Theorem mplidomlem 34092

Description: Lemma for mplidom 34093. (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 7415	. . . 4 ⊢ (𝑖 = ∅ → (𝑖 mPoly 𝑅) = (∅ mPoly 𝑅))
3	2	eleq1d 2845	. . 3 ⊢ (𝑖 = ∅ → ((𝑖 mPoly 𝑅) ∈ IDomn ↔ (∅ mPoly 𝑅) ∈ IDomn))
4		oveq1 7415	. . . 4 ⊢ (𝑖 = 𝑗 → (𝑖 mPoly 𝑅) = (𝑗 mPoly 𝑅))
5	4	eleq1d 2845	. . 3 ⊢ (𝑖 = 𝑗 → ((𝑖 mPoly 𝑅) ∈ IDomn ↔ (𝑗 mPoly 𝑅) ∈ IDomn))
6		oveq1 7415	. . . . 5 ⊢ (𝑖 = (𝑗 ∪ {𝑥}) → (𝑖 mPoly 𝑅) = ((𝑗 ∪ {𝑥}) mPoly 𝑅))
7		mplidomlem.s	. . . . 5 ⊢ 𝑆 = ((𝑗 ∪ {𝑥}) mPoly 𝑅)
8	6, 7	eqtr4di 2813	. . . 4 ⊢ (𝑖 = (𝑗 ∪ {𝑥}) → (𝑖 mPoly 𝑅) = 𝑆)
9	8	eleq1d 2845	. . 3 ⊢ (𝑖 = (𝑗 ∪ {𝑥}) → ((𝑖 mPoly 𝑅) ∈ IDomn ↔ 𝑆 ∈ IDomn))
10		oveq1 7415	. . . 4 ⊢ (𝑖 = 𝐼 → (𝑖 mPoly 𝑅) = (𝐼 mPoly 𝑅))
11	10	eleq1d 2845	. . 3 ⊢ (𝑖 = 𝐼 → ((𝑖 mPoly 𝑅) ∈ IDomn ↔ (𝐼 mPoly 𝑅) ∈ IDomn))
12		eqid 2760	. . . . 5 ⊢ (∅ mPoly 𝑅) = (∅ mPoly 𝑅)
13		0ex 5260	. . . . . 6 ⊢ ∅ ∈ V
14	13	a1i 11	. . . . 5 ⊢ (𝜑 → ∅ ∈ V)
15		mplidom.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ IDomn)
16	15	idomcringd 20939	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing)
17	12, 14, 16	mplcrngd 22292	. . . 4 ⊢ (𝜑 → (∅ mPoly 𝑅) ∈ CRing)
18		eqid 2760	. . . . . 6 ⊢ (Base‘(∅ mPoly 𝑅)) = (Base‘(∅ mPoly 𝑅))
19	15	idomringd 20940	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring)
20	18, 12, 19	0mplric 34080	. . . . 5 ⊢ (𝜑 → (∅ mPoly 𝑅) ≃𝑟 𝑅)
21	15	idomdomd 20938	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Domn)
22		ricdomn 33784	. . . . . 6 ⊢ ((∅ mPoly 𝑅) ≃𝑟 𝑅 → ((∅ mPoly 𝑅) ∈ Domn ↔ 𝑅 ∈ Domn))
23	22	biimpar 483	. . . . 5 ⊢ (((∅ mPoly 𝑅) ≃𝑟 𝑅 ∧ 𝑅 ∈ Domn) → (∅ mPoly 𝑅) ∈ Domn)
24	20, 21, 23	syl2anc 596	. . . 4 ⊢ (𝜑 → (∅ mPoly 𝑅) ∈ Domn)
25		isidom 20937	. . . 4 ⊢ ((∅ mPoly 𝑅) ∈ IDomn ↔ ((∅ mPoly 𝑅) ∈ CRing ∧ (∅ mPoly 𝑅) ∈ Domn))
26	17, 24, 25	sylanbrc 595	. . 3 ⊢ (𝜑 → (∅ mPoly 𝑅) ∈ IDomn)
27		mplidom.i	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ Fin)
28	27	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝐼 ∈ Fin)
29		simpllr 788	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑗 ⊆ 𝐼)
30	28, 29	ssfid 9238	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑗 ∈ Fin)
31		snfi 9049	. . . . . . . . 9 ⊢ {𝑥} ∈ Fin
32	31	a1i 11	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → {𝑥} ∈ Fin)
33	30, 32	unfid 9165	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → (𝑗 ∪ {𝑥}) ∈ Fin)
34	16	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑅 ∈ CRing)
35	7, 33, 34	mplcrngd 22292	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑆 ∈ CRing)
36		domnnzr 20919	. . . . . . . . . 10 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
37	21, 36	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ NzRing)
38	37	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑅 ∈ NzRing)
39	7, 33, 38	mplnzr 34078	. . . . . . 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 745	. . . . . . . . . . . 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 4128	. . . . . . . . . . . . . 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 745	. . . . . . . . . . . 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 2760	. . . . . . . . . . . 12 ⊢ (0g‘𝑄) = (0g‘𝑄)
51		eqid 2760	. . . . . . . . . . . 12 ⊢ (0g‘𝑆) = (0g‘𝑆)
