	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑢 ∈ (ℕ0 ↑m 𝐼))
167	166	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) → 𝑢 ∈ (ℕ0 ↑m 𝐼))
168	162, 163, 167	elmaprd 32750	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) → 𝑢:𝐼⟶ℕ0)
169	168	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → 𝑢:𝐼⟶ℕ0)
170	169	ffnd 6667	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → 𝑢 Fn 𝐼)
171		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → 𝑗 ∈ 𝐼)
172	170, 171	fnfvelrnd 7032	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → (𝑢‘𝑗) ∈ ran 𝑢)
173	161, 172	sseldd 3923	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → (𝑢‘𝑗) ∈ {0, 1})
174	162	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → 𝐼 ∈ Fin)
175	23	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → 0 ∈ ℕ0)
176		suppssdm 8124	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 supp 0) ⊆ dom 𝑢
177	176, 169	fssdm 6685	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → (𝑢 supp 0) ⊆ 𝐼)
178		simplr 769	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → (♯‘(𝑢 supp 0)) = 𝑁)
179	178, 7	eqtr2di 2789	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → (♯‘𝐼) = (♯‘(𝑢 supp 0)))
180	174, 177, 179	phphashd 14425	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → 𝐼 = (𝑢 supp 0))
181	171, 180	eleqtrd 2839	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → 𝑗 ∈ (𝑢 supp 0))
182		elsuppfn 8117	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 Fn 𝐼 ∧ 𝐼 ∈ Fin ∧ 0 ∈ ℕ0) → (𝑗 ∈ (𝑢 supp 0) ↔ (𝑗 ∈ 𝐼 ∧ (𝑢‘𝑗) ≠ 0)))
183	182	simplbda 499	. . . . . . . . . . . . . . 15 ⊢ (((𝑢 Fn 𝐼 ∧ 𝐼 ∈ Fin ∧ 0 ∈ ℕ0) ∧ 𝑗 ∈ (𝑢 supp 0)) → (𝑢‘𝑗) ≠ 0)
184	170, 174, 175, 181, 183	syl31anc 1376	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → (𝑢‘𝑗) ≠ 0)
185		elprn1 4596	. . . . . . . . . . . . . 14 ⊢ (((𝑢‘𝑗) ∈ {0, 1} ∧ (𝑢‘𝑗) ≠ 0) → (𝑢‘𝑗) = 1)
186	173, 184, 185	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → (𝑢‘𝑗) = 1)
187	186	mpteq2dva 5179	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) → (𝑗 ∈ 𝐼 ↦ (𝑢‘𝑗)) = (𝑗 ∈ 𝐼 ↦ 1))
188	168	feqmptd 6906	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) → 𝑢 = (𝑗 ∈ 𝐼 ↦ (𝑢‘𝑗)))
189	88	mpteq2dva 5179	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) = (𝑗 ∈ 𝐼 ↦ 1))
190	189	ad3antrrr 731	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) → (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) = (𝑗 ∈ 𝐼 ↦ 1))
191	187, 188, 190	3eqtr4d 2782	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) → 𝑢 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))))
192	191	anasss 466	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ (ran 𝑢 ⊆ {0, 1} ∧ (♯‘(𝑢 supp 0)) = 𝑁)) → 𝑢 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))))
193	160, 192	impbida 801	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑢 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) ↔ (ran 𝑢 ⊆ {0, 1} ∧ (♯‘(𝑢 supp 0)) = 𝑁)))
194	193	ifbid 4491	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → if(𝑢 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))), (1r‘𝑅), (0g‘𝑅)) = if((ran 𝑢 ⊆ {0, 1} ∧ (♯‘(𝑢 supp 0)) = 𝑁), (1r‘𝑅), (0g‘𝑅)))
195	194	mpteq2dva 5179	. . . . . . 7 ⊢ (𝜑 → (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))), (1r‘𝑅), (0g‘𝑅))) = (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if((ran 𝑢 ⊆ {0, 1} ∧ (♯‘(𝑢 supp 0)) = 𝑁), (1r‘𝑅), (0g‘𝑅))))
