Theorem esplyfval1 34272

Description: The first elementary symmetric polynomial is the sum 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)
esplyfval1.r	⊢ (𝜑 → 𝑅 ∈ Ring)

Assertion

Ref	Expression
esplyfval1	⊢ (𝜑 → (𝐸‘1) = (𝑊 Σg 𝑉))

Proof of Theorem esplyfval1

Dummy variables 𝑓 ℎ 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		esplyfval1.v	. . . . . . . . . . 11 ⊢ 𝑉 = (𝐼 mVar 𝑅)
2		eqid 2763	. . . . . . . . . . . 12 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
3	2	psrbasfsupp 34210	. . . . . . . . . . 11 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
4		eqid 2763	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
5		eqid 2763	. . . . . . . . . . 11 ⊢ (1r‘𝑅) = (1r‘𝑅)
6		esplyfval1.i	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ Fin)
7	6	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝐼 ∈ Fin)
8		esplyfval1.r	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ Ring)
9	8	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑅 ∈ Ring)
10		simplr 781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑖 ∈ 𝐼)
11		simpr 490	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
12	1, 3, 4, 5, 7, 9, 10, 11	mvrval2 22317	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ((𝑉‘𝑖)‘𝑓) = if(𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅)))
13	12	ad4ant14 765	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ((𝑉‘𝑖)‘𝑓) = if(𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅)))
14	13	an52ds 33121	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) → ((𝑉‘𝑖)‘𝑓) = if(𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅)))
15	14	mpteq2dva 5201	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓)) = (𝑖 ∈ 𝐼 ↦ if(𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅))))
16	15	oveq2d 7456	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓))) = (𝑅 Σg (𝑖 ∈ 𝐼 ↦ if(𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅)))))
17		nfv 1949	. . . . . . . . . 10 ⊢ Ⅎ𝑗((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼)
18		nfmpt1 5207	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))
19	18	nfeq2 2943	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))
20		nfv 1949	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑖 = ∪ (𝑓 supp 0)
21	19, 20	nfbi 1938	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)) ↔ 𝑖 = ∪ (𝑓 supp 0))
22		unisnv 4890	. . . . . . . . . . . . 13 ⊢ ∪ {𝑗} = 𝑗
23	22	eqeq2i 2776	. . . . . . . . . . . 12 ⊢ (𝑖 = ∪ {𝑗} ↔ 𝑖 = 𝑗)
24	23	a1i 11	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) → (𝑖 = ∪ {𝑗} ↔ 𝑖 = 𝑗))
25		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) → (𝑓 supp 0) = {𝑗})
26	25	unieqd 4883	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) → ∪ (𝑓 supp 0) = ∪ {𝑗})
27	26	adantllr 732	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) → ∪ (𝑓 supp 0) = ∪ {𝑗})
28	27	eqeq2d 2774	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) → (𝑖 = ∪ (𝑓 supp 0) ↔ 𝑖 = ∪ {𝑗}))
29		simplr 781	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) ∧ 𝑖 = 𝑗) → (𝑓 supp 0) = {𝑗})
30	29	fveq2d 6909	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) ∧ 𝑖 = 𝑗) → ((𝟭‘𝐼)‘(𝑓 supp 0)) = ((𝟭‘𝐼)‘{𝑗}))
31	6	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) → 𝐼 ∈ Fin)
32		ssrab2 4008	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐼)
33	32	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐼))
34	33	sselda 3913	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑓 ∈ (ℕ0 ↑m 𝐼))
35	34	elmaprd 8895	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑓:𝐼⟶ℕ0)
36	35	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) → 𝑓:𝐼⟶ℕ0)
37		ffrn 6742	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝐼⟶ℕ0 → 𝑓:𝐼⟶ran 𝑓)
38	36, 37	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) → 𝑓:𝐼⟶ran 𝑓)
39		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) → ran 𝑓 ⊆ {0, 1})