52		simp-4r 796	. . . . . . . . . . . 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 490	. . . . . . . . . . . 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 34091	. . . . . . . . . . 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 745	. . . . . . . . . . . 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 745	. . . . . . . . . . . 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 788	. . . . . . . . . . . 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 490	. . . . . . . . . . . 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 34091	. . . . . . . . . . 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 798	. . . . . . . . . . . . . . . . . . 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 3911	. . . . . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . 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 7423	. . . . . . . . . . . . . . 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 2807	. . . . . . . . . . . . . 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 796	. . . . . . . . . . . . . . 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 20938	. . . . . . . . . . . . . 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 2860	. . . . . . . . . . . . 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 26403	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ Domn → 𝑄 ∈ Domn)
73	71, 72	syl 18	. . . . . . . . . . . 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 34090	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝐻 ∈ (𝑆 RingHom 𝑄))
76	75	ad3antrrr 743	. . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑄) = (Base‘𝑄)
78	40, 77	rhmf 20676	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ (𝑆 RingHom 𝑄) → 𝐻:𝐶⟶(Base‘𝑄))
79	76, 78	syl 18	. . . . . . . . . . . . 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 788	. . . . . . . . . . . . 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 7073	. . . . . . . . . . . 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 781	. . . . . . . . . . . . 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 7073	. . . . . . . . . . . 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 490	. . . . . . . . . . . . . 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 6877	. . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑆) = (.r‘𝑆)
87		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑄) = (.r‘𝑄)
88	40, 86, 87	rhmmul 20681	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 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 20675	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ (𝑆 RingHom 𝑄) → 𝐻 ∈ (𝑆 GrpHom 𝑄))
91	51, 50	ghmid 19397	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ (𝑆 GrpHom 𝑄) → (𝐻‘(0g‘𝑆)) = (0g‘𝑄))
92	76, 90, 91	3syl 19	. . . . . . . . . . . . 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 2803	. . . . . . . . . . . 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 20921	. . . . . . . . . . . . 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 482	. . . . . . . . . . . 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 1400	. . . . . . . . . . 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 980	. . . . . . . . . 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 418	. . . . . . . . 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 472	. . . . . . . 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 3205	. . . . . . 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 20918	. . . . . . 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 595	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑆 ∈ Domn)
