Theorem mplidomlem 33714

Description: Lemma for mplidom 33715. (Contributed by Thierry Arnoux, 4-May-2026.)

Hypotheses

Ref	Expression
mplidom.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mplidom.i	⊢ (𝜑 → 𝐼 ∈ Fin)
mplidom.r	⊢ (𝜑 → 𝑅 ∈ IDomn)
mplidomlem.j	⊢ 𝐻 = (𝑓 ∈ 𝐶 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑛‘∅)⟩})))
mplidomlem.c	⊢ 𝐶 = (Base‘𝑆)
mplidomlem.s	⊢ 𝑆 = ((𝑗 ∪ {𝑥}) mPoly 𝑅)
mplidomlem.u	⊢ 𝑈 = (((𝑗 ∪ {𝑥}) ∖ {𝑥}) mPoly 𝑅)
mplidomlem.q	⊢ 𝑄 = (Poly1‘𝑈)

Assertion

Ref	Expression
mplidomlem	⊢ (𝜑 → 𝑃 ∈ IDomn)

Distinct variable groups:   𝑓,𝐼,𝑗,𝑛,𝑥   𝑃,𝑓,𝑗,𝑛,𝑥   𝑅,𝑓,𝑗,𝑛,𝑥   𝜑,𝑓,𝑗,𝑛,𝑥   𝐶,𝑓,𝑛   𝑓,𝐻,𝑛   𝑄,𝑓,𝑛   𝑆,𝑓,𝑛   𝑈,𝑓,𝑛

Allowed substitution hints:   𝐶(𝑥,𝑗)   𝑄(𝑥,𝑗)   𝑆(𝑥,𝑗)   𝑈(𝑥,𝑗)   𝐻(𝑥,𝑗)

