Theorem mplidomlem 33826

Description: Lemma for mplidom 33827. (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 7403	. . . 4 ⊢ (𝑖 = ∅ → (𝑖 mPoly 𝑅) = (∅ mPoly 𝑅))
3	2	eleq1d 2848	. . 3 ⊢ (𝑖 = ∅ → ((𝑖 mPoly 𝑅) ∈ IDomn ↔ (∅ mPoly 𝑅) ∈ IDomn))
4		oveq1 7403	. . . 4 ⊢ (𝑖 = 𝑗 → (𝑖 mPoly 𝑅) = (𝑗 mPoly 𝑅))
5	4	eleq1d 2848	. . 3 ⊢ (𝑖 = 𝑗 → ((𝑖 mPoly 𝑅) ∈ IDomn ↔ (𝑗 mPoly 𝑅) ∈ IDomn))
6		oveq1 7403	. . . . 5 ⊢ (𝑖 = (𝑗 ∪ {𝑥}) → (𝑖 mPoly 𝑅) = ((𝑗 ∪ {𝑥}) mPoly 𝑅))
7		mplidomlem.s	. . . . 5 ⊢ 𝑆 = ((𝑗 ∪ {𝑥}) mPoly 𝑅)
8	6, 7	eqtr4di 2816	. . . 4 ⊢ (𝑖 = (𝑗 ∪ {𝑥}) → (𝑖 mPoly 𝑅) = 𝑆)
9	8	eleq1d 2848	. . 3 ⊢ (𝑖 = (𝑗 ∪ {𝑥}) → ((𝑖 mPoly 𝑅) ∈ IDomn ↔ 𝑆 ∈ IDomn))
10		oveq1 7403	. . . 4 ⊢ (𝑖 = 𝐼 → (𝑖 mPoly 𝑅) = (𝐼 mPoly 𝑅))
11	10	eleq1d 2848	. . 3 ⊢ (𝑖 = 𝐼 → ((𝑖 mPoly 𝑅) ∈ IDomn ↔ (𝐼 mPoly 𝑅) ∈ IDomn))
12		eqid 2763	. . . . 5 ⊢ (∅ mPoly 𝑅) = (∅ mPoly 𝑅)
13		0ex 5258	. . . . . 6 ⊢ ∅ ∈ V
14	13	a1i 11	. . . . 5 ⊢ (𝜑 → ∅ ∈ V)
15		mplidom.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ IDomn)
16	15	idomcringd 20786	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing)
17	12, 14, 16	mplcrngd 22082	. . . 4 ⊢ (𝜑 → (∅ mPoly 𝑅) ∈ CRing)
18		eqid 2763	. . . . . 6 ⊢ (Base‘(∅ mPoly 𝑅)) = (Base‘(∅ mPoly 𝑅))
19	15	idomringd 20787	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring)
20	18, 12, 19	0mplric 33814	. . . . 5 ⊢ (𝜑 → (∅ mPoly 𝑅) ≃𝑟 𝑅)
21	15	idomdomd 20785	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Domn)
22		ricdomn 33477	. . . . . 6 ⊢ ((∅ mPoly 𝑅) ≃𝑟 𝑅 → ((∅ mPoly 𝑅) ∈ Domn ↔ 𝑅 ∈ Domn))
23	22	biimpar 481	. . . . 5 ⊢ (((∅ mPoly 𝑅) ≃𝑟 𝑅 ∧ 𝑅 ∈ Domn) → (∅ mPoly 𝑅) ∈ Domn)
24	20, 21, 23	syl2anc 593	. . . 4 ⊢ (𝜑 → (∅ mPoly 𝑅) ∈ Domn)
25		isidom 20784	. . . 4 ⊢ ((∅ mPoly 𝑅) ∈ IDomn ↔ ((∅ mPoly 𝑅) ∈ CRing ∧ (∅ mPoly 𝑅) ∈ Domn))
26	17, 24, 25	sylanbrc 592	. . 3 ⊢ (𝜑 → (∅ mPoly 𝑅) ∈ IDomn)
27		mplidom.i	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ Fin)
28	27	ad3antrrr 740	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝐼 ∈ Fin)
29		simpllr 785	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑗 ⊆ 𝐼)
30	28, 29	ssfid 9213	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑗 ∈ Fin)
31		snfi 9024	. . . . . . . . 9 ⊢ {𝑥} ∈ Fin
32	31	a1i 11	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → {𝑥} ∈ Fin)
33	30, 32	unfid 9140	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → (𝑗 ∪ {𝑥}) ∈ Fin)
34	16	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑅 ∈ CRing)
35	7, 33, 34	mplcrngd 22082	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑆 ∈ CRing)
36		domnnzr 20766	. . . . . . . . . 10 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
37	21, 36	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ NzRing)
38	37	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑅 ∈ NzRing)
39	7, 33, 38	mplnzr 33812	. . . . . . 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 742	. . . . . . . . . . . 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 4623	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ {𝑥}
46		elun2 4136	. . . . . . . . . . . . . 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 742	. . . . . . . . . . . 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 2763	. . . . . . . . . . . 12 ⊢ (0g‘𝑄) = (0g‘𝑄)
51		eqid 2763	. . . . . . . . . . . 12 ⊢ (0g‘𝑆) = (0g‘𝑆)