103		isidom 20937	. . . . . 6 ⊢ (𝑆 ∈ IDomn ↔ (𝑆 ∈ CRing ∧ 𝑆 ∈ Domn))
104	35, 102, 103	sylanbrc 595	. . . . 5 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑆 ∈ IDomn)
105	104	ex 418	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) → ((𝑗 mPoly 𝑅) ∈ IDomn → 𝑆 ∈ IDomn))
106	105	anasss 472	. . 3 ⊢ ((𝜑 ∧ (𝑗 ⊆ 𝐼 ∧ 𝑥 ∈ (𝐼 ∖ 𝑗))) → ((𝑗 mPoly 𝑅) ∈ IDomn → 𝑆 ∈ IDomn))
107	3, 5, 9, 11, 26, 106, 27	findcard2d 9160	. 2 ⊢ (𝜑 → (𝐼 mPoly 𝑅) ∈ IDomn)
108	1, 107	eqeltrid 2864	1 ⊢ (𝜑 → 𝑃 ∈ IDomn)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ∖ cdif 3895   ∪ cun 3896   ∩ cin 3897   ⊆ wss 3898  ∅c0 4278  {csn 4583  ⟨cop 4589   class class class wbr 5102   ↦ cmpt 5185  ⟶wf 6523  ‘cfv 6527  (class class class)co 7408  1_oc1o 8447   ↑_m cmap 8825  Fincfn 8951  ℕ₀cn0 12575  Basecbs 17348  ._rcmulr 17390  0_gc0g 17571   GrpHom cghm 19388  CRingccrg 20421   RingHom crh 20660   ≃_𝑟 cric 20662  NzRingcnzr 20723  Domncdomn 20905  IDomncidom 20906   mPoly cmpl 22175   selectVars cslv 22386  Poly₁cpl1 22456

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11227  ax-resscn 11228  ax-1cn 11229  ax-icn 11230  ax-addcl 11231  ax-addrcl 11232  ax-mulcl 11233  ax-mulrcl 11234  ax-mulcom 11235  ax-addass 11236  ax-mulass 11237  ax-distr 11238  ax-i2m1 11239  ax-1ne0 11240  ax-1rid 11241  ax-rnegex 11242  ax-rrecex 11243  ax-cnre 11244  ax-pre-lttri 11245  ax-pre-lttrn 11246  ax-pre-ltadd 11247  ax-pre-mulgt0 11248  ax-pre-sup 11249  ax-addf 11250

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  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 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-isom 6536  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7676  df-ofr 7677  df-om 7861  df-1st 7984  df-2nd 7985  df-supp 8156  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-2o 8455  df-er 8695  df-map 8827  df-pm 8828  df-ixp 8904  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-fsupp 9332  df-sup 9412  df-oi 9482  df-card 9991  df-pnf 11316  df-mnf 11317  df-xr 11318  df-ltxr 11319  df-le 11320  df-sub 11514  df-neg 11515  df-nn 12305  df-2 12374  df-3 12375  df-4 12376  df-5 12377  df-6 12378  df-7 12379  df-8 12380  df-9 12381  df-n0 12576  df-z 12663  df-dec 12784  df-uz 12935  df-fz 13609  df-fzo 13757  df-seq 14113  df-hash 14442  df-struct 17286  df-sets 17303  df-slot 17321  df-ndx 17333  df-base 17349  df-ress 17370  df-plusg 17402  df-mulr 17403  df-starv 17404  df-sca 17405  df-vsca 17406  df-ip 17407  df-tset 17408  df-ple 17409  df-ds 17411  df-unif 17412  df-hom 17413  df-cco 17414  df-0g 17573  df-gsum 17574  df-prds 17579  df-pws 17581  df-mre 17717  df-mrc 17718  df-acs 17720  df-mgm 18777  df-sgrp 18869  df-mnd 18885  df-mhm 18939  df-submnd 18940  df-grp 19108  df-minusg 19109  df-sbg 19110  df-mulg 19239  df-subg 19294  df-ghm 19389  df-cntz 19492  df-cmn 19957  df-abl 19958  df-mgp 20322  df-rng 20336  df-ur 20369  df-srg 20374  df-ring 20422  df-cring 20423  df-rhm 20663  df-rim 20664  df-ric 20704  df-nzr 20724  df-subrng 20759  df-subrg 20783  df-rlreg 20907  df-domn 20908  df-idom 20909  df-lmod 21098  df-lss 21168  df-lsp 21208  df-cnfld 21640  df-assa 22122  df-asp 22123  df-ascl 22124  df-psr 22178  df-mvr 22179  df-mpl 22180  df-opsr 22182  df-evls 22344  df-evl 22345  df-selv 22387  df-psr1 22459  df-vr1 22460  df-ply1 22461  df-coe1 22462  df-mdeg 26334  df-deg1 26335

This theorem is used by:  mplidom  34093