Proof of Theorem mplidomlem

Dummy variables 𝑝 𝑞 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mplidom.p	. 2 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
2		oveq1 7366	. . . 4 ⊢ (𝑖 = ∅ → (𝑖 mPoly 𝑅) = (∅ mPoly 𝑅))
3	2	eleq1d 2821	. . 3 ⊢ (𝑖 = ∅ → ((𝑖 mPoly 𝑅) ∈ IDomn ↔ (∅ mPoly 𝑅) ∈ IDomn))
4		oveq1 7366	. . . 4 ⊢ (𝑖 = 𝑗 → (𝑖 mPoly 𝑅) = (𝑗 mPoly 𝑅))
5	4	eleq1d 2821	. . 3 ⊢ (𝑖 = 𝑗 → ((𝑖 mPoly 𝑅) ∈ IDomn ↔ (𝑗 mPoly 𝑅) ∈ IDomn))
6		oveq1 7366	. . . . 5 ⊢ (𝑖 = (𝑗 ∪ {𝑥}) → (𝑖 mPoly 𝑅) = ((𝑗 ∪ {𝑥}) mPoly 𝑅))
7		mplidomlem.s	. . . . 5 ⊢ 𝑆 = ((𝑗 ∪ {𝑥}) mPoly 𝑅)
8	6, 7	eqtr4di 2789	. . . 4 ⊢ (𝑖 = (𝑗 ∪ {𝑥}) → (𝑖 mPoly 𝑅) = 𝑆)
9	8	eleq1d 2821	. . 3 ⊢ (𝑖 = (𝑗 ∪ {𝑥}) → ((𝑖 mPoly 𝑅) ∈ IDomn ↔ 𝑆 ∈ IDomn))
10		oveq1 7366	. . . 4 ⊢ (𝑖 = 𝐼 → (𝑖 mPoly 𝑅) = (𝐼 mPoly 𝑅))
11	10	eleq1d 2821	. . 3 ⊢ (𝑖 = 𝐼 → ((𝑖 mPoly 𝑅) ∈ IDomn ↔ (𝐼 mPoly 𝑅) ∈ IDomn))
12		eqid 2736	. . . . 5 ⊢ (∅ mPoly 𝑅) = (∅ mPoly 𝑅)
13		0ex 5232	. . . . . 6 ⊢ ∅ ∈ V
14	13	a1i 11	. . . . 5 ⊢ (𝜑 → ∅ ∈ V)
15		mplidom.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ IDomn)
16	15	idomcringd 20702	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing)
17	12, 14, 16	mplcrngd 22001	. . . 4 ⊢ (𝜑 → (∅ mPoly 𝑅) ∈ CRing)
18		eqid 2736	. . . . . 6 ⊢ (Base‘(∅ mPoly 𝑅)) = (Base‘(∅ mPoly 𝑅))
19	15	idomringd 20703	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring)
20	18, 12, 19	0mplric 33702	. . . . 5 ⊢ (𝜑 → (∅ mPoly 𝑅) ≃𝑟 𝑅)
21	15	idomdomd 20701	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Domn)
22		ricdomn 33374	. . . . . 6 ⊢ ((∅ mPoly 𝑅) ≃𝑟 𝑅 → ((∅ mPoly 𝑅) ∈ Domn ↔ 𝑅 ∈ Domn))
23	22	biimpar 478	. . . . 5 ⊢ (((∅ mPoly 𝑅) ≃𝑟 𝑅 ∧ 𝑅 ∈ Domn) → (∅ mPoly 𝑅) ∈ Domn)
24	20, 21, 23	syl2anc 586	. . . 4 ⊢ (𝜑 → (∅ mPoly 𝑅) ∈ Domn)
25		isidom 20700	. . . 4 ⊢ ((∅ mPoly 𝑅) ∈ IDomn ↔ ((∅ mPoly 𝑅) ∈ CRing ∧ (∅ mPoly 𝑅) ∈ Domn))
26	17, 24, 25	sylanbrc 585	. . 3 ⊢ (𝜑 → (∅ mPoly 𝑅) ∈ IDomn)
27		mplidom.i	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ Fin)
28	27	ad3antrrr 732	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝐼 ∈ Fin)
29		simpllr 777	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑗 ⊆ 𝐼)
30	28, 29	ssfid 9172	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑗 ∈ Fin)
31		snfi 8983	. . . . . . . . 9 ⊢ {𝑥} ∈ Fin
32	31	a1i 11	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → {𝑥} ∈ Fin)
33	30, 32	unfid 9099	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → (𝑗 ∪ {𝑥}) ∈ Fin)
34	16	ad3antrrr 732	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑅 ∈ CRing)
35	7, 33, 34	mplcrngd 22001	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑆 ∈ CRing)
36		domnnzr 20681	. . . . . . . . . 10 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
37	21, 36	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ NzRing)
38	37	ad3antrrr 732	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑅 ∈ NzRing)
39	7, 33, 38	mplnzr 33700	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑆 ∈ NzRing)
40		mplidomlem.c	. . . . . . . . . . . 12 ⊢ 𝐶 = (Base‘𝑆)
41		mplidomlem.u	. . . . . . . . . . . 12 ⊢ 𝑈 = (((𝑗 ∪ {𝑥}) ∖ {𝑥}) mPoly 𝑅)
42		mplidomlem.q	. . . . . . . . . . . 12 ⊢ 𝑄 = (Poly1‘𝑈)
43		mplidomlem.j	. . . . . . . . . . . 12 ⊢ 𝐻 = (𝑓 ∈ 𝐶 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑛‘∅)⟩})))
44	33	ad4antr 734	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) ∧ (𝐻‘𝑝) = (0g‘𝑄)) → (𝑗 ∪ {𝑥}) ∈ Fin)
45		vsnid 4598	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ {𝑥}
46		elun2 4115	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {𝑥} → 𝑥 ∈ (𝑗 ∪ {𝑥}))
47	45, 46	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ (𝑗 ∪ {𝑥})