196		rneq 5889	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑓 → ran 𝑢 = ran 𝑓)
197	196	sseq1d 3954	. . . . . . . . . 10 ⊢ (𝑢 = 𝑓 → (ran 𝑢 ⊆ {0, 1} ↔ ran 𝑓 ⊆ {0, 1}))
198		oveq1 7371	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑓 → (𝑢 supp 0) = (𝑓 supp 0))
199	198	fveqeq2d 6846	. . . . . . . . . 10 ⊢ (𝑢 = 𝑓 → ((♯‘(𝑢 supp 0)) = 𝑁 ↔ (♯‘(𝑓 supp 0)) = 𝑁))
200	197, 199	anbi12d 633	. . . . . . . . 9 ⊢ (𝑢 = 𝑓 → ((ran 𝑢 ⊆ {0, 1} ∧ (♯‘(𝑢 supp 0)) = 𝑁) ↔ (ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝑁)))
201	200	ifbid 4491	. . . . . . . 8 ⊢ (𝑢 = 𝑓 → if((ran 𝑢 ⊆ {0, 1} ∧ (♯‘(𝑢 supp 0)) = 𝑁), (1r‘𝑅), (0g‘𝑅)) = if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝑁), (1r‘𝑅), (0g‘𝑅)))
202	201	cbvmptv 5190	. . . . . . 7 ⊢ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if((ran 𝑢 ⊆ {0, 1} ∧ (♯‘(𝑢 supp 0)) = 𝑁), (1r‘𝑅), (0g‘𝑅))) = (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝑁), (1r‘𝑅), (0g‘𝑅)))
203	195, 202	eqtrdi 2788	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))), (1r‘𝑅), (0g‘𝑅))) = (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝑁), (1r‘𝑅), (0g‘𝑅))))
204	46, 203	sylan9eqr 2794	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))))) → (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅))) = (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝑁), (1r‘𝑅), (0g‘𝑅))))
205		breq1 5089	. . . . . 6 ⊢ (ℎ = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) → (ℎ finSupp 0 ↔ (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) finSupp 0))
206	17	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℕ0 ∈ V)
207	111	fmpttd 7065	. . . . . . 7 ⊢ (𝜑 → (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))):𝐼⟶ℕ0)
208	206, 4, 207	elmapdd 8785	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) ∈ (ℕ0 ↑m 𝐼))
209	23	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℕ0)
210	207, 4, 209	fidmfisupp 9282	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) finSupp 0)
211	205, 208, 210	elrabd 3637	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
212		ovex 7397	. . . . . . . 8 ⊢ (ℕ0 ↑m 𝐼) ∈ V
213	212	rabex 5279	. . . . . . 7 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∈ V
214	213	a1i 11	. . . . . 6 ⊢ (𝜑 → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∈ V)
215	214	mptexd 7176	. . . . 5 ⊢ (𝜑 → (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝑁), (1r‘𝑅), (0g‘𝑅))) ∈ V)
216	43, 204, 211, 215	fvmptd2 6954	. . . 4 ⊢ (𝜑 → ((𝑡 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅))))‘(𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))))) = (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝑁), (1r‘𝑅), (0g‘𝑅))))
217	42, 216	eqtrd 2772	. . 3 ⊢ (𝜑 → (𝑀 Σg ((𝑡 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅)))) ∘ (𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖})))) = (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝑁), (1r‘𝑅), (0g‘𝑅))))
218		indval 12159	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ Fin ∧ {𝑖} ⊆ 𝐼) → ((𝟭‘𝐼)‘{𝑖}) = (𝑗 ∈ 𝐼 ↦ if(𝑗 ∈ {𝑖}, 1, 0)))
219	4, 20, 218	syl2an 597	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}) = (𝑗 ∈ 𝐼 ↦ if(𝑗 ∈ {𝑖}, 1, 0)))
220		velsn 4584	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ {𝑖} ↔ 𝑗 = 𝑖)
221	220	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (𝑗 ∈ {𝑖} ↔ 𝑗 = 𝑖))
222	221	ifbid 4491	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → if(𝑗 ∈ {𝑖}, 1, 0) = if(𝑗 = 𝑖, 1, 0))