52		simp-4r 793	. . . . . . . . . . . 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 488	. . . . . . . . . . . 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 33825	. . . . . . . . . . 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 742	. . . . . . . . . . . 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 742	. . . . . . . . . . . 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 785	. . . . . . . . . . . 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 488	. . . . . . . . . . . 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 33825	. . . . . . . . . . 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 795	. . . . . . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . . . . . 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 4671	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ 𝑗)
64	62, 63	sylibr 236	. . . . . . . . . . . . . . . . 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 4439	. . . . . . . . . . . . . . . . 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 7411	. . . . . . . . . . . . . . 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 2810	. . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . . 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 20785	. . . . . . . . . . . . . 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 2863	. . . . . . . . . . . . 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 26191	. . . . . . . . . . . . 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 33824	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝐻 ∈ (𝑆 RingHom 𝑄))
76	75	ad3antrrr 740	. . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑄) = (Base‘𝑄)
78	40, 77	rhmf 20543	. . . . . . . . . . . . . 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 785	. . . . . . . . . . . . 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 7066	. . . . . . . . . . . 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 778	. . . . . . . . . . . . 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 7066	. . . . . . . . . . . 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 488	. . . . . . . . . . . . . 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 6871	. . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑆) = (.r‘𝑆)
87		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑄) = (.r‘𝑄)
88	40, 86, 87	rhmmul 20545	. . . . . . . . . . . . . 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 1392	. . . . . . . . . . . . 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 20542	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ (𝑆 RingHom 𝑄) → 𝐻 ∈ (𝑆 GrpHom 𝑄))
91	51, 50	ghmid 19272	. . . . . . . . . . . . . 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 2806	. . . . . . . . . . . 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 20768	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 1394	. . . . . . . . . . 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 978	. . . . . . . . . 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 416	. . . . . . . . 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 470	. . . . . . . 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 3206	. . . . . . 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 20765	. . . . . . 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 592	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑆 ∈ Domn)
103		isidom 20784	. . . . . 6 ⊢ (𝑆 ∈ IDomn ↔ (𝑆 ∈ CRing ∧ 𝑆 ∈ Domn))
