tdeglem4 - Metamath Proof Explorer

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Theorem tdeglem4 23624

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 2978	. . . . 5 ⊢ (∃𝑥 ∈ 𝐼 ¬ (𝑋‘𝑥) = 0 ↔ ¬ ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = 0)
2		df-ne 2782	. . . . . . 7 ⊢ ((𝑋‘𝑥) ≠ 0 ↔ ¬ (𝑋‘𝑥) = 0)
3		oveq2 6557	. . . . . . . . . . . 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 6577	. . . . . . . . . . . 12 ⊢ (ℂ_fld Σ_g 𝑋) ∈ V
6	3, 4, 5	fvmpt 6191	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝐴 → (𝐻‘𝑋) = (ℂ_fld Σ_g 𝑋))
7	6	ad2antlr 759	. . . . . . . . . 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 19186	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑋:𝐼⟶ℕ₀)
10	9	feqmptd 6159	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑋 = (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)))
11	10	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → 𝑋 = (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)))
12	11	oveq2d 6565	. . . . . . . . . 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 19571	. . . . . . . . . . 11 ⊢ ℂ = (Base‘ℂ_fld)
14		cnfld0 19589	. . . . . . . . . . 11 ⊢ 0 = (0_g‘ℂ_fld)
15		cnfldadd 19572	. . . . . . . . . . 11 ⊢ + = (+_g‘ℂ_fld)
16		cnring 19587	. . . . . . . . . . . 12 ⊢ ℂ_fld ∈ Ring
17		ringcmn 18404	. . . . . . . . . . . 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 786	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → 𝐼 ∈ 𝑉)
20	9	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → 𝑋:𝐼⟶ℕ₀)
21	20	ffvelrnda 6267	. . . . . . . . . . . 12 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) ∧ 𝑦 ∈ 𝐼) → (𝑋‘𝑦) ∈ ℕ₀)
22	21	nn0cnd 11230	. . . . . . . . . . 11 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) ∧ 𝑦 ∈ 𝐼) → (𝑋‘𝑦) ∈ ℂ)
23	8	psrbagfsupp 19330	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐼 ∈ 𝑉) → 𝑋 finSupp 0)
24	23	ancoms 468	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑋 finSupp 0)
25	24	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → 𝑋 finSupp 0)
26	11, 25	eqbrtrrd 4607	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) finSupp 0)
27		incom 3767	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∖ {𝑥}) ∩ {𝑥}) = ({𝑥} ∩ (𝐼 ∖ {𝑥}))
28		disjdif 3992	. . . . . . . . . . . . 13 ⊢ ({𝑥} ∩ (𝐼 ∖ {𝑥})) = ∅
29	27, 28	eqtri 2632	. . . . . . . . . . . 12 ⊢ ((𝐼 ∖ {𝑥}) ∩ {𝑥}) = ∅
30	29	a1i 11	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ((𝐼 ∖ {𝑥}) ∩ {𝑥}) = ∅)
31		difsnid 4282	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐼 → ((𝐼 ∖ {𝑥}) ∪ {𝑥}) = 𝐼)
32	31	eqcomd 2616	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐼 → 𝐼 = ((𝐼 ∖ {𝑥}) ∪ {𝑥}))
33	32	ad2antrl 760	. . . . . . . . . . 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 18152	. . . . . . . . . 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 2648	. . . . . . . . 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 4735	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ 𝑉 → (𝐼 ∖ {𝑥}) ∈ V)
37	36	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝐼 ∖ {𝑥}) ∈ V)
38		nn0subm 19620	. . . . . . . . . . . . 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 3694	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝐼 ∖ {𝑥}) → 𝑦 ∈ 𝐼)
41		ffvelrn 6265	. . . . . . . . . . . . . 14 ⊢ ((𝑋:𝐼⟶ℕ₀ ∧ 𝑦 ∈ 𝐼) → (𝑋‘𝑦) ∈ ℕ₀)
42	20, 40, 41	syl2an 493	. . . . . . . . . . . . 13 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) ∧ 𝑦 ∈ (𝐼 ∖ {𝑥})) → (𝑋‘𝑦) ∈ ℕ₀)
