tdeglem4 - Metamath Proof Explorer

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Theorem tdeglem4 23820

Description: There is only one multi-index with total degree 0. (Contributed by Stefan O'Rear, 29-Mar-2015.)

**Hypotheses**
Ref	Expression
tdeglem.a	⊢ 𝐴 = {𝑚 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑚 “ ℕ) ∈ Fin}
tdeglem.h	⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂ_fld Σ_g ℎ))

**Assertion**
Ref	Expression
tdeglem4	⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ((𝐻‘𝑋) = 0 ↔ 𝑋 = (𝐼 × {0})))

Distinct variable groups: 𝐴,ℎ ℎ,𝐼,𝑚 ℎ,𝑉 ℎ,𝑋,𝑚 Allowed substitution hints: 𝐴(𝑚) 𝐻(ℎ,𝑚) 𝑉(𝑚)

Proof of Theorem tdeglem4 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexnal 2995	. . . . 5 ⊢ (∃𝑥 ∈ 𝐼 ¬ (𝑋‘𝑥) = 0 ↔ ¬ ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = 0)
2		df-ne 2795	. . . . . . 7 ⊢ ((𝑋‘𝑥) ≠ 0 ↔ ¬ (𝑋‘𝑥) = 0)
3		oveq2 6658	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑋 → (ℂ_fld Σ_g ℎ) = (ℂ_fld Σ_g 𝑋))
4		tdeglem.h	. . . . . . . . . . . 12 ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂ_fld Σ_g ℎ))
5		ovex 6678	. . . . . . . . . . . 12 ⊢ (ℂ_fld Σ_g 𝑋) ∈ V
6	3, 4, 5	fvmpt 6282	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝐴 → (𝐻‘𝑋) = (ℂ_fld Σ_g 𝑋))
7	6	ad2antlr 763	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝐻‘𝑋) = (ℂ_fld Σ_g 𝑋))
8		tdeglem.a	. . . . . . . . . . . . . 14 ⊢ 𝐴 = {𝑚 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑚 “ ℕ) ∈ Fin}
9	8	psrbagf 19365	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑋:𝐼⟶ℕ₀)
10	9	feqmptd 6249	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑋 = (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)))
11	10	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → 𝑋 = (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)))
12	11	oveq2d 6666	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (ℂ_fld Σ_g 𝑋) = (ℂ_fld Σ_g (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦))))
13		cnfldbas 19750	. . . . . . . . . . 11 ⊢ ℂ = (Base‘ℂ_fld)
14		cnfld0 19770	. . . . . . . . . . 11 ⊢ 0 = (0_g‘ℂ_fld)
15		cnfldadd 19751	. . . . . . . . . . 11 ⊢ + = (+_g‘ℂ_fld)
16		cnring 19768	. . . . . . . . . . . 12 ⊢ ℂ_fld ∈ Ring
17		ringcmn 18581	. . . . . . . . . . . 12 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ CMnd)
18	16, 17	mp1i 13	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ℂ_fld ∈ CMnd)
19		simpll 790	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → 𝐼 ∈ 𝑉)
20	9	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → 𝑋:𝐼⟶ℕ₀)
21	20	ffvelrnda 6359	. . . . . . . . . . . 12 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) ∧ 𝑦 ∈ 𝐼) → (𝑋‘𝑦) ∈ ℕ₀)
22	21	nn0cnd 11353	. . . . . . . . . . 11 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) ∧ 𝑦 ∈ 𝐼) → (𝑋‘𝑦) ∈ ℂ)
23	8	psrbagfsupp 19509	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐼 ∈ 𝑉) → 𝑋 finSupp 0)
24	23	ancoms 469	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑋 finSupp 0)
25	24	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → 𝑋 finSupp 0)
26	11, 25	eqbrtrrd 4677	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) finSupp 0)
27		incom 3805	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∖ {𝑥}) ∩ {𝑥}) = ({𝑥} ∩ (𝐼 ∖ {𝑥}))
28		disjdif 4040	. . . . . . . . . . . . 13 ⊢ ({𝑥} ∩ (𝐼 ∖ {𝑥})) = ∅
29	27, 28	eqtri 2644	. . . . . . . . . . . 12 ⊢ ((𝐼 ∖ {𝑥}) ∩ {𝑥}) = ∅
30	29	a1i 11	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ((𝐼 ∖ {𝑥}) ∩ {𝑥}) = ∅)
31		difsnid 4341	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐼 → ((𝐼 ∖ {𝑥}) ∪ {𝑥}) = 𝐼)
32	31	eqcomd 2628	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐼 → 𝐼 = ((𝐼 ∖ {𝑥}) ∪ {𝑥}))
