Theorem esplyfvaln 34273

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 6906	. 2 ⊢ (𝐸‘𝑁) = ((𝐼eSymPoly𝑅)‘𝑁)
3		eqid 2763	. . . 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 20500	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
7		esplyfvaln.n	. . . . 5 ⊢ 𝑁 = (♯‘𝐼)
8		hashcl 14524	. . . . . 6 ⊢ (𝐼 ∈ Fin → (♯‘𝐼) ∈ ℕ0)
9	4, 8	syl 18	. . . . 5 ⊢ (𝜑 → (♯‘𝐼) ∈ ℕ0)
10	7, 9	eqeltrid 2868	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
11		eqid 2763	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
12		eqid 2763	. . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅)
13	3, 4, 6, 10, 11, 12	esplyfval3 34271	. . 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 2763	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
16		breq1 5109	. . . . . . 7 ⊢ (ℎ = ((𝟭‘𝐼)‘{𝑖}) → (ℎ finSupp 0 ↔ ((𝟭‘𝐼)‘{𝑖}) finSupp 0))
17		nn0ex 12636	. . . . . . . . 9 ⊢ ℕ0 ∈ V
18	17	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ℕ0 ∈ V)
19	4	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ Fin)
20		snssi 4746	. . . . . . . . . 10 ⊢ (𝑖 ∈ 𝐼 → {𝑖} ⊆ 𝐼)
21		indf 12350	. . . . . . . . . 10 ⊢ ((𝐼 ∈ Fin ∧ {𝑖} ⊆ 𝐼) → ((𝟭‘𝐼)‘{𝑖}):𝐼⟶{0, 1})
22	4, 20, 21	syl2an 608	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}):𝐼⟶{0, 1})
23		0nn0 12645	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ0
24	23	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 0 ∈ ℕ0)
25		1nn0 12646	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ0
26	25	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 1 ∈ ℕ0)
27	24, 26	prssd 4781	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → {0, 1} ⊆ ℕ0)
28	22, 27	fssd 6746	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}):𝐼⟶ℕ0)
29	18, 19, 28	elmapdd 8886	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}) ∈ (ℕ0 ↑m 𝐼))
30	22, 19, 24	fdmfifsupp 9391	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}) finSupp 0)
31	16, 29, 30	elrabd 3650	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
32	31	fmpttd 7136	. . . . 5 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖})):𝐼⟶{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
33		esplyfvaln.m	. . . . 5 ⊢ 𝑀 = (mulGrp‘𝑊)
34		eqeq2 2775	. . . . . . . . 9 ⊢ (𝑡 = 𝑦 → (𝑢 = 𝑡 ↔ 𝑢 = 𝑦))
35	34	ifbid 4509	. . . . . . . 8 ⊢ (𝑡 = 𝑦 → if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅)) = if(𝑢 = 𝑦, (1r‘𝑅), (0g‘𝑅)))
36	35	mpteq2dv 5202	. . . . . . 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 2767	. . . . . . . . 9 ⊢ (𝑢 = 𝑧 → (𝑢 = 𝑦 ↔ 𝑧 = 𝑦))
38	37	ifbid 4509	. . . . . . . 8 ⊢ (𝑢 = 𝑧 → if(𝑢 = 𝑦, (1r‘𝑅), (0g‘𝑅)) = if(𝑧 = 𝑦, (1r‘𝑅), (0g‘𝑅)))
39	38	cbvmptv 5212	. . . . . . 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 2814	. . . . . 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 5212	. . . . 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 34253	. . . 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 2763	. . . . 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 2775	. . . . . . . 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 4509	. . . . . . 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 5202	. . . . . 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 490	. . . . . . . . . . . . 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 5926	. . . . . . . . . . . 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 1949	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
50		eqid 2763	. . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖})) = (𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))
52		sneq 4594	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑘 → {𝑖} = {𝑘})
53	52	fveq2d 6909	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑘 → ((𝟭‘𝐼)‘{𝑖}) = ((𝟭‘𝐼)‘{𝑘}))
54		simpr 490	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼)
55		fvexd 6920	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑘}) ∈ V)
56	51, 53, 54, 55	fvmptd3 7037	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → ((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘) = ((𝟭‘𝐼)‘{𝑘}))
57	56	fveq1d 6907	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗) = (((𝟭‘𝐼)‘{𝑘})‘𝑗))