43		eqid 2610	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) = (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))
44	42, 43	fmptd 6292	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)):(𝐼 ∖ {𝑥})⟶ℕ₀)
45		mptexg 6389	. . . . . . . . . . . . . . 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 758	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦)) ∈ V)
48		funmpt 5840	. . . . . . . . . . . . . 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 5840	. . . . . . . . . . . . . . 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 3699	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ∖ {𝑥}) ⊆ 𝐼
53		resmpt 5369	. . . . . . . . . . . . . . . . 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 5342	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) ↾ (𝐼 ∖ {𝑥})) ⊆ (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦))
56	54, 55	eqsstr3i 3599	. . . . . . . . . . . . . . 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 6389	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ 𝑉 → (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) ∈ V)
59	58	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑦 ∈ 𝐼 ↦ (𝑋‘𝑦)) ∈ V)
60		funsssuppss 7208	. . . . . . . . . . . . . 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 1318	. . . . . . . . . . . . 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 8174	. . . . . . . . . . . . 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 1319	. . . . . . . . . . . 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 18142	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (ℂ_fld Σ_g (𝑦 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑋‘𝑦))) ∈ ℕ₀)
65		ringmnd 18379	. . . . . . . . . . . . . 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 790	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → 𝑥 ∈ 𝐼)
68	20, 67	ffvelrnd 6268	. . . . . . . . . . . . . 14 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑋‘𝑥) ∈ ℕ₀)
69	68	nn0cnd 11230	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑋‘𝑥) ∈ ℂ)
70		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑋‘𝑦) = (𝑋‘𝑥))
71	13, 70	gsumsn 18177	. . . . . . . . . . . . 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 1318	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (ℂ_fld Σ_g (𝑦 ∈ {𝑥} ↦ (𝑋‘𝑦))) = (𝑋‘𝑥))
73		simprr 792	. . . . . . . . . . . . . 14 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑋‘𝑥) ≠ 0)
74	73, 2	sylib 207	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ¬ (𝑋‘𝑥) = 0)
75		elnn0 11171	. . . . . . . . . . . . . 14 ⊢ ((𝑋‘𝑥) ∈ ℕ₀ ↔ ((𝑋‘𝑥) ∈ ℕ ∨ (𝑋‘𝑥) = 0))
76	68, 75	sylib 207	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → ((𝑋‘𝑥) ∈ ℕ ∨ (𝑋‘𝑥) = 0))
77		orel2 397	. . . . . . . . . . . . 13 ⊢ (¬ (𝑋‘𝑥) = 0 → (((𝑋‘𝑥) ∈ ℕ ∨ (𝑋‘𝑥) = 0) → (𝑋‘𝑥) ∈ ℕ))
78	74, 76, 77	sylc 63	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝑋‘𝑥) ∈ ℕ)
79	72, 78	eqeltrd 2688	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (ℂ_fld Σ_g (𝑦 ∈ {𝑥} ↦ (𝑋‘𝑦))) ∈ ℕ)
80		nn0nnaddcl 11201	. . . . . . . . . . 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 691	. . . . . . . . . 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 10942	. . . . . . . . 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 2849	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ (𝑥 ∈ 𝐼 ∧ (𝑋‘𝑥) ≠ 0)) → (𝐻‘𝑋) ≠ 0)
84	83	expr 641	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ 𝐼) → ((𝑋‘𝑥) ≠ 0 → (𝐻‘𝑋) ≠ 0))
85	2, 84	syl5bir 232	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 ∈ 𝐼) → (¬ (𝑋‘𝑥) = 0 → (𝐻‘𝑋) ≠ 0))