33	32	ad2antrl 764	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → 𝐼 = ((𝐼 ∖ {𝑥}) ∪ {𝑥}))
34	13, 14, 15, 18, 19, 22, 26, 30, 33	gsumsplit2 18329	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (ℂ_fld Σ_g (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦))) = ((ℂ_fld Σ_g (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))) + (ℂ_fld Σ_g (𝑦 ∈ {𝑥} ↦ (𝑋‘𝑦)))))
35	7, 12, 34	3eqtrd 2660	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝐻‘𝑋) = ((ℂ_fld Σ_g (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))) + (ℂ_fld Σ_g (𝑦 ∈ {𝑥} ↦ (𝑋‘𝑦)))))
36		difexg 4808	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ 𝑉 → (𝐼 ∖ {𝑥}) ∈ V)
37	36	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝐼 ∖ {𝑥}) ∈ V)
38		nn0subm 19801	. . . . . . . . . . . . 13 ⊢ ℕ₀ ∈ (SubMnd‘ℂ_fld)
39	38	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ℕ₀ ∈ (SubMnd‘ℂ_fld))
40		eldifi 3732	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝐼 ∖ {𝑥}) → 𝑦 ∈ 𝐼)
41		ffvelrn 6357	. . . . . . . . . . . . . 14 ⊢ ((𝑋:𝐼⟶ℕ₀ ∧ 𝑦 ∈ 𝐼) → (𝑋‘𝑦) ∈ ℕ₀)
42	20, 40, 41	syl2an 494	. . . . . . . . . . . . 13 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) ∧ 𝑦 ∈ (𝐼 ∖ {𝑥})) → (𝑋‘𝑦) ∈ ℕ₀)
43		eqid 2622	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) = (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))
44	42, 43	fmptd 6385	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)):(𝐼 ∖ {𝑥})⟶ℕ₀)
45		mptexg 6484	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∖ {𝑥}) ∈ V → (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) ∈ V)
46	36, 45	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ 𝑉 → (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) ∈ V)
47	46	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) ∈ V)
48		funmpt 5926	. . . . . . . . . . . . . 14 ⊢ Fun (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))
49	48	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → Fun (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)))
50		funmpt 5926	. . . . . . . . . . . . . . 15 ⊢ Fun (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦))
51	50	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → Fun (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)))
52		difss 3737	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ∖ {𝑥}) ⊆ 𝐼
53		resmpt 5449	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ∖ {𝑥}) ⊆ 𝐼 → ((𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) ↾ (𝐼 ∖ {𝑥})) = (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)))
54	52, 53	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) ↾ (𝐼 ∖ {𝑥})) = (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))
55		resss 5422	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) ↾ (𝐼 ∖ {𝑥})) ⊆ (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦))
56	54, 55	eqsstr3i 3636	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) ⊆ (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦))
57	56	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) ⊆ (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)))
58		mptexg 6484	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ 𝑉 → (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) ∈ V)
59	58	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) ∈ V)
60		funsssuppss 7321	. . . . . . . . . . . . . 14 ⊢ ((Fun (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) ∧ (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) ⊆ (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) ∧ (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) ∈ V) → ((𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) supp 0) ⊆ ((𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) supp 0))