58	4	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → 𝐼 ∈ Fin)
59	54	snssd 4747	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → {𝑘} ⊆ 𝐼)
60		simplr 781	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → 𝑗 ∈ 𝐼)
61		indfval 12351	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼 ∈ Fin ∧ {𝑘} ⊆ 𝐼 ∧ 𝑗 ∈ 𝐼) → (((𝟭‘𝐼)‘{𝑘})‘𝑗) = if(𝑗 ∈ {𝑘}, 1, 0))
62	58, 59, 60, 61	syl3anc 1399	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (((𝟭‘𝐼)‘{𝑘})‘𝑗) = if(𝑗 ∈ {𝑘}, 1, 0))
63		velsn 4600	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ {𝑘} ↔ 𝑗 = 𝑘)
64		equcom 2053	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑘 ↔ 𝑘 = 𝑗)
65	63, 64	bitri 278	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ {𝑘} ↔ 𝑘 = 𝑗)
66	65	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑗 ∈ {𝑘} ↔ 𝑘 = 𝑗))
67	66	ifbid 4509	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → if(𝑗 ∈ {𝑘}, 1, 0) = if(𝑘 = 𝑗, 1, 0))
68	57, 62, 67	3eqtrd 2802	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗) = if(𝑘 = 𝑗, 1, 0))
69	68	mpteq2dva 5201	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1, 0)))
70	69	oveq2d 7456	. . . . . . . . . . . . . . . . 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 21729	. . . . . . . . . . . . . . . . . 18 ⊢ 0 = (0g‘ℂfld)
72		cnfldfld 33970	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℂfld ∈ Field
73		id 23	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℂfld ∈ Field → ℂfld ∈ Field)
74	73	fldcrngd 21022	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℂfld ∈ Field → ℂfld ∈ CRing)
75		crngring 20499	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℂfld ∈ CRing → ℂfld ∈ Ring)
76		ringcmn 20538	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℂfld ∈ Ring → ℂfld ∈ CMnd)
77	74, 75, 76	3syl 19	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℂfld ∈ Field → ℂfld ∈ CMnd)
78	72, 77	mp1i 14	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → ℂfld ∈ CMnd)
79	78	cmnmndd 20045	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → ℂfld ∈ Mnd)
80	4	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → 𝐼 ∈ Fin)
81		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → 𝑗 ∈ 𝐼)
82		eqid 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1, 0))
83		ax-1cn 11282	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℂ
84		cnfldbas 21709	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℂ = (Base‘ℂfld)
85	83, 84	eleqtri 2862	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ (Base‘ℂfld)
86	85	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → 1 ∈ (Base‘ℂfld))
87	71, 79, 80, 81, 82, 86	gsummptif1n0 20207	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1, 0))) = 1)
88	70, 87	eqtrd 2798	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))) = 1)
89		1elpr01 11328	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ {0, 1}
90	88, 89	eqeltrdi 2872	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))) ∈ {0, 1})
91	90	adantlr 728	. . . . . . . . . . . . . 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})
92	49, 50, 91	rnmptssd 7145	. . . . . . . . . . . . 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})
93	92	adantr 486	. . . . . . . . . . . 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})
94	48, 93	eqsstrd 3947	. . . . . . . . . . 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})
95	47	oveq1d 7455	. . . . . . . . . . . . . 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))
96		suppssdm 8219	. . . . . . . . . . . . . . . . 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))))
97		nn0subm 21755	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℕ0 ∈ (SubMnd‘ℂfld)
98	97	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → ℕ0 ∈ (SubMnd‘ℂfld))
99	23	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → 0 ∈ ℕ0)
100	25	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → 1 ∈ ℕ0)
101	99, 100	prssd 4781	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → {0, 1} ⊆ ℕ0)
102		indf 12350	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐼 ∈ Fin ∧ {𝑘} ⊆ 𝐼) → ((𝟭‘𝐼)‘{𝑘}):𝐼⟶{0, 1})
103	58, 59, 102	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑘}):𝐼⟶{0, 1})