40	38, 39	fssd 6746	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) → 𝑓:𝐼⟶{0, 1})
41	31, 40	indfsid 33503	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) → 𝑓 = ((𝟭‘𝐼)‘(𝑓 supp 0)))
42	41	ad5antr 747	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) ∧ 𝑖 = 𝑗) → 𝑓 = ((𝟭‘𝐼)‘(𝑓 supp 0)))
43		sneq 4594	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑗 → {𝑖} = {𝑗})
44	43	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) ∧ 𝑖 = 𝑗) → {𝑖} = {𝑗})
45	44	fveq2d 6909	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) ∧ 𝑖 = 𝑗) → ((𝟭‘𝐼)‘{𝑖}) = ((𝟭‘𝐼)‘{𝑗}))
46	30, 42, 45	3eqtr4d 2808	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) ∧ 𝑖 = 𝑗) → 𝑓 = ((𝟭‘𝐼)‘{𝑖}))
47		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) ∧ 𝑓 = ((𝟭‘𝐼)‘{𝑖})) → 𝑓 = ((𝟭‘𝐼)‘{𝑖}))
48	47	oveq1d 7455	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) ∧ 𝑓 = ((𝟭‘𝐼)‘{𝑖})) → (𝑓 supp 0) = (((𝟭‘𝐼)‘{𝑖}) supp 0))
49		simplr 781	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) ∧ 𝑓 = ((𝟭‘𝐼)‘{𝑖})) → (𝑓 supp 0) = {𝑗})
50	6	ad3antrrr 743	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → 𝐼 ∈ Fin)
51	50	ad4antr 745	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) ∧ 𝑓 = ((𝟭‘𝐼)‘{𝑖})) → 𝐼 ∈ Fin)
52		snssi 4746	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ 𝐼 → {𝑖} ⊆ 𝐼)
53	52	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) → {𝑖} ⊆ 𝐼)
54	53	ad3antrrr 743	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) ∧ 𝑓 = ((𝟭‘𝐼)‘{𝑖})) → {𝑖} ⊆ 𝐼)
55		indsupp 33501	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ Fin ∧ {𝑖} ⊆ 𝐼) → (((𝟭‘𝐼)‘{𝑖}) supp 0) = {𝑖})
56	51, 54, 55	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) ∧ 𝑓 = ((𝟭‘𝐼)‘{𝑖})) → (((𝟭‘𝐼)‘{𝑖}) supp 0) = {𝑖})
57	48, 49, 56	3eqtr3rd 2807	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) ∧ 𝑓 = ((𝟭‘𝐼)‘{𝑖})) → {𝑖} = {𝑗})
58		vex 3458	. . . . . . . . . . . . . . 15 ⊢ 𝑖 ∈ V
59	58	sneqr 4798	. . . . . . . . . . . . . 14 ⊢ ({𝑖} = {𝑗} → 𝑖 = 𝑗)
60	57, 59	syl 18	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) ∧ 𝑓 = ((𝟭‘𝐼)‘{𝑖})) → 𝑖 = 𝑗)
61	46, 60	impbida 813	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) → (𝑖 = 𝑗 ↔ 𝑓 = ((𝟭‘𝐼)‘{𝑖})))
62		indsn 33497	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ Fin ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}) = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
63	50, 62	sylan 592	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}) = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
64	63	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) → ((𝟭‘𝐼)‘{𝑖}) = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
65	64	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) → (𝑓 = ((𝟭‘𝐼)‘{𝑖}) ↔ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))))
66	61, 65	bitr2d 283	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) → (𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)) ↔ 𝑖 = 𝑗))
67	24, 28, 66	3bitr4rd 315	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) → (𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)) ↔ 𝑖 = ∪ (𝑓 supp 0)))
68		ovexd 7475	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → (𝑓 supp 0) ∈ V)
69		simpr 490	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → (♯‘(𝑓 supp 0)) = 1)
70		hash1snb 14588	. . . . . . . . . . . . . 14 ⊢ ((𝑓 supp 0) ∈ V → ((♯‘(𝑓 supp 0)) = 1 ↔ ∃𝑗(𝑓 supp 0) = {𝑗}))
71	70	biimpa 482	. . . . . . . . . . . . 13 ⊢ (((𝑓 supp 0) ∈ V ∧ (♯‘(𝑓 supp 0)) = 1) → ∃𝑗(𝑓 supp 0) = {𝑗})
72	68, 69, 71	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → ∃𝑗(𝑓 supp 0) = {𝑗})
73		exsnrex 4641	. . . . . . . . . . . 12 ⊢ (∃𝑗(𝑓 supp 0) = {𝑗} ↔ ∃𝑗 ∈ (𝑓 supp 0)(𝑓 supp 0) = {𝑗})
74	72, 73	sylib 221	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → ∃𝑗 ∈ (𝑓 supp 0)(𝑓 supp 0) = {𝑗})
75	74	adantr 486	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) → ∃𝑗 ∈ (𝑓 supp 0)(𝑓 supp 0) = {𝑗})
