Theorem esplyfval1 33929

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 2761	. . . . . . . . . . . 12 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
3	2	psrbasfsupp 33867	. . . . . . . . . . 11 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
4		eqid 2761	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
5		eqid 2761	. . . . . . . . . . 11 ⊢ (1r‘𝑅) = (1r‘𝑅)
6		esplyfval1.i	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ Fin)
7	6	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝐼 ∈ Fin)
8		esplyfval1.r	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ Ring)
9	8	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑅 ∈ Ring)
10		simplr 780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑖 ∈ 𝐼)
11		simpr 489	. . . . . . . . . . 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 22111	. . . . . . . . . 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 764	. . . . . . . . 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 32768	. . . . . . . 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 5203	. . . . . . 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 7426	. . . . . 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 1942	. . . . . . . . . 10 ⊢ Ⅎ𝑗((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼)
18		nfmpt1 5209	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))
19	18	nfeq2 2940	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))
20		nfv 1942	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑖 = ∪ (𝑓 supp 0)
21	19, 20	nfbi 1931	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)) ↔ 𝑖 = ∪ (𝑓 supp 0))
22		unisnv 4891	. . . . . . . . . . . . 13 ⊢ ∪ {𝑗} = 𝑗
23	22	eqeq2i 2774	. . . . . . . . . . . 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 489	. . . . . . . . . . . . . 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 4884	. . . . . . . . . . . . 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 731	. . . . . . . . . . . 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 2772	. . . . . . . . . . 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 780	. . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) → 𝐼 ∈ Fin)
32	6	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝐼 ∈ Fin)
33		nn0ex 12509	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℕ0 ∈ V
34	33	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ℕ0 ∈ V)
35		ssrab2 4033	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐼)
36	35	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐼))
37	36	sselda 3936	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑓 ∈ (ℕ0 ↑m 𝐼))
38	32, 34, 37	elmaprd 32991	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑓:𝐼⟶ℕ0)
39	38	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) → 𝑓:𝐼⟶ℕ0)
40		ffrn 6719	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝐼⟶ℕ0 → 𝑓:𝐼⟶ran 𝑓)
41	39, 40	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) → 𝑓:𝐼⟶ran 𝑓)
42		simpr 489	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) → ran 𝑓 ⊆ {0, 1})
43	41, 42	fssd 6723	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) → 𝑓:𝐼⟶{0, 1})
44	31, 43	indfsid 33155	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) → 𝑓 = ((𝟭‘𝐼)‘(𝑓 supp 0)))
45	44	ad5antr 746	. . . . . . . . . . . . . 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)))
46		sneq 4598	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑗 → {𝑖} = {𝑗})
47	46	adantl 486	. . . . . . . . . . . . . . 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})
48	47	fveq2d 6885	. . . . . . . . . . . . . 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}))
49	30, 45, 48	3eqtr4d 2806	. . . . . . . . . . . . 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}))
50		simpr 489	. . . . . . . . . . . . . . . 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}))
51	50	oveq1d 7425	. . . . . . . . . . . . . . 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))
52		simplr 780	. . . . . . . . . . . . . . 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})
53	6	ad3antrrr 742	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → 𝐼 ∈ Fin)
54	53	ad4antr 744	. . . . . . . . . . . . . . . 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)
55		snssi 4750	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ 𝐼 → {𝑖} ⊆ 𝐼)
56	55	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) → {𝑖} ⊆ 𝐼)