104	103, 60	ffvelcdmd 7106	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (((𝟭‘𝐼)‘{𝑘})‘𝑗) ∈ {0, 1})
105	101, 104	sseldd 3914	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (((𝟭‘𝐼)‘{𝑘})‘𝑗) ∈ ℕ0)
106	57, 105	eqeltrd 2864	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗) ∈ ℕ0)
107	106	fmpttd 7136	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)):𝐼⟶ℕ0)
108	23	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → 0 ∈ ℕ0)
109	107, 80, 108	fdmfifsupp 9391	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)) finSupp 0)
110	71, 78, 80, 98, 107, 109	gsumsubmcl 20160	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))) ∈ ℕ0)
111	50, 110	dmmptd 6703	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) = 𝐼)
112	96, 111	sseqtrid 3955	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) supp 0) ⊆ 𝐼)
113		nfv 1949	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑖 ∈ 𝐼)
114		ovexd 7475	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗))) ∈ V)
115		eqid 2763	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗)))) = (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗))))
116	113, 114, 115	fnmptd 6699	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗)))) Fn 𝐼)
117		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
118		fveq2 6905	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑖 → (((𝟭‘𝐼)‘{𝑘})‘𝑗) = (((𝟭‘𝐼)‘{𝑘})‘𝑖))
119	118	mpteq2dv 5202	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 𝑖 → (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗)) = (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑖)))
120	119	oveq2d 7456	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑖 → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗))) = (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑖))))
121		ovexd 7475	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑖))) ∈ V)
122	115, 120, 117, 121	fvmptd3 7037	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗))))‘𝑖) = (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑖))))
123	4	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → 𝐼 ∈ Fin)
124		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼)
125	124	snssd 4747	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → {𝑘} ⊆ 𝐼)
126		simplr 781	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → 𝑖 ∈ 𝐼)
127		indfval 12351	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐼 ∈ Fin ∧ {𝑘} ⊆ 𝐼 ∧ 𝑖 ∈ 𝐼) → (((𝟭‘𝐼)‘{𝑘})‘𝑖) = if(𝑖 ∈ {𝑘}, 1, 0))
128	123, 125, 126, 127	syl3anc 1399	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (((𝟭‘𝐼)‘{𝑘})‘𝑖) = if(𝑖 ∈ {𝑘}, 1, 0))
129		velsn 4600	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 ∈ {𝑘} ↔ 𝑖 = 𝑘)
130		equcom 2053	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 = 𝑘 ↔ 𝑘 = 𝑖)
131	129, 130	bitri 278	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ {𝑘} ↔ 𝑘 = 𝑖)
132	131	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑖 ∈ {𝑘} ↔ 𝑘 = 𝑖))
133	132	ifbid 4509	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → if(𝑖 ∈ {𝑘}, 1, 0) = if(𝑘 = 𝑖, 1, 0))
134	128, 133	eqtrd 2798	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (((𝟭‘𝐼)‘{𝑘})‘𝑖) = if(𝑘 = 𝑖, 1, 0))
135	134	mpteq2dva 5201	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑖)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑖, 1, 0)))
136	135	oveq2d 7456	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑖))) = (ℂfld Σg (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑖, 1, 0))))
137	72, 77	mp1i 14	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ℂfld ∈ CMnd)
138	137	cmnmndd 20045	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ℂfld ∈ Mnd)
139		eqid 2763	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑖, 1, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑖, 1, 0))
140	85	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 1 ∈ (Base‘ℂfld))
141	71, 138, 19, 117, 139, 140	gsummptif1n0 20207	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑖, 1, 0))) = 1)
142	122, 136, 141	3eqtrd 2802	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗))))‘𝑖) = 1)