223	222	mpteq2dva 5179	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑗 ∈ 𝐼 ↦ if(𝑗 ∈ {𝑖}, 1, 0)) = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
224	219, 223	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}) = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
225	224	eqeq2d 2748	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑢 = ((𝟭‘𝐼)‘{𝑖}) ↔ 𝑢 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))))
226	225	ifbid 4491	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → if(𝑢 = ((𝟭‘𝐼)‘{𝑖}), (1r‘𝑅), (0g‘𝑅)) = if(𝑢 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅)))
227	226	mpteq2dv 5180	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = ((𝟭‘𝐼)‘{𝑖}), (1r‘𝑅), (0g‘𝑅))) = (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅))))
228		eqeq1 2741	. . . . . . . . 9 ⊢ (𝑡 = 𝑢 → (𝑡 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)) ↔ 𝑢 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))))
229	228	ifbid 4491	. . . . . . . 8 ⊢ (𝑡 = 𝑢 → if(𝑡 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅)) = if(𝑢 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅)))
230	229	cbvmptv 5190	. . . . . . 7 ⊢ (𝑡 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑡 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅))) = (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅)))
231	227, 230	eqtr4di 2790	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = ((𝟭‘𝐼)‘{𝑖}), (1r‘𝑅), (0g‘𝑅))) = (𝑡 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑡 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅))))
232	231	mpteq2dva 5179	. . . . 5 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = ((𝟭‘𝐼)‘{𝑖}), (1r‘𝑅), (0g‘𝑅)))) = (𝑖 ∈ 𝐼 ↦ (𝑡 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑡 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅)))))
233		eqidd 2738	. . . . . 6 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖})) = (𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖})))
234		eqidd 2738	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅)))) = (𝑡 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅)))))
235		eqeq2 2749	. . . . . . . 8 ⊢ (𝑡 = ((𝟭‘𝐼)‘{𝑖}) → (𝑢 = 𝑡 ↔ 𝑢 = ((𝟭‘𝐼)‘{𝑖})))
236	235	ifbid 4491	. . . . . . 7 ⊢ (𝑡 = ((𝟭‘𝐼)‘{𝑖}) → if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅)) = if(𝑢 = ((𝟭‘𝐼)‘{𝑖}), (1r‘𝑅), (0g‘𝑅)))
237	236	mpteq2dv 5180	. . . . . 6 ⊢ (𝑡 = ((𝟭‘𝐼)‘{𝑖}) → (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅))) = (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = ((𝟭‘𝐼)‘{𝑖}), (1r‘𝑅), (0g‘𝑅))))
238	31, 233, 234, 237	fmptco 7080	. . . . 5 ⊢ (𝜑 → ((𝑡 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅)))) ∘ (𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))) = (𝑖 ∈ 𝐼 ↦ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = ((𝟭‘𝐼)‘{𝑖}), (1r‘𝑅), (0g‘𝑅)))))
239		esplyfval1.v	. . . . . 6 ⊢ 𝑉 = (𝐼 mVar 𝑅)
240	3	psrbasfsupp 33669	. . . . . 6 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
241	239, 240, 11, 12, 4, 5	mvrfval 21956	. . . . 5 ⊢ (𝜑 → 𝑉 = (𝑖 ∈ 𝐼 ↦ (𝑡 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑡 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅)))))
242	232, 238, 241	3eqtr4d 2782	. . . 4 ⊢ (𝜑 → ((𝑡 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅)))) ∘ (𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))) = 𝑉)