61	51, 57, 59, 60	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ((𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) supp 0) ⊆ ((𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) supp 0))
62		fsuppsssupp 8291	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) ∈ V ∧ Fun (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))) ∧ ((𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) finSupp 0 ∧ ((𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) supp 0) ⊆ ((𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) supp 0))) → (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) finSupp 0)
63	47, 49, 26, 61, 62	syl22anc 1327	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) finSupp 0)
64	14, 18, 37, 39, 44, 63	gsumsubmcl 18319	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (ℂ_fld Σ_g (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))) ∈ ℕ₀)
65		ringmnd 18556	. . . . . . . . . . . . . 14 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ Mnd)
66	16, 65	mp1i 13	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ℂ_fld ∈ Mnd)
67		simprl 794	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → 𝑥 ∈ 𝐼)
68	20, 67	ffvelrnd 6360	. . . . . . . . . . . . . 14 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑋‘𝑥) ∈ ℕ₀)
69	68	nn0cnd 11353	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑋‘𝑥) ∈ ℂ)
70		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑋‘𝑦) = (𝑋‘𝑥))
71	13, 70	gsumsn 18354	. . . . . . . . . . . . 13 ⊢ ((ℂ_fld ∈ Mnd ∧ 𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ∈ ℂ) → (ℂ_fld Σ_g (𝑦 ∈ {𝑥} ↦ (𝑋‘𝑦))) = (𝑋‘𝑥))
72	66, 67, 69, 71	syl3anc 1326	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (ℂ_fld Σ_g (𝑦 ∈ {𝑥} ↦ (𝑋‘𝑦))) = (𝑋‘𝑥))
73		simprr 796	. . . . . . . . . . . . . 14 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑋‘𝑥) ≠ 0)
74	73, 2	sylib 208	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ¬ (𝑋‘𝑥) = 0)
75		elnn0 11294	. . . . . . . . . . . . . 14 ⊢ ((𝑋‘𝑥) ∈ ℕ₀ ↔ ((𝑋‘𝑥) ∈ ℕ ∨ (𝑋‘𝑥) = 0))
76	68, 75	sylib 208	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ((𝑋‘𝑥) ∈ ℕ ∨ (𝑋‘𝑥) = 0))
77		orel2 398	. . . . . . . . . . . . 13 ⊢ (¬ (𝑋‘𝑥) = 0 → (((𝑋‘𝑥) ∈ ℕ ∨ (𝑋‘𝑥) = 0) → (𝑋‘𝑥) ∈ ℕ))
78	74, 76, 77	sylc 65	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑋‘𝑥) ∈ ℕ)
79	72, 78	eqeltrd 2701	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (ℂ_fld Σ_g (𝑦 ∈ {𝑥} ↦ (𝑋‘𝑦))) ∈ ℕ)
80		nn0nnaddcl 11324	. . . . . . . . . . 11 ⊢ (((ℂ_fld Σ_g (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))) ∈ ℕ₀ ∧ (ℂ_fld Σ_g (𝑦 ∈ {𝑥} ↦ (𝑋‘𝑦))) ∈ ℕ) → ((ℂ_fld Σ_g (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))) + (ℂ_fld Σ_g (𝑦 ∈ {𝑥} ↦ (𝑋‘𝑦)))) ∈ ℕ)
81	64, 79, 80	syl2anc 693	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ((ℂ_fld Σ_g (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))) + (ℂ_fld Σ_g (𝑦 ∈ {𝑥} ↦ (𝑋‘𝑦)))) ∈ ℕ)
82	81	nnne0d 11065	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ((ℂ_fld Σ_g (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))) + (ℂ_fld Σ_g (𝑦 ∈ {𝑥} ↦ (𝑋‘𝑦)))) ≠ 0)
83	35, 82	eqnetrd 2861	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝐻‘𝑋) ≠ 0)
84	83	expr 643	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ 𝐼) → ((𝑋‘𝑥) ≠ 0 → (𝐻‘𝑋) ≠ 0))
85	2, 84	syl5bir 233	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ 𝐼) → (¬ (𝑋‘𝑥) = 0 → (𝐻‘𝑋) ≠ 0))
86	85	rexlimdva 3031	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (∃𝑥 ∈ 𝐼 ¬ (𝑋‘𝑥) = 0 → (𝐻‘𝑋) ≠ 0))
