Theorem esplyfval1 33722

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 2737	. . . . . . . . . . . 12 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
3	2	psrbasfsupp 33677	. . . . . . . . . . 11 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
4		eqid 2737	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
5		eqid 2737	. . . . . . . . . . 11 ⊢ (1r‘𝑅) = (1r‘𝑅)
6		esplyfval1.i	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ Fin)
7	6	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝐼 ∈ Fin)
8		esplyfval1.r	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ Ring)
9	8	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑅 ∈ Ring)
10		simplr 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑖 ∈ 𝐼)
11		simpr 484	. . . . . . . . . . 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 21939	. . . . . . . . . 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 753	. . . . . . . . 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 32510	. . . . . . . 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 5179	. . . . . . 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 7374	. . . . . 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 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑗((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼)
18		nfmpt1 5185	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))
19	18	nfeq2 2917	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))
20		nfv 1916	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑖 = ∪ (𝑓 supp 0)
21	19, 20	nfbi 1905	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)) ↔ 𝑖 = ∪ (𝑓 supp 0))
22		unisnv 4871	. . . . . . . . . . . . 13 ⊢ ∪ {𝑗} = 𝑗
23	22	eqeq2i 2750	. . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 4864	. . . . . . . . . . . . 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 720	. . . . . . . . . . . 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 2748	. . . . . . . . . . 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 769	. . . . . . . . . . . . . . 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 6836	. . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) → 𝐼 ∈ Fin)
32	6	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝐼 ∈ Fin)
33		nn0ex 12408	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℕ0 ∈ V
34	33	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ℕ0 ∈ V)
35		ssrab2 4021	. . . . . . . . . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑓 ∈ (ℕ0 ↑m 𝐼))
38	32, 34, 37	elmaprd 32742	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑓:𝐼⟶ℕ0)
39	38	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) → 𝑓:𝐼⟶ℕ0)
40		ffrn 6673	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝐼⟶ℕ0 → 𝑓:𝐼⟶ran 𝑓)
41	39, 40	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) → 𝑓:𝐼⟶ran 𝑓)
42		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) → ran 𝑓 ⊆ {0, 1})
43	41, 42	fssd 6677	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) → 𝑓:𝐼⟶{0, 1})
44	31, 43	indfsid 32934	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) → 𝑓 = ((𝟭‘𝐼)‘(𝑓 supp 0)))
45	44	ad5antr 735	. . . . . . . . . . . . . 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 4578	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑗 → {𝑖} = {𝑗})
47	46	adantl 481	. . . . . . . . . . . . . . 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 6836	. . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 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 7373	. . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → 𝐼 ∈ Fin)
54	53	ad4antr 733	. . . . . . . . . . . . . . . 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 4752	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ 𝐼 → {𝑖} ⊆ 𝐼)
56	55	adantl 481	. . . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . 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 32932	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ Fin ∧ {𝑖} ⊆ 𝐼) → (((𝟭‘𝐼)‘{𝑖}) supp 0) = {𝑖})
59	54, 57, 58	syl2anc 585	. . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . 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 3434	. . . . . . . . . . . . . . 15 ⊢ 𝑖 ∈ V
62	61	sneqr 4784	. . . . . . . . . . . . . 14 ⊢ ({𝑖} = {𝑗} → 𝑖 = 𝑗)
63	60, 62	syl 17	. . . . . . . . . . . . 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 801	. . . . . . . . . . . 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 32928	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ Fin ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}) = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
66	53, 65	sylan 581	. . . . . . . . . . . . . 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 727	. . . . . . . . . . . . 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 2748	. . . . . . . . . . . 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 280	. . . . . . . . . . 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 312	. . . . . . . . . 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 7393	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → (𝑓 supp 0) ∈ V)
