Theorem mplidomlem 33768

Description: Lemma for mplidom 33769. (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 7388	. . . 4 ⊢ (𝑖 = ∅ → (𝑖 mPoly 𝑅) = (∅ mPoly 𝑅))
3	2	eleq1d 2837	. . 3 ⊢ (𝑖 = ∅ → ((𝑖 mPoly 𝑅) ∈ IDomn ↔ (∅ mPoly 𝑅) ∈ IDomn))
4		oveq1 7388	. . . 4 ⊢ (𝑖 = 𝑗 → (𝑖 mPoly 𝑅) = (𝑗 mPoly 𝑅))
5	4	eleq1d 2837	. . 3 ⊢ (𝑖 = 𝑗 → ((𝑖 mPoly 𝑅) ∈ IDomn ↔ (𝑗 mPoly 𝑅) ∈ IDomn))
6		oveq1 7388	. . . . 5 ⊢ (𝑖 = (𝑗 ∪ {𝑥}) → (𝑖 mPoly 𝑅) = ((𝑗 ∪ {𝑥}) mPoly 𝑅))
7		mplidomlem.s	. . . . 5 ⊢ 𝑆 = ((𝑗 ∪ {𝑥}) mPoly 𝑅)
8	6, 7	eqtr4di 2805	. . . 4 ⊢ (𝑖 = (𝑗 ∪ {𝑥}) → (𝑖 mPoly 𝑅) = 𝑆)
9	8	eleq1d 2837	. . 3 ⊢ (𝑖 = (𝑗 ∪ {𝑥}) → ((𝑖 mPoly 𝑅) ∈ IDomn ↔ 𝑆 ∈ IDomn))
10		oveq1 7388	. . . 4 ⊢ (𝑖 = 𝐼 → (𝑖 mPoly 𝑅) = (𝐼 mPoly 𝑅))
11	10	eleq1d 2837	. . 3 ⊢ (𝑖 = 𝐼 → ((𝑖 mPoly 𝑅) ∈ IDomn ↔ (𝐼 mPoly 𝑅) ∈ IDomn))
12		eqid 2752	. . . . 5 ⊢ (∅ mPoly 𝑅) = (∅ mPoly 𝑅)
13		0ex 5247	. . . . . 6 ⊢ ∅ ∈ V
14	13	a1i 11	. . . . 5 ⊢ (𝜑 → ∅ ∈ V)
15		mplidom.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ IDomn)
16	15	idomcringd 20745	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing)
17	12, 14, 16	mplcrngd 22044	. . . 4 ⊢ (𝜑 → (∅ mPoly 𝑅) ∈ CRing)
18		eqid 2752	. . . . . 6 ⊢ (Base‘(∅ mPoly 𝑅)) = (Base‘(∅ mPoly 𝑅))
19	15	idomringd 20746	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring)
20	18, 12, 19	0mplric 33756	. . . . 5 ⊢ (𝜑 → (∅ mPoly 𝑅) ≃𝑟 𝑅)
21	15	idomdomd 20744	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Domn)
22		ricdomn 33420	. . . . . 6 ⊢ ((∅ mPoly 𝑅) ≃𝑟 𝑅 → ((∅ mPoly 𝑅) ∈ Domn ↔ 𝑅 ∈ Domn))
23	22	biimpar 480	. . . . 5 ⊢ (((∅ mPoly 𝑅) ≃𝑟 𝑅 ∧ 𝑅 ∈ Domn) → (∅ mPoly 𝑅) ∈ Domn)
24	20, 21, 23	syl2anc 592	. . . 4 ⊢ (𝜑 → (∅ mPoly 𝑅) ∈ Domn)
25		isidom 20743	. . . 4 ⊢ ((∅ mPoly 𝑅) ∈ IDomn ↔ ((∅ mPoly 𝑅) ∈ CRing ∧ (∅ mPoly 𝑅) ∈ Domn))
26	17, 24, 25	sylanbrc 591	. . 3 ⊢ (𝜑 → (∅ mPoly 𝑅) ∈ IDomn)
27		mplidom.i	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ Fin)
28	27	ad3antrrr 738	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝐼 ∈ Fin)
29		simpllr 783	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑗 ⊆ 𝐼)
30	28, 29	ssfid 9198	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑗 ∈ Fin)
31		snfi 9009	. . . . . . . . 9 ⊢ {𝑥} ∈ Fin
32	31	a1i 11	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → {𝑥} ∈ Fin)
33	30, 32	unfid 9125	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → (𝑗 ∪ {𝑥}) ∈ Fin)
34	16	ad3antrrr 738	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑅 ∈ CRing)
35	7, 33, 34	mplcrngd 22044	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑆 ∈ CRing)
36		domnnzr 20724	. . . . . . . . . 10 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
37	21, 36	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ NzRing)
38	37	ad3antrrr 738	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑅 ∈ NzRing)
39	7, 33, 38	mplnzr 33754	. . . . . . 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 740	. . . . . . . . . . . 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 4612	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ {𝑥}
46		elun2 4126	. . . . . . . . . . . . . 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 740	. . . . . . . . . . . 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 2752	. . . . . . . . . . . 12 ⊢ (0g‘𝑄) = (0g‘𝑄)
51		eqid 2752	. . . . . . . . . . . 12 ⊢ (0g‘𝑆) = (0g‘𝑆)