86	85	rexlimdva 3013	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (∃𝑥 ∈ 𝐼 ¬ (𝑋‘𝑥) = 0 → (𝐻‘𝑋) ≠ 0))
87	1, 86	syl5bir 232	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (¬ ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = 0 → (𝐻‘𝑋) ≠ 0))
88	87	necon4bd 2802	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ((𝐻‘𝑋) = 0 → ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = 0))
89		ffn 5958	. . . . . 6 ⊢ (𝑋:𝐼⟶ℕ₀ → 𝑋 Fn 𝐼)
90	9, 89	syl 17	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑋 Fn 𝐼)
91		0nn0 11184	. . . . . 6 ⊢ 0 ∈ ℕ₀
92		fnconstg 6006	. . . . . 6 ⊢ (0 ∈ ℕ₀ → (𝐼 × {0}) Fn 𝐼)
93	91, 92	mp1i 13	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝐼 × {0}) Fn 𝐼)
94		eqfnfv 6219	. . . . 5 ⊢ ((𝑋 Fn 𝐼 ∧ (𝐼 × {0}) Fn 𝐼) → (𝑋 = (𝐼 × {0}) ↔ ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = ((𝐼 × {0})‘𝑥)))
95	90, 93, 94	syl2anc 691	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝑋 = (𝐼 × {0}) ↔ ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = ((𝐼 × {0})‘𝑥)))
96		c0ex 9913	. . . . . . 7 ⊢ 0 ∈ V
97	96	fvconst2 6374	. . . . . 6 ⊢ (𝑥 ∈ 𝐼 → ((𝐼 × {0})‘𝑥) = 0)
98	97	eqeq2d 2620	. . . . 5 ⊢ (𝑥 ∈ 𝐼 → ((𝑋‘𝑥) = ((𝐼 × {0})‘𝑥) ↔ (𝑋‘𝑥) = 0))
99	98	ralbiia 2962	. . . 4 ⊢ (∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = ((𝐼 × {0})‘𝑥) ↔ ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = 0)
100	95, 99	syl6bb 275	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝑋 = (𝐼 × {0}) ↔ ∀𝑥 ∈ 𝐼 (𝑋‘𝑥) = 0))
101	88, 100	sylibrd 248	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ((𝐻‘𝑋) = 0 → 𝑋 = (𝐼 × {0})))
102	8	psrbag0 19315	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) ∈ 𝐴)
103	102	adantr 480	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝐼 × {0}) ∈ 𝐴)
104		oveq2 6557	. . . . . 6 ⊢ (ℎ = (𝐼 × {0}) → (ℂ_fld Σ_g ℎ) = (ℂ_fld Σ_g (𝐼 × {0})))
105		ovex 6577	. . . . . 6 ⊢ (ℂ_fld Σ_g (𝐼 × {0})) ∈ V
106	104, 4, 105	fvmpt 6191	. . . . 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 5085	. . . . . 6 ⊢ (𝐼 × {0}) = (𝑥 ∈ 𝐼 ↦ 0)
109	108	oveq2i 6560	. . . . 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 17197	. . . . . . 7 ⊢ ((ℂ_fld ∈ Mnd ∧ 𝐼 ∈ 𝑉) → (ℂ_fld Σ_g (𝑥 ∈ 𝐼 ↦ 0)) = 0)
112	110, 111	mpan 702	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (ℂ_fld Σ_g (𝑥 ∈ 𝐼 ↦ 0)) = 0)
113	112	adantr 480	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (ℂ_fld Σ_g (𝑥 ∈ 𝐼 ↦ 0)) = 0)
114	109, 113	syl5eq 2656	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (ℂ_fld Σ_g (𝐼 × {0})) = 0)
115	107, 114	eqtrd 2644	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝐻‘(𝐼 × {0})) = 0)
116		fveq2 6103	. . . 4 ⊢ (𝑋 = (𝐼 × {0}) → (𝐻‘𝑋) = (𝐻‘(𝐼 × {0})))
117	116	eqeq1d 2612	. . 3 ⊢ (𝑋 = (𝐼 × {0}) → ((𝐻‘𝑋) = 0 ↔ (𝐻‘(𝐼 × {0})) = 0))
118	115, 117	syl5ibrcom 236	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝑋 = (𝐼 × {0}) → (𝐻‘𝑋) = 0))
119	101, 118	impbid 201	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ((𝐻‘𝑋) = 0 ↔ 𝑋 = (𝐼 × {0})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ^◡ccnv 5037 ↾ cres 5040 “ cima 5041 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 supp csupp 7182 ↑_𝑚 cmap 7744 Fincfn 7841 finSupp cfsupp 8158 ℂcc 9813 0cc0 9815 + caddc 9818 ℕcn 10897 ℕ₀cn0 11169 Σ_g cgsu 15924 Mndcmnd 17117 SubMndcsubmnd 17157 CMndccmn 18016 Ringcrg 18370 ℂ_fldccnfld 19567

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-gsum 15926 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-cnfld 19568

This theorem is referenced by: mdegle0 23641

W3C validator