72		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → (♯‘(𝑓 supp 0)) = 1)
73		hash1snb 14343	. . . . . . . . . . . . . 14 ⊢ ((𝑓 supp 0) ∈ V → ((♯‘(𝑓 supp 0)) = 1 ↔ ∃𝑗(𝑓 supp 0) = {𝑗}))
74	73	biimpa 476	. . . . . . . . . . . . 13 ⊢ (((𝑓 supp 0) ∈ V ∧ (♯‘(𝑓 supp 0)) = 1) → ∃𝑗(𝑓 supp 0) = {𝑗})
75	71, 72, 74	syl2anc 585	. . . . . . . . . . . 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 4625	. . . . . . . . . . . 12 ⊢ (∃𝑗(𝑓 supp 0) = {𝑗} ↔ ∃𝑗 ∈ (𝑓 supp 0)(𝑓 supp 0) = {𝑗})
77	75, 76	sylib 218	. . . . . . . . . . 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 480	. . . . . . . . . 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 3246	. . . . . . . . 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 4491	. . . . . . . 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 5179	. . . . . . 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 7374	. . . . . 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 20182	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
84	8, 83	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Mnd)
85	84	ad3antrrr 731	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → 𝑅 ∈ Mnd)
86		suppssdm 8118	. . . . . . . . . . . 12 ⊢ (𝑓 supp 0) ⊆ dom 𝑓
87	38	fdmd 6670	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → dom 𝑓 = 𝐼)
88	87	ad4antr 733	. . . . . . . . . . . 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 3965	. . . . . . . . . . 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 769	. . . . . . . . . . 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 3923	. . . . . . . . . 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 2841	. . . . . . . . 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 2837	. . . . . . . 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 3146	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ran 𝑓 ⊆ {0, 1}) ∧ (♯‘(𝑓 supp 0)) = 1) → ∪ (𝑓 supp 0) ∈ 𝐼)
95		eqid 2737	. . . . . . 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 2737	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
97	96, 5, 8	ringidcld 20205	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
98	97	ad3antrrr 731	. . . . . . 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 19899	. . . . . 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 2777	. . . . 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 466	. . . 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 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → 𝑅 ∈ Mnd)
103	6	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) → 𝐼 ∈ Fin)
104	4	gsumz 18762	. . . . . . . 8 ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ Fin) → (𝑅 Σg (𝑖 ∈ 𝐼 ↦ (0g‘𝑅))) = (0g‘𝑅))
105	102, 103, 104	syl2anc 585	. . . . . . 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 653	. . . . . . . . . . 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 716	. . . . . . . . . 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 484	. . . . . . . . . . . . . . 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 5885	. . . . . . . . . . . . . 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 1916	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼)
111	110, 19	nfan 1901	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗(((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼) ∧ 𝑓 = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
112		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)) = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0))
113		1nn0 12418	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℕ0
114		prid2g 4706	. . . . . . . . . . . . . . . . 17 ⊢ (1 ∈ ℕ0 → 1 ∈ {0, 1})
115	113, 114	mp1i 13	. . . . . . . . . . . . . . . 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 12417	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℕ0
117		prid1g 4705	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ ℕ0 → 0 ∈ {0, 1})
118	116, 117	mp1i 13	. . . . . . . . . . . . . . . 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 4514	. . . . . . . . . . . . . . 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 7068	. . . . . . . . . . . . . 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 3957	. . . . . . . . . . . . 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 720	. . . . . . . . . . . 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 776	. . . . . . . . . . . 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 817	. . . . . . . . . . 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 4478	. . . . . . . . . 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 2773	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ ran 𝑓 ⊆ {0, 1}) ∧ 𝑖 ∈ 𝐼) → (0g‘𝑅) = ((𝑉‘𝑖)‘𝑓))