143		ax-1ne0 11293	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ≠ 0
144	143	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 1 ≠ 0)
145	142, 144	eqnetrd 3026	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗))))‘𝑖) ≠ 0)
146	116, 19, 24, 117, 145	elsuppfnd 33343	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗)))) supp 0))
147	146	ex 418	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑖 ∈ 𝐼 → 𝑖 ∈ ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗)))) supp 0)))
148	147	ssrdv 3919	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐼 ⊆ ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗)))) supp 0))
149	57	mpteq2dva 5201	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)) = (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗)))
150	149	oveq2d 7456	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗))) = (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝟭‘𝐼)‘{𝑘})‘𝑗))))
151	150	mpteq2dva 5201	. . . . . . . . . . . . . . . . . 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)))))
152	151	oveq1d 7455	. . . . . . . . . . . . . . . . 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))
153	148, 152	sseqtrrd 3950	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ⊆ ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) supp 0))
154	112, 153	eqssd 3930	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) supp 0) = 𝐼)
155	154	ad2antrr 739	. . . . . . . . . . . . . 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)
156	95, 155	eqtrd 2798	. . . . . . . . . . . . 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)
157	156	fveq2d 6909	. . . . . . . . . . . 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))
158	157, 7	eqtr4di 2816	. . . . . . . . . . 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)
159	94, 158	jca 521	. . . . . . . . . 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))
160		simpllr 788	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → ran 𝑢 ⊆ {0, 1})
161		ssrab2 4008	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐼)
162	161	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐼))
163	162	sselda 3913	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑢 ∈ (ℕ0 ↑m 𝐼))
164	163	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) → 𝑢 ∈ (ℕ0 ↑m 𝐼))
165	164	elmaprd 8895	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) → 𝑢:𝐼⟶ℕ0)
166	165	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → 𝑢:𝐼⟶ℕ0)
167	166	ffnd 6729	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → 𝑢 Fn 𝐼)
168		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → 𝑗 ∈ 𝐼)
169	167, 168	fnfvelrnd 7103	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → (𝑢‘𝑗) ∈ ran 𝑢)
170	160, 169	sseldd 3914	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → (𝑢‘𝑗) ∈ {0, 1})
171	4	ad3antrrr 743	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) → 𝐼 ∈ Fin)
172	171	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → 𝐼 ∈ Fin)
173	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)
174		suppssdm 8219	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 supp 0) ⊆ dom 𝑢
175	174, 166	fssdm 6748	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → (𝑢 supp 0) ⊆ 𝐼)
176		simplr 781	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → (♯‘(𝑢 supp 0)) = 𝑁)
177	176, 7	eqtr2di 2815	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → (♯‘𝐼) = (♯‘(𝑢 supp 0)))
178	172, 175, 177	phphashd 14635	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → 𝐼 = (𝑢 supp 0))
179	168, 178	eleqtrd 2866	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → 𝑗 ∈ (𝑢 supp 0))
180		elsuppfn 8212	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 Fn 𝐼 ∧ 𝐼 ∈ Fin ∧ 0 ∈ ℕ0) → (𝑗 ∈ (𝑢 supp 0) ↔ (𝑗 ∈ 𝐼 ∧ (𝑢‘𝑗) ≠ 0)))
181	180	simplbda 505	. . . . . . . . . . . . . . 15 ⊢ (((𝑢 Fn 𝐼 ∧ 𝐼 ∈ Fin ∧ 0 ∈ ℕ0) ∧ 𝑗 ∈ (𝑢 supp 0)) → (𝑢‘𝑗) ≠ 0)
182	167, 172, 173, 179, 181	syl31anc 1401	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → (𝑢‘𝑗) ≠ 0)
183		elprn1 4612	. . . . . . . . . . . . . 14 ⊢ (((𝑢‘𝑗) ∈ {0, 1} ∧ (𝑢‘𝑗) ≠ 0) → (𝑢‘𝑗) = 1)
184	170, 182, 183	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) ∧ 𝑗 ∈ 𝐼) → (𝑢‘𝑗) = 1)