243	242	oveq2d 7380	. . 3 ⊢ (𝜑 → (𝑀 Σg ((𝑡 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅)))) ∘ (𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖})))) = (𝑀 Σg 𝑉))
244	13, 217, 243	3eqtr2d 2778	. 2 ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝑁) = (𝑀 Σg 𝑉))
245	2, 244	eqtrid 2784	1 ⊢ (𝜑 → (𝐸‘𝑁) = (𝑀 Σg 𝑉))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  {crab 3390  Vcvv 3430   ⊆ wss 3890  ifcif 4467  {csn 4568  {cpr 4570   class class class wbr 5086   ↦ cmpt 5167  dom cdm 5628  ran crn 5629   ∘ ccom 5632   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7364   supp csupp 8107   ↑_m cmap 8770  Fincfn 8890   finSupp cfsupp 9271  ℂcc 11033  0cc0 11035  1c1 11036  𝟭cind 12156  ℕ₀cn0 12434  ♯chash 14289  Basecbs 17176  0_gc0g 17399   Σ_g cgsu 17400  SubMndcsubmnd 18747  CMndccmn 19752  mulGrpcmgp 20118  1_rcur 20159  Ringcrg 20211  CRingccrg 20212  Fieldcfield 20704  ℂ_fldccnfld 21349   mVar cmvr 21882   mPoly cmpl 21883  eSymPolycesply 33697

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5306  ax-pr 5374  ax-un 7686  ax-cnex 11091  ax-resscn 11092  ax-1cn 11093  ax-icn 11094  ax-addcl 11095  ax-addrcl 11096  ax-mulcl 11097  ax-mulrcl 11098  ax-mulcom 11099  ax-addass 11100  ax-mulass 11101  ax-distr 11102  ax-i2m1 11103  ax-1ne0 11104  ax-1rid 11105  ax-rnegex 11106  ax-rrecex 11107  ax-cnre 11108  ax-pre-lttri 11109  ax-pre-lttrn 11110  ax-pre-ltadd 11111  ax-pre-mulgt0 11112  ax-addf 11114  ax-mulf 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5523  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5581  df-se 5582  df-we 5583  df-xp 5634  df-rel 5635  df-cnv 5636  df-co 5637  df-dm 5638  df-rn 5639  df-res 5640  df-ima 5641  df-pred 6263  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7321  df-ov 7367  df-oprab 7368  df-mpo 7369  df-of 7628  df-ofr 7629  df-om 7815  df-1st 7939  df-2nd 7940  df-supp 8108  df-tpos 8173  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-oadd 8406  df-er 8640  df-map 8772  df-pm 8773  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fsupp 9272  df-sup 9352  df-oi 9422  df-card 9860  df-pnf 11178  df-mnf 11179  df-xr 11180  df-ltxr 11181  df-le 11182  df-sub 11376  df-neg 11377  df-div 11805  df-ind 12157  df-nn 12172  df-2 12241  df-3 12242  df-4 12243  df-5 12244  df-6 12245  df-7 12246  df-8 12247  df-9 12248  df-n0 12435  df-xnn0 12508  df-z 12522  df-dec 12642  df-uz 12786  df-fz 13459  df-fzo 13606  df-seq 13961  df-hash 14290  df-struct 17114  df-sets 17131  df-slot 17149  df-ndx 17161  df-base 17177  df-ress 17198  df-plusg 17230  df-mulr 17231  df-starv 17232  df-sca 17233  df-vsca 17234  df-ip 17235  df-tset 17236  df-ple 17237  df-ds 17239  df-unif 17240  df-hom 17241  df-cco 17242  df-0g 17401  df-gsum 17402  df-prds 17407  df-pws 17409  df-mre 17545  df-mrc 17546  df-acs 17548  df-mgm 18605  df-sgrp 18684  df-mnd 18700  df-mhm 18748  df-submnd 18749  df-grp 18909  df-minusg 18910  df-mulg 19041  df-subg 19096  df-ghm 19185  df-cntz 19289  df-cmn 19754  df-abl 19755  df-mgp 20119  df-rng 20131  df-ur 20160  df-ring 20213  df-cring 20214  df-oppr 20314  df-dvdsr 20334  df-unit 20335  df-invr 20365  df-dvr 20378  df-rhm 20449  df-subrng 20520  df-subrg 20544  df-drng 20705  df-field 20706  df-cnfld 21350  df-zring 21424  df-zrh 21480  df-psr 21886  df-mvr 21887  df-mpl 21888  df-esply 33699

This theorem is referenced by: (None)