127	126	mpteq2dva 5179	. . . . . . . 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 7374	. . . . . . 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 2774	. . . . . 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 716	. . . . 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 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) → 𝑅 ∈ Mnd)
132	6	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) → 𝐼 ∈ Fin)
133	131, 132, 104	syl2anc 585	. . . . . . 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 716	. . . . . . . . . 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 484	. . . . . . . . . . . . . . . . 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 581	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝟭‘𝐼)‘{𝑖}) = (𝑗 ∈ 𝐼 ↦ if(𝑗 = 𝑖, 1, 0)))
137	136	ad5ant14 758	. . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . . 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 7373	. . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . 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 6836	. . . . . . . . . . . . 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 14293	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ 𝐼 → (♯‘{𝑖}) = 1)
146	145	ad2antlr 728	. . . . . . . . . . . . 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 2772	. . . . . . . . . . . 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 776	. . . . . . . . . . . 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 817	. . . . . . . . . . 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 4478	. . . . . . . . . 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 2773	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ ¬ (♯‘(𝑓 supp 0)) = 1) ∧ 𝑖 ∈ 𝐼) → (0g‘𝑅) = ((𝑉‘𝑖)‘𝑓))
152	151	mpteq2dva 5179	. . . . . . . 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 7374	. . . . . . 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 2774	. . . . . 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 716	. . . . 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		pm3.13 997	. . . . . 6 ⊢ (¬ (ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 1) → (¬ ran 𝑓 ⊆ {0, 1} ∨ ¬ (♯‘(𝑓 supp 0)) = 1))
157	156	adantl 481	. . . . 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 961	. . . 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 4504	. . 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 5179	. 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 6833	. . 3 ⊢ (𝐸‘1) = ((𝐼eSymPoly𝑅)‘1)
163	113	a1i 11	. . . 4 ⊢ (𝜑 → 1 ∈ ℕ0)
164	2, 6, 8, 163, 4, 5	esplyfval3 33721	. . 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 2784	. 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 2737	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
168	166, 1, 167, 6, 8	mvrf2 21949	. . 3 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘𝑊))
169	166, 167, 8, 6, 2, 6, 168	mplgsum 33702	. 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 2782	1 ⊢ (𝜑 → (𝐸‘1) = (𝑊 Σg 𝑉))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3062  {crab 3390  Vcvv 3430   ⊆ wss 3890  ifcif 4467  {csn 4568  {cpr 4570  ∪ cuni 4851   class class class wbr 5086   ↦ cmpt 5167  dom cdm 5622  ran crn 5623  ⟶wf 6486  ‘cfv 6490  (class class class)co 7358   supp csupp 8101   ↑_m cmap 8764  Fincfn 8884   finSupp cfsupp 9265  0cc0 11027  1c1 11028  ℕ₀cn0 12402  ♯chash 14254  Basecbs 17137  0_gc0g 17360   Σ_g cgsu 17361  Mndcmnd 18660  1_rcur 20120  Ringcrg 20172   mVar cmvr 21862   mPoly cmpl 21863  𝟭cind 32912  eSymPolycesply 33705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104  ax-addf 11106  ax-mulf 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-ofr 7623  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8102  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-2o 8397  df-oadd 8400  df-er 8634  df-map 8766  df-pm 8767  df-ixp 8837  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-fsupp 9266  df-sup 9346  df-oi 9416  df-dju 9814  df-card 9852  df-pnf 11169  df-mnf 11170  df-xr 11171  df-ltxr 11172  df-le 11173  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-3 12210  df-4 12211  df-5 12212  df-6 12213  df-7 12214  df-8 12215  df-9 12216  df-n0 12403  df-xnn0 12476  df-z 12490  df-dec 12609  df-uz 12753  df-fz 13425  df-fzo 13572  df-seq 13926  df-hash 14255  df-struct 17075  df-sets 17092  df-slot 17110  df-ndx 17122  df-base 17138  df-ress 17159  df-plusg 17191  df-mulr 17192  df-starv 17193  df-sca 17194  df-vsca 17195  df-ip 17196  df-tset 17197  df-ple 17198  df-ds 17200  df-unif 17201  df-hom 17202  df-cco 17203  df-0g 17362  df-gsum 17363  df-prds 17368  df-pws 17370  df-mre 17506  df-mrc 17507  df-acs 17509  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-mhm 18709  df-submnd 18710  df-grp 18870  df-minusg 18871  df-mulg 19002  df-subg 19057  df-ghm 19146  df-cntz 19250  df-cmn 19715  df-abl 19716  df-mgp 20080  df-rng 20092  df-ur 20121  df-ring 20174  df-cring 20175  df-rhm 20410  df-subrng 20481  df-subrg 20505  df-cnfld 21312  df-zring 21404  df-zrh 21460  df-psr 21866  df-mvr 21867  df-mpl 21868  df-ind 32913  df-esply 33707

This theorem is referenced by: (None)