76	17, 21, 67, 75	r19.29af2 3274	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) → (𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)) ↔ 𝑖 = ∪ (𝑓 supp 0)))
77	76	ifbid 4509	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) → if(𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅)) = if(𝑖 = ∪ (𝑓 supp 0), (1r‘𝑅), (0g‘𝑅)))
78	77	mpteq2dva 5201	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → (𝑖 ∈ 𝐼 ↦ if(𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅))) = (𝑖 ∈ 𝐼 ↦ if(𝑖 = ∪ (𝑓 supp 0), (1r‘𝑅), (0g‘𝑅))))
79	78	oveq2d 7456	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → (𝑅 Σg (𝑖 ∈ 𝐼 ↦ if(𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅)))) = (𝑅 Σg (𝑖 ∈ 𝐼 ↦ if(𝑖 = ∪ (𝑓 supp 0), (1r‘𝑅), (0g‘𝑅)))))
80		ringmnd 20497	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
81	8, 80	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Mnd)
82	81	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → 𝑅 ∈ Mnd)
83		suppssdm 8219	. . . . . . . . . . . 12 ⊢ (𝑓 supp 0) ⊆ dom 𝑓
84	35	fdmd 6739	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → dom 𝑓 = 𝐼)
85	84	ad4antr 745	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) → dom 𝑓 = 𝐼)
86	83, 85	sseqtrid 3955	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) → (𝑓 supp 0) ⊆ 𝐼)
87		simplr 781	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) → 𝑗 ∈ (𝑓 supp 0))
88	86, 87	sseldd 3914	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) → 𝑗 ∈ 𝐼)
89	22, 88	eqeltrid 2868	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) → ∪ {𝑗} ∈ 𝐼)
90	26, 89	eqeltrd 2864	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑗 ∈ (𝑓 supp 0)) ∧ (𝑓 supp 0) = {𝑗}) → ∪ (𝑓 supp 0) ∈ 𝐼)
91	90, 74	r19.29a 3174	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → ∪ (𝑓 supp 0) ∈ 𝐼)
92		eqid 2763	. . . . . . 7 ⊢ (𝑖 ∈ 𝐼 ↦ if(𝑖 = ∪ (𝑓 supp 0), (1r‘𝑅), (0g‘𝑅))) = (𝑖 ∈ 𝐼 ↦ if(𝑖 = ∪ (𝑓 supp 0), (1r‘𝑅), (0g‘𝑅)))
93		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
94	93, 5, 8	ringidcld 20522	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
95	94	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → (1r‘𝑅) ∈ (Base‘𝑅))
96	4, 82, 50, 91, 92, 95	gsummptif1n0 20207	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → (𝑅 Σg (𝑖 ∈ 𝐼 ↦ if(𝑖 = ∪ (𝑓 supp 0), (1r‘𝑅), (0g‘𝑅)))) = (1r‘𝑅))
97	16, 79, 96	3eqtrrd 2803	. . . . 5 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → (1r‘𝑅) = (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓))))
98	97	anasss 472	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ (ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 1)) → (1r‘𝑅) = (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓))))
99	81	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → 𝑅 ∈ Mnd)
100	6	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → 𝐼 ∈ Fin)
101	4	gsumz 19058	. . . . . . . 8 ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ Fin) → (𝑅 Σg (𝑖 ∈ 𝐼 ↦ (0g‘𝑅))) = (0g‘𝑅))
102	99, 100, 101	syl2anc 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → (𝑅 Σg (𝑖 ∈ 𝐼 ↦ (0g‘𝑅))) = (0g‘𝑅))
103	12	an32s 665	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼) → ((𝑉‘𝑖)‘𝑓) = if(𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅)))
104	103	adantlr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) ∧ 𝑖 ∈ 𝐼) → ((𝑉‘𝑖)‘𝑓) = if(𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅)))
105		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) → 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
106	105	rneqd 5926	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) → ran 𝑓 = ran (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
107		nfv 1949	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼)
108	107, 19	nfan 1934	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗(((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