57	56	ad3antrrr 742	. . . . . . . . . . . . . . . 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)
58		indsupp 33153	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ Fin ∧ {𝑖} ⊆ 𝐼) → (((𝟭‘𝐼)‘{𝑖}) supp 0) = {𝑖})
59	54, 57, 58	syl2anc 595	. . . . . . . . . . . . . . 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})
60	51, 52, 59	3eqtr3rd 2805	. . . . . . . . . . . . . 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})
61		vex 3457	. . . . . . . . . . . . . . 15 ⊢ 𝑖 ∈ V
62	61	sneqr 4804	. . . . . . . . . . . . . 14 ⊢ ({𝑖} = {𝑗} → 𝑖 = 𝑗)
63	60, 62	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)
64	49, 63	impbida 812	. . . . . . . . . . . 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})))
65		indsn 33149	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ Fin ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}) = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
66	53, 65	sylan 591	. . . . . . . . . . . . . 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)))
67	66	ad2antrr 738	. . . . . . . . . . . . 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)))
68	67	eqeq2d 2772	. . . . . . . . . . . 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))))
69	64, 68	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))
70	24, 28, 69	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)))
71		ovexd 7445	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → (𝑓 supp 0) ∈ V)
72		simpr 489	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → (♯‘(𝑓 supp 0)) = 1)
73		hash1snb 14455	. . . . . . . . . . . . . 14 ⊢ ((𝑓 supp 0) ∈ V → ((♯‘(𝑓 supp 0)) = 1 ↔ ∃𝑗(𝑓 supp 0) = {𝑗}))
74	73	biimpa 481	. . . . . . . . . . . . 13 ⊢ (((𝑓 supp 0) ∈ V ∧ (♯‘(𝑓 supp 0)) = 1) → ∃𝑗(𝑓 supp 0) = {𝑗})
75	71, 72, 74	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → ∃𝑗(𝑓 supp 0) = {𝑗})
76		exsnrex 4645	. . . . . . . . . . . 12 ⊢ (∃𝑗(𝑓 supp 0) = {𝑗} ↔ ∃𝑗 ∈ (𝑓 supp 0)(𝑓 supp 0) = {𝑗})
77	75, 76	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})
78	77	adantr 485	. . . . . . . . . 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})
79	17, 21, 70, 78	r19.29af2 3271	. . . . . . . . 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)))
80	79	ifbid 4510	. . . . . . . 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)))
81	80	mpteq2dva 5203	. . . . . . 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))))
82	81	oveq2d 7426	. . . . . 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)))))
83		ringmnd 20324	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
84	8, 83	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Mnd)
85	84	ad3antrrr 742	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → 𝑅 ∈ Mnd)
86		suppssdm 8172	. . . . . . . . . . . 12 ⊢ (𝑓 supp 0) ⊆ dom 𝑓
87	38	fdmd 6716	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → dom 𝑓 = 𝐼)
88	87	ad4antr 744	. . . . . . . . . . . 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)
89	86, 88	sseqtrid 3978	. . . . . . . . . . 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)
90		simplr 780	. . . . . . . . . . 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))
91	89, 90	sseldd 3937	. . . . . . . . . 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)
92	22, 91	eqeltrid 2865	. . . . . . . . 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)
93	26, 92	eqeltrd 2861	. . . . . . . 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)
94	93, 77	r19.29a 3171	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → ∪ (𝑓 supp 0) ∈ 𝐼)
95		eqid 2761	. . . . . . 7 ⊢ (𝑖 ∈ 𝐼 ↦ if(𝑖 = ∪ (𝑓 supp 0), (1r‘𝑅), (0g‘𝑅))) = (𝑖 ∈ 𝐼 ↦ if(𝑖 = ∪ (𝑓 supp 0), (1r‘𝑅), (0g‘𝑅)))
96		eqid 2761	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
97	96, 5, 8	ringidcld 20348	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
98	97	ad3antrrr 742	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → (1r‘𝑅) ∈ (Base‘𝑅))
99	4, 85, 53, 94, 95, 98	gsummptif1n0 20035	. . . . . 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))
100	16, 82, 99	3eqtrrd 2801	. . . . 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))))
101	100	anasss 471	. . . 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))))
102	84	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → 𝑅 ∈ Mnd)
103	6	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → 𝐼 ∈ Fin)