48	47	a1i 11	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) ∧ (𝐻‘𝑝) = (0g‘𝑄)) → 𝑥 ∈ (𝑗 ∪ {𝑥}))
49	34	ad4antr 734	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) ∧ (𝐻‘𝑝) = (0g‘𝑄)) → 𝑅 ∈ CRing)
50		eqid 2736	. . . . . . . . . . . 12 ⊢ (0g‘𝑄) = (0g‘𝑄)
51		eqid 2736	. . . . . . . . . . . 12 ⊢ (0g‘𝑆) = (0g‘𝑆)
52		simp-4r 785	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) ∧ (𝐻‘𝑝) = (0g‘𝑄)) → 𝑝 ∈ 𝐶)
53		simpr 485	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) ∧ (𝐻‘𝑝) = (0g‘𝑄)) → (𝐻‘𝑝) = (0g‘𝑄))
54	40, 7, 41, 42, 43, 44, 48, 49, 50, 51, 52, 53	selvply1rhm0 33713	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) ∧ (𝐻‘𝑝) = (0g‘𝑄)) → 𝑝 = (0g‘𝑆))
55	33	ad4antr 734	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) ∧ (𝐻‘𝑞) = (0g‘𝑄)) → (𝑗 ∪ {𝑥}) ∈ Fin)
56	47	a1i 11	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) ∧ (𝐻‘𝑞) = (0g‘𝑄)) → 𝑥 ∈ (𝑗 ∪ {𝑥}))
57	34	ad4antr 734	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) ∧ (𝐻‘𝑞) = (0g‘𝑄)) → 𝑅 ∈ CRing)
58		simpllr 777	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) ∧ (𝐻‘𝑞) = (0g‘𝑄)) → 𝑞 ∈ 𝐶)
59		simpr 485	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) ∧ (𝐻‘𝑞) = (0g‘𝑄)) → (𝐻‘𝑞) = (0g‘𝑄))
60	40, 7, 41, 42, 43, 55, 56, 57, 50, 51, 58, 59	selvply1rhm0 33713	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) ∧ (𝐻‘𝑞) = (0g‘𝑄)) → 𝑞 = (0g‘𝑆))
61		simp-5r 787	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → 𝑥 ∈ (𝐼 ∖ 𝑗))
62	61	eldifbd 3899	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → ¬ 𝑥 ∈ 𝑗)
63		disjsn 4646	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ 𝑗)
64	62, 63	sylibr 235	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → (𝑗 ∩ {𝑥}) = ∅)
65		undif5 4415	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∩ {𝑥}) = ∅ → ((𝑗 ∪ {𝑥}) ∖ {𝑥}) = 𝑗)
66	64, 65	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → ((𝑗 ∪ {𝑥}) ∖ {𝑥}) = 𝑗)
67	66	oveq1d 7374	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → (((𝑗 ∪ {𝑥}) ∖ {𝑥}) mPoly 𝑅) = (𝑗 mPoly 𝑅))
68	41, 67	eqtrid 2783	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → 𝑈 = (𝑗 mPoly 𝑅))
69		simp-4r 785	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → (𝑗 mPoly 𝑅) ∈ IDomn)
70	69	idomdomd 20701	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → (𝑗 mPoly 𝑅) ∈ Domn)
71	68, 70	eqeltrd 2836	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → 𝑈 ∈ Domn)
72	42	ply1domn 26110	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ Domn → 𝑄 ∈ Domn)
73	71, 72	syl 17	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → 𝑄 ∈ Domn)
74	47	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑥 ∈ (𝑗 ∪ {𝑥}))
75	40, 7, 41, 42, 43, 33, 74, 34	selvply1rhm 33712	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝐻 ∈ (𝑆 RingHom 𝑄))
76	75	ad3antrrr 732	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → 𝐻 ∈ (𝑆 RingHom 𝑄))
77		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑄) = (Base‘𝑄)
78	40, 77	rhmf 20458	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ (𝑆 RingHom 𝑄) → 𝐻:𝐶⟶(Base‘𝑄))
79	76, 78	syl 17	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → 𝐻:𝐶⟶(Base‘𝑄))
80		simpllr 777	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → 𝑝 ∈ 𝐶)
81	79, 80	ffvelcdmd 7029	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → (𝐻‘𝑝) ∈ (Base‘𝑄))
82		simplr 770	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → 𝑞 ∈ 𝐶)
83	79, 82	ffvelcdmd 7029	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → (𝐻‘𝑞) ∈ (Base‘𝑄))
84		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆))
85	84	fveq2d 6834	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → (𝐻‘(𝑝(.r‘𝑆)𝑞)) = (𝐻‘(0g‘𝑆)))