109		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)) = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))
110		1nn0 12646	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℕ0
111		prid2g 4722	. . . . . . . . . . . . . . . . 17 ⊢ (1 ∈ ℕ0 → 1 ∈ {0, 1})
112	110, 111	mp1i 14	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) ∧ 𝑗 ∈ 𝐼) → 1 ∈ {0, 1})
113		0nn0 12645	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℕ0
114		prid1g 4721	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ ℕ0 → 0 ∈ {0, 1})
115	113, 114	mp1i 14	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) ∧ 𝑗 ∈ 𝐼) → 0 ∈ {0, 1})
116	112, 115	ifcld 4532	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) ∧ 𝑗 ∈ 𝐼) → if(𝑗 = 𝑖, 1, 0) ∈ {0, 1})
117	108, 109, 116	rnmptssd 7145	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) → ran (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)) ⊆ {0, 1})
118	106, 117	eqsstrd 3947	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) → ran 𝑓 ⊆ {0, 1})
119	118	adantllr 732	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) → ran 𝑓 ⊆ {0, 1})
120		simpllr 788	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) → ¬ ran 𝑓 ⊆ {0, 1})
121	119, 120	pm2.65da 829	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) ∧ 𝑖 ∈ 𝐼) → ¬ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
122	121	iffalsed 4496	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) ∧ 𝑖 ∈ 𝐼) → if(𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅))
123	104, 122	eqtr2d 2799	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) ∧ 𝑖 ∈ 𝐼) → (0g‘𝑅) = ((𝑉‘𝑖)‘𝑓))
124	123	mpteq2dva 5201	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → (𝑖 ∈ 𝐼 ↦ (0g‘𝑅)) = (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓)))
125	124	oveq2d 7456	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → (𝑅 Σg (𝑖 ∈ 𝐼 ↦ (0g‘𝑅))) = (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓))))
126	102, 125	eqtr3d 2800	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → (0g‘𝑅) = (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓))))
127	126	adantlr 728	. . . . 5 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 1)) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → (0g‘𝑅) = (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓))))
128	81	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) → 𝑅 ∈ Mnd)
129	6	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) → 𝐼 ∈ Fin)
130	128, 129, 101	syl2anc 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) → (𝑅 Σg (𝑖 ∈ 𝐼 ↦ (0g‘𝑅))) = (0g‘𝑅))
131	103	adantlr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) → ((𝑉‘𝑖)‘𝑓) = if(𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅)))
132		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) → 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
133	6, 62	sylan 592	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}) = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
134	133	ad5ant14 770	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) → ((𝟭‘𝐼)‘{𝑖}) = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
135	132, 134	eqtr4d 2801	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) → 𝑓 = ((𝟭‘𝐼)‘{𝑖}))
136	135	oveq1d 7455	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) → (𝑓 supp 0) = (((𝟭‘𝐼)‘{𝑖}) supp 0))
137	129	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) → 𝐼 ∈ Fin)
138	52	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) → {𝑖} ⊆ 𝐼)
139	137, 138, 55	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) → (((𝟭‘𝐼)‘{𝑖}) supp 0) = {𝑖})
140	136, 139	eqtrd 2798	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) → (𝑓 supp 0) = {𝑖})
141	140	fveq2d 6909	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) → (♯‘(𝑓 supp 0)) = (♯‘{𝑖}))