185	184	mpteq2dva 5201	. . . . . . . . . . . 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))
186	165	feqmptd 6973	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑢 ⊆ {0, 1}) ∧ (♯‘(𝑢 supp 0)) = 𝑁) → 𝑢 = (𝑗 ∈ 𝐼 ↦ (𝑢‘𝑗)))
187	88	mpteq2dva 5201	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) = (𝑗 ∈ 𝐼 ↦ 1))
188	187	ad3antrrr 743	. . . . . . . . . . . 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))
189	185, 186, 188	3eqtr4d 2808	. . . . . . . . . . 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)))))
190	189	anasss 472	. . . . . . . . . 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)))))
191	159, 190	impbida 813	. . . . . . . . 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)))
192	191	ifbid 4509	. . . . . . . 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)))
193	192	mpteq2dva 5201	. . . . . . 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))))
194		rneq 5924	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑓 → ran 𝑢 = ran 𝑓)
195	194	sseq1d 3944	. . . . . . . . . 10 ⊢ (𝑢 = 𝑓 → (ran 𝑢 ⊆ {0, 1} ↔ ran 𝑓 ⊆ {0, 1}))
196		oveq1 7447	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑓 → (𝑢 supp 0) = (𝑓 supp 0))
197	196	fveqeq2d 6913	. . . . . . . . . 10 ⊢ (𝑢 = 𝑓 → ((♯‘(𝑢 supp 0)) = 𝑁 ↔ (♯‘(𝑓 supp 0)) = 𝑁))
198	195, 197	anbi12d 644	. . . . . . . . 9 ⊢ (𝑢 = 𝑓 → ((ran 𝑢 ⊆ {0, 1} ∧ (♯‘(𝑢 supp 0)) = 𝑁) ↔ (ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝑁)))
199	198	ifbid 4509	. . . . . . . 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)))
200	199	cbvmptv 5212	. . . . . . 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)))
201	193, 200	eqtrdi 2814	. . . . . 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))))
202	46, 201	sylan9eqr 2820	. . . . 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))))
203		breq1 5109	. . . . . 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))
204	17	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℕ0 ∈ V)
205	110	fmpttd 7136	. . . . . . 7 ⊢ (𝜑 → (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))):𝐼⟶ℕ0)
206	204, 4, 205	elmapdd 8886	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) ∈ (ℕ0 ↑m 𝐼))
207	23	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℕ0)
208	205, 4, 207	fdmfifsupp 9391	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) finSupp 0)
209	203, 206, 208	elrabd 3650	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ 𝐼 ↦ (ℂfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖}))‘𝑘)‘𝑗)))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
210		ovex 7473	. . . . . . . 8 ⊢ (ℕ0 ↑m 𝐼) ∈ V
211	210	rabex 5304	. . . . . . 7 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∈ V
212	211	a1i 11	. . . . . 6 ⊢ (𝜑 → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∈ V)
213	212	mptexd 7250	. . . . 5 ⊢ (𝜑 → (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝑁), (1r‘𝑅), (0g‘𝑅))) ∈ V)
214	43, 202, 209, 213	fvmptd2 7022	. . . 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))))
215	42, 214	eqtrd 2798	. . 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))))
216		indval 12347	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ Fin ∧ {𝑖} ⊆ 𝐼) → ((𝟭‘𝐼)‘{𝑖}) = (𝑗 ∈ 𝐼 ↦ if(𝑗 ∈ {𝑖}, 1, 0)))
217	4, 20, 216	syl2an 608	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}) = (𝑗 ∈ 𝐼 ↦ if(𝑗 ∈ {𝑖}, 1, 0)))
218		velsn 4600	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ {𝑖} ↔ 𝑗 = 𝑖)
219	218	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → (𝑗 ∈ {𝑖} ↔ 𝑗 = 𝑖))
220	219	ifbid 4509	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼) → if(𝑗 ∈ {𝑖}, 1, 0) = if(𝑗 = 𝑖, 1, 0))
221	220	mpteq2dva 5201	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑗 ∈ 𝐼 ↦ if(𝑗 ∈ {𝑖}, 1, 0)) = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
222	217, 221	eqtrd 2798	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}) = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
223	222	eqeq2d 2774	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑢 = ((𝟭‘𝐼)‘{𝑖}) ↔ 𝑢 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))))
224	223	ifbid 4509	. . . . . . . 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)))