104	35, 102, 103	sylanbrc 592	. . . . 5 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑆 ∈ IDomn)
105	104	ex 416	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) → ((𝑗 mPoly 𝑅) ∈ IDomn → 𝑆 ∈ IDomn))
106	105	anasss 470	. . 3 ⊢ ((𝜑 ∧ (𝑗 ⊆ 𝐼 ∧ 𝑥 ∈ (𝐼 ∖ 𝑗))) → ((𝑗 mPoly 𝑅) ∈ IDomn → 𝑆 ∈ IDomn))
107	3, 5, 9, 11, 26, 106, 27	findcard2d 9135	. 2 ⊢ (𝜑 → (𝐼 mPoly 𝑅) ∈ IDomn)
108	1, 107	eqeltrid 2867	1 ⊢ (𝜑 → 𝑃 ∈ IDomn)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 858   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  ∀wral 3077  Vcvv 3455   ∖ cdif 3902   ∪ cun 3903   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  {csn 4583  ⟨cop 4589   class class class wbr 5101   ↦ cmpt 5182  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396  1_oc1o 8430   ↑_m cmap 8808  Fincfn 8927  ℕ₀cn0 12491  Basecbs 17255  ._rcmulr 17297  0_gc0g 17478   GrpHom cghm 19263  CRingccrg 20294   RingHom crh 20528   ≃_𝑟 cric 20530  NzRingcnzr 20572  Domncdomn 20752  IDomncidom 20753   mPoly cmpl 21965   selectVars cslv 22176  Poly₁cpl1 22246

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718  ax-cnex 11140  ax-resscn 11141  ax-1cn 11142  ax-icn 11143  ax-addcl 11144  ax-addrcl 11145  ax-mulcl 11146  ax-mulrcl 11147  ax-mulcom 11148  ax-addass 11149  ax-mulass 11150  ax-distr 11151  ax-i2m1 11152  ax-1ne0 11153  ax-1rid 11154  ax-rnegex 11155  ax-rrecex 11156  ax-cnre 11157  ax-pre-lttri 11158  ax-pre-lttrn 11159  ax-pre-ltadd 11160  ax-pre-mulgt0 11161  ax-pre-sup 11162  ax-addf 11163

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-se 5602  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7660  df-ofr 7661  df-om 7847  df-1st 7970  df-2nd 7971  df-supp 8141  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8678  df-map 8810  df-pm 8811  df-ixp 8880  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9306  df-sup 9386  df-oi 9456  df-card 9909  df-pnf 11229  df-mnf 11230  df-xr 11231  df-ltxr 11232  df-le 11233  df-sub 11427  df-neg 11428  df-nn 12221  df-2 12290  df-3 12291  df-4 12292  df-5 12293  df-6 12294  df-7 12295  df-8 12296  df-9 12297  df-n0 12492  df-z 12579  df-dec 12699  df-uz 12850  df-fz 13523  df-fzo 13670  df-seq 14025  df-hash 14354  df-struct 17193  df-sets 17210  df-slot 17228  df-ndx 17240  df-base 17256  df-ress 17277  df-plusg 17309  df-mulr 17310  df-starv 17311  df-sca 17312  df-vsca 17313  df-ip 17314  df-tset 17315  df-ple 17316  df-ds 17318  df-unif 17319  df-hom 17320  df-cco 17321  df-0g 17480  df-gsum 17481  df-prds 17486  df-pws 17488  df-mre 17624  df-mrc 17625  df-acs 17627  df-mgm 18684  df-sgrp 18763  df-mnd 18779  df-mhm 18827  df-submnd 18828  df-grp 18988  df-minusg 18989  df-sbg 18990  df-mulg 19120  df-subg 19175  df-ghm 19264  df-cntz 19367  df-cmn 19832  df-abl 19833  df-mgp 20197  df-rng 20209  df-ur 20242  df-srg 20247  df-ring 20295  df-cring 20296  df-rhm 20531  df-rim 20532  df-ric 20534  df-nzr 20573  df-subrng 20606  df-subrg 20630  df-rlreg 20754  df-domn 20755  df-idom 20756  df-lmod 20936  df-lss 21006  df-lsp 21046  df-cnfld 21432  df-assa 21912  df-asp 21913  df-ascl 21914  df-psr 21968  df-mvr 21969  df-mpl 21970  df-opsr 21972  df-evls 22134  df-evl 22135  df-selv 22177  df-psr1 22249  df-vr1 22250  df-ply1 22251  df-coe1 22252  df-mdeg 26122  df-deg1 26123

This theorem is referenced by:  mplidom  33827