86		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑆) = (.r‘𝑆)
87		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑄) = (.r‘𝑄)
88	40, 86, 87	rhmmul 20460	. . . . . . . . . . . . . 14 ⊢ ((𝐻 ∈ (𝑆 RingHom 𝑄) ∧ 𝑝 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶) → (𝐻‘(𝑝(.r‘𝑆)𝑞)) = ((𝐻‘𝑝)(.r‘𝑄)(𝐻‘𝑞)))
89	76, 80, 82, 88	syl3anc 1375	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → (𝐻‘(𝑝(.r‘𝑆)𝑞)) = ((𝐻‘𝑝)(.r‘𝑄)(𝐻‘𝑞)))
90		rhmghm 20457	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ (𝑆 RingHom 𝑄) → 𝐻 ∈ (𝑆 GrpHom 𝑄))
91	51, 50	ghmid 19191	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ (𝑆 GrpHom 𝑄) → (𝐻‘(0g‘𝑆)) = (0g‘𝑄))
92	76, 90, 91	3syl 18	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → (𝐻‘(0g‘𝑆)) = (0g‘𝑄))
93	85, 89, 92	3eqtr3d 2779	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → ((𝐻‘𝑝)(.r‘𝑄)(𝐻‘𝑞)) = (0g‘𝑄))
94	77, 87, 50	domneq0 20683	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ Domn ∧ (𝐻‘𝑝) ∈ (Base‘𝑄) ∧ (𝐻‘𝑞) ∈ (Base‘𝑄)) → (((𝐻‘𝑝)(.r‘𝑄)(𝐻‘𝑞)) = (0g‘𝑄) ↔ ((𝐻‘𝑝) = (0g‘𝑄) ∨ (𝐻‘𝑞) = (0g‘𝑄))))
95	94	biimpa 477	. . . . . . . . . . . 12 ⊢ (((𝑄 ∈ Domn ∧ (𝐻‘𝑝) ∈ (Base‘𝑄) ∧ (𝐻‘𝑞) ∈ (Base‘𝑄)) ∧ ((𝐻‘𝑝)(.r‘𝑄)(𝐻‘𝑞)) = (0g‘𝑄)) → ((𝐻‘𝑝) = (0g‘𝑄) ∨ (𝐻‘𝑞) = (0g‘𝑄)))
96	73, 81, 83, 93, 95	syl31anc 1377	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → ((𝐻‘𝑝) = (0g‘𝑄) ∨ (𝐻‘𝑞) = (0g‘𝑄)))
97	54, 60, 96	orim12da 32548	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) ∧ (𝑝(.r‘𝑆)𝑞) = (0g‘𝑆)) → (𝑝 = (0g‘𝑆) ∨ 𝑞 = (0g‘𝑆)))
98	97	ex 413	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ 𝑝 ∈ 𝐶) ∧ 𝑞 ∈ 𝐶) → ((𝑝(.r‘𝑆)𝑞) = (0g‘𝑆) → (𝑝 = (0g‘𝑆) ∨ 𝑞 = (0g‘𝑆))))
99	98	anasss 467	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) ∧ (𝑝 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶)) → ((𝑝(.r‘𝑆)𝑞) = (0g‘𝑆) → (𝑝 = (0g‘𝑆) ∨ 𝑞 = (0g‘𝑆))))
100	99	ralrimivva 3179	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → ∀𝑝 ∈ 𝐶 ∀𝑞 ∈ 𝐶 ((𝑝(.r‘𝑆)𝑞) = (0g‘𝑆) → (𝑝 = (0g‘𝑆) ∨ 𝑞 = (0g‘𝑆))))
101	40, 86, 51	isdomn 20680	. . . . . . 7 ⊢ (𝑆 ∈ Domn ↔ (𝑆 ∈ NzRing ∧ ∀𝑝 ∈ 𝐶 ∀𝑞 ∈ 𝐶 ((𝑝(.r‘𝑆)𝑞) = (0g‘𝑆) → (𝑝 = (0g‘𝑆) ∨ 𝑞 = (0g‘𝑆)))))
102	39, 100, 101	sylanbrc 585	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑆 ∈ Domn)
103		isidom 20700	. . . . . 6 ⊢ (𝑆 ∈ IDomn ↔ (𝑆 ∈ CRing ∧ 𝑆 ∈ Domn))
104	35, 102, 103	sylanbrc 585	. . . . 5 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑆 ∈ IDomn)
105	104	ex 413	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) → ((𝑗 mPoly 𝑅) ∈ IDomn → 𝑆 ∈ IDomn))
106	105	anasss 467	. . 3 ⊢ ((𝜑 ∧ (𝑗 ⊆ 𝐼 ∧ 𝑥 ∈ (𝐼 ∖ 𝑗))) → ((𝑗 mPoly 𝑅) ∈ IDomn → 𝑆 ∈ IDomn))
107	3, 5, 9, 11, 26, 106, 27	findcard2d 9094	. 2 ⊢ (𝜑 → (𝐼 mPoly 𝑅) ∈ IDomn)
108	1, 107	eqeltrid 2840	1 ⊢ (𝜑 → 𝑃 ∈ IDomn)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 849   ∧ w3a 1088   = wceq 1543   ∈ wcel 2115  ∀wral 3050  Vcvv 3428   ∖ cdif 3883   ∪ cun 3884   ∩ cin 3885   ⊆ wss 3886  ∅c0 4264  {csn 4558  ⟨cop 4564   class class class wbr 5075   ↦ cmpt 5156  ⟶wf 6484  ‘cfv 6488  (class class class)co 7359  1_oc1o 8391   ↑_m cmap 8766  Fincfn 8886  ℕ₀cn0 12431  Basecbs 17173  ._rcmulr 17215  0_gc0g 17396   GrpHom cghm 19181  CRingccrg 20209   RingHom crh 20443   ≃_𝑟 cric 20445  NzRingcnzr 20487  Domncdomn 20667  IDomncidom 20668   mPoly cmpl 21884   selectVars cslv 22095  Poly₁cpl1 22165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-8 2117  ax-9 2125  ax-10 2148  ax-11 2164  ax-12 2185  ax-ext 2708  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7681  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109  ax-pre-sup 11110  ax-addf 11111