225	224	mpteq2dv 5202	. . . . . . 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))))
226		eqeq1 2767	. . . . . . . . 9 ⊢ (𝑡 = 𝑢 → (𝑡 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)) ↔ 𝑢 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))))
227	226	ifbid 4509	. . . . . . . 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)))
228	227	cbvmptv 5212	. . . . . . 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)))
229	225, 228	eqtr4di 2816	. . . . . 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))))
230	229	mpteq2dva 5201	. . . . 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)))))
231		eqidd 2764	. . . . . 6 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖})) = (𝑖 ∈ 𝐼 ↦ ((𝟭‘𝐼)‘{𝑖})))
232		eqidd 2764	. . . . . 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)))))
233		eqeq2 2775	. . . . . . . 8 ⊢ (𝑡 = ((𝟭‘𝐼)‘{𝑖}) → (𝑢 = 𝑡 ↔ 𝑢 = ((𝟭‘𝐼)‘{𝑖})))
234	233	ifbid 4509	. . . . . . 7 ⊢ (𝑡 = ((𝟭‘𝐼)‘{𝑖}) → if(𝑢 = 𝑡, (1r‘𝑅), (0g‘𝑅)) = if(𝑢 = ((𝟭‘𝐼)‘{𝑖}), (1r‘𝑅), (0g‘𝑅)))
235	234	mpteq2dv 5202	. . . . . 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))))
236	31, 231, 232, 235	fmptco 7151	. . . . 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)))))
237		esplyfval1.v	. . . . . 6 ⊢ 𝑉 = (𝐼 mVar 𝑅)
238	3	psrbasfsupp 34210	. . . . . 6 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
239	237, 238, 11, 12, 4, 5	mvrfval 22315	. . . . 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)))))
240	230, 236, 239	3eqtr4d 2808	. . . 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)
241	240	oveq2d 7456	. . 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))
242	13, 215, 241	3eqtr2d 2804	. 2 ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝑁) = (𝑀 Σg 𝑉))
243	2, 242	eqtrid 2810	1 ⊢ (𝜑 → (𝐸‘𝑁) = (𝑀 Σg 𝑉))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  {crab 3416  Vcvv 3454   ⊆ wss 3900  ifcif 4485  {csn 4588  {cpr 4590   class class class wbr 5106   ↦ cmpt 5189  dom cdm 5655  ran crn 5656   ∘ ccom 5659   Fn wfn 6550  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440   supp csupp 8201   ↑_m cmap 8872  Fincfn 8998   finSupp cfsupp 9378  ℂcc 11222  0cc0 11224  1c1 11225  𝟭cind 12344  ℕ₀cn0 12630  ♯chash 14498  Basecbs 17413  0_gc0g 17636   Σ_g cgsu 17637  SubMndcsubmnd 19003  CMndccmn 20021  mulGrpcmgp 20387  1_rcur 20434  Ringcrg 20486  CRingccrg 20487  Fieldcfield 21008  ℂ_fldccnfld 21705   mVar cmvr 22240   mPoly cmpl 22241  eSymPolycesply 34255

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301  ax-addf 11303  ax-mulf 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-of 7714  df-ofr 7715  df-om 7899  df-1st 8022  df-2nd 8023  df-supp 8202  df-tpos 8268  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-oadd 8505  df-er 8742  df-map 8874  df-pm 8875  df-ixp 8951  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-fsupp 9379  df-sup 9458  df-oi 9528  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-div 11998  df-ind 12345  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-9 12436  df-n0 12631  df-xnn0 12704  df-z 12718  df-dec 12839  df-uz 12990  df-fz 13664  df-fzo 13813  df-seq 14169  df-hash 14499  df-struct 17351  df-sets 17368  df-slot 17386  df-ndx 17398  df-base 17414  df-ress 17435  df-plusg 17467  df-mulr 17468  df-starv 17469  df-sca 17470  df-vsca 17471  df-ip 17472  df-tset 17473  df-ple 17474  df-ds 17476  df-unif 17477  df-hom 17478  df-cco 17479  df-0g 17638  df-gsum 17639  df-prds 17644  df-pws 17646  df-mre 17782  df-mrc 17783  df-acs 17785  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-mhm 19004  df-submnd 19005  df-grp 19173  df-minusg 19174  df-mulg 19304  df-subg 19359  df-ghm 19454  df-cntz 19558  df-cmn 20023  df-abl 20024  df-mgp 20388  df-rng 20402  df-ur 20435  df-ring 20488  df-cring 20489  df-oppr 20594  df-dvdsr 20614  df-unit 20615  df-invr 20645  df-dvr 20658  df-rhm 20729  df-subrng 20825  df-subrg 20849  df-drng 21009  df-field 21010  df-cnfld 21706  df-zring 21780  df-zrh 21836  df-psr 22244  df-mvr 22245  df-mpl 22246  df-esply 34257

This theorem is used by: (None)