142		hashsng 14537	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ 𝐼 → (♯‘{𝑖}) = 1)
143	142	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) → (♯‘{𝑖}) = 1)
144	141, 143	eqtrd 2798	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) → (♯‘(𝑓 supp 0)) = 1)
145		simpllr 788	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) → ¬ (♯‘(𝑓 supp 0)) = 1)
146	144, 145	pm2.65da 829	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) → ¬ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
147	146	iffalsed 4496	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) → if(𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)), (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅))
148	131, 147	eqtr2d 2799	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) → (0g‘𝑅) = ((𝑉‘𝑖)‘𝑓))
149	148	mpteq2dva 5201	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) → (𝑖 ∈ 𝐼 ↦ (0g‘𝑅)) = (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓)))
150	149	oveq2d 7456	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) → (𝑅 Σg (𝑖 ∈ 𝐼 ↦ (0g‘𝑅))) = (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓))))
151	130, 150	eqtr3d 2800	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) → (0g‘𝑅) = (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓))))
152	151	adantlr 728	. . . . 5 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 1)) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) → (0g‘𝑅) = (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓))))
153		pm3.13 1010	. . . . . 6 ⊢ (¬ (ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 1) → (¬ ran 𝑓 ⊆ {0, 1} ∨ ¬ (♯‘(𝑓 supp 0)) = 1))
154	153	adantl 487	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 1)) → (¬ ran 𝑓 ⊆ {0, 1} ∨ ¬ (♯‘(𝑓 supp 0)) = 1))
155	127, 152, 154	mpjaodan 973	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 1)) → (0g‘𝑅) = (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓))))
156	98, 155	ifeqda 4522	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 1), (1r‘𝑅), (0g‘𝑅)) = (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓))))
157	156	mpteq2dva 5201	. 2 ⊢ (𝜑 → (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 1), (1r‘𝑅), (0g‘𝑅))) = (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓)))))
158		esplyfval1.e	. . . 4 ⊢ 𝐸 = (𝐼eSymPoly𝑅)
159	158	fveq1i 6906	. . 3 ⊢ (𝐸‘1) = ((𝐼eSymPoly𝑅)‘1)
160	110	a1i 11	. . . 4 ⊢ (𝜑 → 1 ∈ ℕ0)
161	2, 6, 8, 160, 4, 5	esplyfval3 34271	. . 3 ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘1) = (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 1), (1r‘𝑅), (0g‘𝑅))))
162	159, 161	eqtrid 2810	. 2 ⊢ (𝜑 → (𝐸‘1) = (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 1), (1r‘𝑅), (0g‘𝑅))))
163		esplyfval1.w	. . 3 ⊢ 𝑊 = (𝐼 mPoly 𝑅)
164		eqid 2763	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
165	163, 1, 164, 6, 8	mvrf2 22327	. . 3 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘𝑊))
166	163, 164, 8, 6, 2, 6, 165	mplgsum 34252	. 2 ⊢ (𝜑 → (𝑊 Σg 𝑉) = (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓)))))
167	157, 162, 166	3eqtr4d 2808	1 ⊢ (𝜑 → (𝐸‘1) = (𝑊 Σg 𝑉))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900  ifcif 4485  {csn 4588  {cpr 4590  ∪ cuni 4870   class class class wbr 5106   ↦ cmpt 5189  dom cdm 5655  ran crn 5656  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440   supp csupp 8201   ↑_m cmap 8872  Fincfn 8998   finSupp cfsupp 9378  0cc0 11224  1c1 11225  𝟭cind 12344  ℕ₀cn0 12630  ♯chash 14498  Basecbs 17413  0_gc0g 17636   Σ_g cgsu 17637  Mndcmnd 18949  1_rcur 20434  Ringcrg 20486   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-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-dju 10006  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-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-rhm 20729  df-subrng 20825  df-subrg 20849  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)