104	4	gsumz 18894	. . . . . . . 8 ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ Fin) → (𝑅 Σg (𝑖 ∈ 𝐼 ↦ (0g‘𝑅))) = (0g‘𝑅))
105	102, 103, 104	syl2anc 595	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → (𝑅 Σg (𝑖 ∈ 𝐼 ↦ (0g‘𝑅))) = (0g‘𝑅))
106	12	an32s 664	. . . . . . . . . . 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)))
107	106	adantlr 727	. . . . . . . . . 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)))
108		simpr 489	. . . . . . . . . . . . . . 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)))
109	108	rneqd 5928	. . . . . . . . . . . . . 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)))
110		nfv 1942	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼)
111	110, 19	nfan 1927	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗(((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
112		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)) = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))
113		1nn0 12519	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℕ0
114		prid2g 4726	. . . . . . . . . . . . . . . . 17 ⊢ (1 ∈ ℕ0 → 1 ∈ {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) → 1 ∈ {0, 1})
116		0nn0 12518	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℕ0
117		prid1g 4725	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ ℕ0 → 0 ∈ {0, 1})
118	116, 117	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})
119	115, 118	ifcld 4533	. . . . . . . . . . . . . . 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})
120	111, 112, 119	rnmptssd 7119	. . . . . . . . . . . . . 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})
121	109, 120	eqsstrd 3970	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))) → ran 𝑓 ⊆ {0, 1})
122	121	adantllr 731	. . . . . . . . . . . 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})
123		simpllr 787	. . . . . . . . . . . 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})
124	122, 123	pm2.65da 828	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) ∧ 𝑖 ∈ 𝐼) → ¬ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
125	124	iffalsed 4497	. . . . . . . . . 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))
126	107, 125	eqtr2d 2797	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) ∧ 𝑖 ∈ 𝐼) → (0g‘𝑅) = ((𝑉‘𝑖)‘𝑓))
127	126	mpteq2dva 5203	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → (𝑖 ∈ 𝐼 ↦ (0g‘𝑅)) = (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓)))
128	127	oveq2d 7426	. . . . . . 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))))
129	105, 128	eqtr3d 2798	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → (0g‘𝑅) = (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓))))
130	129	adantlr 727	. . . . 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))))
131	84	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) → 𝑅 ∈ Mnd)
132	6	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) → 𝐼 ∈ Fin)
133	131, 132, 104	syl2anc 595	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) → (𝑅 Σg (𝑖 ∈ 𝐼 ↦ (0g‘𝑅))) = (0g‘𝑅))
134	106	adantlr 727	. . . . . . . . . 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)))
135		simpr 489	. . . . . . . . . . . . . . . . 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)))
136	6, 65	sylan 591	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}) = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
137	136	ad5ant14 769	. . . . . . . . . . . . . . . . 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)))
138	135, 137	eqtr4d 2799	. . . . . . . . . . . . . . . 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}))
139	138	oveq1d 7425	. . . . . . . . . . . . . . 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))
140	132	ad2antrr 738	. . . . . . . . . . . . . . . 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)
141	55	ad2antlr 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} ⊆ 𝐼)
142	140, 141, 58	syl2anc 595	. . . . . . . . . . . . . . 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})
143	139, 142	eqtrd 2796	. . . . . . . . . . . . . 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})
144	143	fveq2d 6885	. . . . . . . . . . . . 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}))
145		hashsng 14404	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ 𝐼 → (♯‘{𝑖}) = 1)
