Theorem mplidomlem 34229

Description: Lemma for mplidom 34230. (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 7450	. . . 4 ⊢ (𝑖 = ∅ → (𝑖 mPoly 𝑅) = (∅ mPoly 𝑅))
3	2	eleq1d 2849	. . 3 ⊢ (𝑖 = ∅ → ((𝑖 mPoly 𝑅) ∈ IDomn ↔ (∅ mPoly 𝑅) ∈ IDomn))
4		oveq1 7450	. . . 4 ⊢ (𝑖 = 𝑗 → (𝑖 mPoly 𝑅) = (𝑗 mPoly 𝑅))
5	4	eleq1d 2849	. . 3 ⊢ (𝑖 = 𝑗 → ((𝑖 mPoly 𝑅) ∈ IDomn ↔ (𝑗 mPoly 𝑅) ∈ IDomn))
6		oveq1 7450	. . . . 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 2849	. . 3 ⊢ (𝑖 = (𝑗 ∪ {𝑥}) → ((𝑖 mPoly 𝑅) ∈ IDomn ↔ 𝑆 ∈ IDomn))
10		oveq1 7450	. . . 4 ⊢ (𝑖 = 𝐼 → (𝑖 mPoly 𝑅) = (𝐼 mPoly 𝑅))
11	10	eleq1d 2849	. . 3 ⊢ (𝑖 = 𝐼 → ((𝑖 mPoly 𝑅) ∈ IDomn ↔ (𝐼 mPoly 𝑅) ∈ IDomn))
12		eqid 2763	. . . . 5 ⊢ (∅ mPoly 𝑅) = (∅ mPoly 𝑅)
13		0ex 5264	. . . . . 6 ⊢ ∅ ∈ V
14	13	a1i 11	. . . . 5 ⊢ (𝜑 → ∅ ∈ V)
15		mplidom.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ IDomn)
16	15	idomcringd 21008	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing)
17	12, 14, 16	mplcrngd 22361	. . . 4 ⊢ (𝜑 → (∅ mPoly 𝑅) ∈ CRing)
18		eqid 2763	. . . . . 6 ⊢ (Base‘(∅ mPoly 𝑅)) = (Base‘(∅ mPoly 𝑅))
19	15	idomringd 21009	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring)
20	18, 12, 19	0mplric 34217	. . . . 5 ⊢ (𝜑 → (∅ mPoly 𝑅) ≃𝑟 𝑅)
21	15	idomdomd 21007	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Domn)
22		ricdomn 33921	. . . . . 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 21006	. . . 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 9288	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑗 ∈ Fin)
31		snfi 9099	. . . . . . . . 9 ⊢ {𝑥} ∈ Fin
32	31	a1i 11	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → {𝑥} ∈ Fin)
33	30, 32	unfid 9215	. . . . . . 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 22361	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑆 ∈ CRing)
36		domnnzr 20988	. . . . . . . . . 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 34215	. . . . . . 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 4624	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ {𝑥}
46		elun2 4184	. . . . . . . . . . . . . 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 2763	. . . . . . . . . . . 12 ⊢ (0g‘𝑄) = (0g‘𝑄)
51		eqid 2763	. . . . . . . . . . . 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 34228	. . . . . . . . . . 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 34228	. . . . . . . . . . 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 4055	. . . . . . . . . . . . . . . . . 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 4672	. . . . . . . . . . . . . . . . . 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 4443	. . . . . . . . . . . . . . . . 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 7458	. . . . . . . . . . . . . . 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 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 21007	. . . . . . . . . . . . . 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 2864	. . . . . . . . . . . . 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 26472	. . . . . . . . . . . . 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 34227	. . . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑄) = (Base‘𝑄)
78	40, 77	rhmf 20745	. . . . . . . . . . . . . 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 7109	. . . . . . . . . . . 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 7109	. . . . . . . . . . . 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 6912	. . . . . . . . . . . . 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 20750	. . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . 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 20744	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ (𝑆 RingHom 𝑄) → 𝐻 ∈ (𝑆 GrpHom 𝑄))
91	51, 50	ghmid 19465	. . . . . . . . . . . . . 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 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 20990	. . . . . . . . . . . . 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 1401	. . . . . . . . . . 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 3209	. . . . . . 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 20987	. . . . . . 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 21006	. . . . . 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 9210	. 2 ⊢ (𝜑 → (𝐼 mPoly 𝑅) ∈ IDomn)
108	1, 107	eqeltrid 2868	1 ⊢ (𝜑 → 𝑃 ∈ IDomn)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  {csn 4588  ⟨cop 4838   class class class wbr 5106   ↦ cmpt 5189  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443  1_oc1o 8497   ↑_m cmap 8875  Fincfn 9001  ℕ₀cn0 12633  Basecbs 17416  ._rcmulr 17458  0_gc0g 17639   GrpHom cghm 19456  CRingccrg 20490   RingHom crh 20729   ≃_𝑟 cric 20731  NzRingcnzr 20792  Domncdomn 20974  IDomncidom 20975   mPoly cmpl 22244   selectVars cslv 22455  Poly₁cpl1 22525

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304  ax-pre-sup 11305  ax-addf 11306

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-of 7717  df-ofr 7718  df-om 7902  df-1st 8025  df-2nd 8026  df-supp 8205  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-er 8745  df-map 8877  df-pm 8878  df-ixp 8954  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-fsupp 9382  df-sup 9461  df-oi 9531  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-z 12721  df-dec 12842  df-uz 12993  df-fz 13667  df-fzo 13816  df-seq 14172  df-hash 14502  df-struct 17354  df-sets 17371  df-slot 17389  df-ndx 17401  df-base 17417  df-ress 17438  df-plusg 17470  df-mulr 17471  df-starv 17472  df-sca 17473  df-vsca 17474  df-ip 17475  df-tset 17476  df-ple 17477  df-ds 17479  df-unif 17480  df-hom 17481  df-cco 17482  df-0g 17641  df-gsum 17642  df-prds 17647  df-pws 17649  df-mre 17785  df-mrc 17786  df-acs 17788  df-mgm 18845  df-sgrp 18937  df-mnd 18953  df-mhm 19007  df-submnd 19008  df-grp 19176  df-minusg 19177  df-sbg 19178  df-mulg 19307  df-subg 19362  df-ghm 19457  df-cntz 19561  df-cmn 20026  df-abl 20027  df-mgp 20391  df-rng 20405  df-ur 20438  df-srg 20443  df-ring 20491  df-cring 20492  df-rhm 20732  df-rim 20733  df-ric 20773  df-nzr 20793  df-subrng 20828  df-subrg 20852  df-rlreg 20976  df-domn 20977  df-idom 20978  df-lmod 21167  df-lss 21237  df-lsp 21277  df-cnfld 21709  df-assa 22191  df-asp 22192  df-ascl 22193  df-psr 22247  df-mvr 22248  df-mpl 22249  df-opsr 22251  df-evls 22413  df-evl 22414  df-selv 22456  df-psr1 22528  df-vr1 22529  df-ply1 22530  df-coe1 22531  df-mdeg 26403  df-deg1 26404

This theorem is used by:  mplidom  34230