87	1, 86	syl5bir 233	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (¬ ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = 0 → (𝐻‘𝑋) ≠ 0))
88	87	necon4bd 2814	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ((𝐻‘𝑋) = 0 → ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = 0))
89		ffn 6045	. . . . . 6 ⊢ (𝑋:𝐼⟶ℕ₀ → 𝑋 Fn 𝐼)
90	9, 89	syl 17	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑋 Fn 𝐼)
91		0nn0 11307	. . . . . 6 ⊢ 0 ∈ ℕ₀
92		fnconstg 6093	. . . . . 6 ⊢ (0 ∈ ℕ₀ → (𝐼 × {0}) Fn 𝐼)
93	91, 92	mp1i 13	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝐼 × {0}) Fn 𝐼)
94		eqfnfv 6311	. . . . 5 ⊢ ((𝑋 Fn 𝐼 ∧ (𝐼 × {0}) Fn 𝐼) → (𝑋 = (𝐼 × {0}) ↔ ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = ((𝐼 × {0})‘𝑥)))
95	90, 93, 94	syl2anc 693	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝑋 = (𝐼 × {0}) ↔ ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = ((𝐼 × {0})‘𝑥)))
96		c0ex 10034	. . . . . . 7 ⊢ 0 ∈ V
97	96	fvconst2 6469	. . . . . 6 ⊢ (𝑥 ∈ 𝐼 → ((𝐼 × {0})‘𝑥) = 0)
98	97	eqeq2d 2632	. . . . 5 ⊢ (𝑥 ∈ 𝐼 → ((𝑋‘𝑥) = ((𝐼 × {0})‘𝑥) ↔ (𝑋‘𝑥) = 0))
99	98	ralbiia 2979	. . . 4 ⊢ (∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = ((𝐼 × {0})‘𝑥) ↔ ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = 0)
100	95, 99	syl6bb 276	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝑋 = (𝐼 × {0}) ↔ ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = 0))
101	88, 100	sylibrd 249	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ((𝐻‘𝑋) = 0 → 𝑋 = (𝐼 × {0})))
102	8	psrbag0 19494	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) ∈ 𝐴)
103	102	adantr 481	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝐼 × {0}) ∈ 𝐴)
104		oveq2 6658	. . . . . 6 ⊢ (ℎ = (𝐼 × {0}) → (ℂ_fld Σ_g ℎ) = (ℂ_fld Σ_g (𝐼 × {0})))
105		ovex 6678	. . . . . 6 ⊢ (ℂ_fld Σ_g (𝐼 × {0})) ∈ V
106	104, 4, 105	fvmpt 6282	. . . . 5 ⊢ ((𝐼 × {0}) ∈ 𝐴 → (𝐻‘(𝐼 × {0})) = (ℂ_fld Σ_g (𝐼 × {0})))
107	103, 106	syl 17	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝐻‘(𝐼 × {0})) = (ℂ_fld Σ_g (𝐼 × {0})))
108		fconstmpt 5163	. . . . . 6 ⊢ (𝐼 × {0}) = (𝑥 ∈ 𝐼 ↦ 0)
109	108	oveq2i 6661	. . . . 5 ⊢ (ℂ_fld Σ_g (𝐼 × {0})) = (ℂ_fld Σ_g (𝑥 ∈ 𝐼 ↦ 0))
110	16, 65	ax-mp 5	. . . . . . 7 ⊢ ℂ_fld ∈ Mnd
111	14	gsumz 17374	. . . . . . 7 ⊢ ((ℂ_fld ∈ Mnd ∧ 𝐼 ∈ 𝑉) → (ℂ_fld Σ_g (𝑥 ∈ 𝐼 ↦ 0)) = 0)
112	110, 111	mpan 706	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (ℂ_fld Σ_g (𝑥 ∈ 𝐼 ↦ 0)) = 0)
113	112	adantr 481	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (ℂ_fld Σ_g (𝑥 ∈ 𝐼 ↦ 0)) = 0)
114	109, 113	syl5eq 2668	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (ℂ_fld Σ_g (𝐼 × {0})) = 0)
115	107, 114	eqtrd 2656	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝐻‘(𝐼 × {0})) = 0)
116		fveq2 6191	. . . 4 ⊢ (𝑋 = (𝐼 × {0}) → (𝐻‘𝑋) = (𝐻‘(𝐼 × {0})))
117	116	eqeq1d 2624	. . 3 ⊢ (𝑋 = (𝐼 × {0}) → ((𝐻‘𝑋) = 0 ↔ (𝐻‘(𝐼 × {0})) = 0))
118	115, 117	syl5ibrcom 237	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝑋 = (𝐼 × {0}) → (𝐻‘𝑋) = 0))
119	101, 118	impbid 202	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ((𝐻‘𝑋) = 0 ↔ 𝑋 = (𝐼 × {0})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 ^◡ccnv 5113 ↾ cres 5116 “ cima 5117 Fun wfun 5882 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 supp csupp 7295 ↑_𝑚 cmap 7857 Fincfn 7955 finSupp cfsupp 8275 ℂcc 9934 0cc0 9936 + caddc 9939 ℕcn 11020 ℕ₀cn0 11292 Σ_g cgsu 16101 Mndcmnd 17294 SubMndcsubmnd 17334 CMndccmn 18193 Ringcrg 18547 ℂ_fldccnfld 19746

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-gsum 16103 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-cnfld 19747

This theorem is referenced by: mdegle0 23837

W3C validator