146	145	ad2antlr 739	. . . . . . . . . . . . 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)
147	144, 146	eqtrd 2796	. . . . . . . . . . . 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)
148		simpllr 787	. . . . . . . . . . . 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)
149	147, 148	pm2.65da 828	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) → ¬ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
150	149	iffalsed 4497	. . . . . . . . . 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))
151	134, 150	eqtr2d 2797	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) → (0g‘𝑅) = ((𝑉‘𝑖)‘𝑓))
152	151	mpteq2dva 5203	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) → (𝑖 ∈ 𝐼 ↦ (0g‘𝑅)) = (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓)))
153	152	oveq2d 7426	. . . . . . 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))))
154	133, 153	eqtr3d 2798	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) → (0g‘𝑅) = (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓))))
155	154	adantlr 727	. . . . 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))))
156		pm3.13 1010	. . . . . 6 ⊢ (¬ (ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 1) → (¬ ran 𝑓 ⊆ {0, 1} ∨ ¬ (♯‘(𝑓 supp 0)) = 1))
157	156	adantl 486	. . . . 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))
158	130, 155, 157	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))))
159	101, 158	ifeqda 4523	. . 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))))
160	159	mpteq2dva 5203	. 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)))))
161		esplyfval1.e	. . . 4 ⊢ 𝐸 = (𝐼eSymPoly𝑅)
162	161	fveq1i 6882	. . 3 ⊢ (𝐸‘1) = ((𝐼eSymPoly𝑅)‘1)
163	113	a1i 11	. . . 4 ⊢ (𝜑 → 1 ∈ ℕ0)
164	2, 6, 8, 163, 4, 5	esplyfval3 33928	. . 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))))
165	162, 164	eqtrid 2808	. 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))))
166		esplyfval1.w	. . 3 ⊢ 𝑊 = (𝐼 mPoly 𝑅)
167		eqid 2761	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
168	166, 1, 167, 6, 8	mvrf2 22121	. . 3 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘𝑊))
169	166, 167, 8, 6, 2, 6, 168	mplgsum 33909	. 2 ⊢ (𝜑 → (𝑊 Σg 𝑉) = (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑉‘𝑖)‘𝑓)))))
170	160, 165, 169	3eqtr4d 2806	1 ⊢ (𝜑 → (𝐸‘1) = (𝑊 Σg 𝑉))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1568  ∃wex 1807   ∈ wcel 2141  ∃wrex 3087  {crab 3414  Vcvv 3453   ⊆ wss 3904  ifcif 4486  {csn 4588  {cpr 4590  ∪ cuni 4871   class class class wbr 5108   ↦ cmpt 5191  dom cdm 5661  ran crn 5662  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   supp csupp 8155   ↑_m cmap 8823  Fincfn 8942   finSupp cfsupp 9320  0cc0 11099  1c1 11100  𝟭cind 12217  ℕ₀cn0 12503  ♯chash 14365  Basecbs 17268  0_gc0g 17491   Σ_g cgsu 17492  Mndcmnd 18791  1_rcur 20262  Ringcrg 20314   mVar cmvr 22034   mPoly cmpl 22035  eSymPolycesply 33912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-addf 11178  ax-mulf 11179

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-ofr 7675  df-om 7862  df-1st 7985  df-2nd 7986  df-supp 8156  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-oadd 8456  df-er 8693  df-map 8825  df-pm 8826  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fsupp 9321  df-sup 9401  df-oi 9471  df-dju 9886  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-ind 12218  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-9 12309  df-n0 12504  df-xnn0 12577  df-z 12591  df-dec 12711  df-uz 12862  df-fz 13535  df-fzo 13682  df-seq 14037  df-hash 14366  df-struct 17206  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-ress 17290  df-plusg 17322  df-mulr 17323  df-starv 17324  df-sca 17325  df-vsca 17326  df-ip 17327  df-tset 17328  df-ple 17329  df-ds 17331  df-unif 17332  df-hom 17333  df-cco 17334  df-0g 17493  df-gsum 17494  df-prds 17499  df-pws 17501  df-mre 17637  df-mrc 17638  df-acs 17640  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-mhm 18840  df-submnd 18841  df-grp 19002  df-minusg 19003  df-mulg 19133  df-subg 19188  df-ghm 19283  df-cntz 19386  df-cmn 19851  df-abl 19852  df-mgp 20216  df-rng 20230  df-ur 20263  df-ring 20316  df-cring 20317  df-rhm 20553  df-subrng 20630  df-subrg 20654  df-cnfld 21502  df-zring 21576  df-zrh 21632  df-psr 22038  df-mvr 22039  df-mpl 22040  df-esply 33914

This theorem is referenced by: (None)