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 850  df-3or 1089  df-3an 1090  df-tru 1546  df-fal 1556  df-ex 1783  df-nf 1787  df-sb 2070  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3061  df-rmo 3341  df-reu 3342  df-rab 3389  df-v 3430  df-sbc 3727  df-csb 3835  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-pss 3906  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-iin 4927  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-isom 6497  df-riota 7316  df-ov 7362  df-oprab 7363  df-mpo 7364  df-of 7623  df-ofr 7624  df-om 7810  df-1st 7934  df-2nd 7935  df-supp 8104  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-er 8636  df-map 8768  df-pm 8769  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9268  df-sup 9348  df-oi 9418  df-card 9857  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-9 12245  df-n0 12432  df-z 12519  df-dec 12639  df-uz 12783  df-fz 13456  df-fzo 13603  df-seq 13958  df-hash 14287  df-struct 17111  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-ress 17195  df-plusg 17227  df-mulr 17228  df-starv 17229  df-sca 17230  df-vsca 17231  df-ip 17232  df-tset 17233  df-ple 17234  df-ds 17236  df-unif 17237  df-hom 17238  df-cco 17239  df-0g 17398  df-gsum 17399  df-prds 17404  df-pws 17406  df-mre 17542  df-mrc 17543  df-acs 17545  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-mhm 18745  df-submnd 18746  df-grp 18906  df-minusg 18907  df-sbg 18908  df-mulg 19038  df-subg 19093  df-ghm 19182  df-cntz 19286  df-cmn 19751  df-abl 19752  df-mgp 20116  df-rng 20128  df-ur 20157  df-srg 20162  df-ring 20210  df-cring 20211  df-rhm 20446  df-rim 20447  df-ric 20449  df-nzr 20488  df-subrng 20521  df-subrg 20545  df-rlreg 20669  df-domn 20670  df-idom 20671  df-lmod 20855  df-lss 20925  df-lsp 20965  df-cnfld 21351  df-assa 21831  df-asp 21832  df-ascl 21833  df-psr 21887  df-mvr 21888  df-mpl 21889  df-opsr 21891  df-evls 22053  df-evl 22054  df-selv 22096  df-psr1 22168  df-vr1 22169  df-ply1 22170  df-coe1 22171  df-mdeg 26041  df-deg1 26042

This theorem is referenced by:  mplidom  33715