52		simp-4r 791	. . . . . . . . . . . 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 487	. . . . . . . . . . . 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 33767	. . . . . . . . . . 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 740	. . . . . . . . . . . 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 740	. . . . . . . . . . . 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 783	. . . . . . . . . . . 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 487	. . . . . . . . . . . 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 33767	. . . . . . . . . . 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 793	. . . . . . . . . . . . . . . . . . 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 3908	. . . . . . . . . . . . . . . . . 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 4660	. . . . . . . . . . . . . . . . . 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 4428	. . . . . . . . . . . . . . . . 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 7396	. . . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . . 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 791	. . . . . . . . . . . . . . 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 20744	. . . . . . . . . . . . . 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 2852	. . . . . . . . . . . . 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 26153	. . . . . . . . . . . . 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 33766	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝐻 ∈ (𝑆 RingHom 𝑄))
76	75	ad3antrrr 738	. . . . . . . . . . . . . 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 2752	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑄) = (Base‘𝑄)
78	40, 77	rhmf 20501	. . . . . . . . . . . . . 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 783	. . . . . . . . . . . . 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 7051	. . . . . . . . . . . 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 776	. . . . . . . . . . . . 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 7051	. . . . . . . . . . . 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 487	. . . . . . . . . . . . . 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 6856	. . . . . . . . . . . . 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 2752	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑆) = (.r‘𝑆)
87		eqid 2752	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑄) = (.r‘𝑄)
88	40, 86, 87	rhmmul 20503	. . . . . . . . . . . . . 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 1382	. . . . . . . . . . . . 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 20500	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ (𝑆 RingHom 𝑄) → 𝐻 ∈ (𝑆 GrpHom 𝑄))
91	51, 50	ghmid 19234	. . . . . . . . . . . . . 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 2795	. . . . . . . . . . . 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 20726	. . . . . . . . . . . . 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 479	. . . . . . . . . . . 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 1384	. . . . . . . . . . 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 32594	. . . . . . . . . 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 415	. . . . . . . . 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 469	. . . . . . . 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 3195	. . . . . . 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 20723	. . . . . . 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 591	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑆 ∈ Domn)
103		isidom 20743	. . . . . 6 ⊢ (𝑆 ∈ IDomn ↔ (𝑆 ∈ CRing ∧ 𝑆 ∈ Domn))
104	35, 102, 103	sylanbrc 591	. . . . 5 ⊢ ((((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) ∧ (𝑗 mPoly 𝑅) ∈ IDomn) → 𝑆 ∈ IDomn)
105	104	ex 415	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ⊆ 𝐼) ∧ 𝑥 ∈ (𝐼 ∖ 𝑗)) → ((𝑗 mPoly 𝑅) ∈ IDomn → 𝑆 ∈ IDomn))
106	105	anasss 469	. . 3 ⊢ ((𝜑 ∧ (𝑗 ⊆ 𝐼 ∧ 𝑥 ∈ (𝐼 ∖ 𝑗))) → ((𝑗 mPoly 𝑅) ∈ IDomn → 𝑆 ∈ IDomn))
107	3, 5, 9, 11, 26, 106, 27	findcard2d 9120	. 2 ⊢ (𝜑 → (𝐼 mPoly 𝑅) ∈ IDomn)
108	1, 107	eqeltrid 2856	1 ⊢ (𝜑 → 𝑃 ∈ IDomn)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 856   ∧ w3a 1095   = wceq 1550   ∈ wcel 2132  ∀wral 3066  Vcvv 3444   ∖ cdif 3892   ∪ cun 3893   ∩ cin 3894   ⊆ wss 3895  ∅c0 4276  {csn 4572  ⟨cop 4578   class class class wbr 5090   ↦ cmpt 5171  ⟶wf 6502  ‘cfv 6506  (class class class)co 7381  1_oc1o 8414   ↑_m cmap 8792  Fincfn 8912  ℕ₀cn0 12467  Basecbs 17217  ._rcmulr 17259  0_gc0g 17440   GrpHom cghm 19225  CRingccrg 20252   RingHom crh 20486   ≃_𝑟 cric 20488  NzRingcnzr 20530  Domncdomn 20710  IDomncidom 20711   mPoly cmpl 21927   selectVars cslv 22138  Poly₁cpl1 22208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137  ax-addf 11138

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-nel 3052  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-tp 4577  df-op 4579  df-uni 4856  df-int 4896  df-iun 4941  df-iin 4942  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-se 5590  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-isom 6515  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-of 7645  df-ofr 7646  df-om 7832  df-1st 7955  df-2nd 7956  df-supp 8125  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-1o 8421  df-2o 8422  df-er 8662  df-map 8794  df-pm 8795  df-ixp 8865  df-en 8913  df-dom 8914  df-sdom 8915  df-fin 8916  df-fsupp 9294  df-sup 9374  df-oi 9444  df-card 9883  df-pnf 11204  df-mnf 11205  df-xr 11206  df-ltxr 11207  df-le 11208  df-sub 11402  df-neg 11403  df-nn 12197  df-2 12266  df-3 12267  df-4 12268  df-5 12269  df-6 12270  df-7 12271  df-8 12272  df-9 12273  df-n0 12468  df-z 12555  df-dec 12675  df-uz 12826  df-fz 13499  df-fzo 13646  df-seq 14001  df-hash 14330  df-struct 17155  df-sets 17172  df-slot 17190  df-ndx 17202  df-base 17218  df-ress 17239  df-plusg 17271  df-mulr 17272  df-starv 17273  df-sca 17274  df-vsca 17275  df-ip 17276  df-tset 17277  df-ple 17278  df-ds 17280  df-unif 17281  df-hom 17282  df-cco 17283  df-0g 17442  df-gsum 17443  df-prds 17448  df-pws 17450  df-mre 17586  df-mrc 17587  df-acs 17589  df-mgm 18646  df-sgrp 18725  df-mnd 18741  df-mhm 18789  df-submnd 18790  df-grp 18950  df-minusg 18951  df-sbg 18952  df-mulg 19082  df-subg 19137  df-ghm 19226  df-cntz 19329  df-cmn 19794  df-abl 19795  df-mgp 20159  df-rng 20171  df-ur 20200  df-srg 20205  df-ring 20253  df-cring 20254  df-rhm 20489  df-rim 20490  df-ric 20492  df-nzr 20531  df-subrng 20564  df-subrg 20588  df-rlreg 20712  df-domn 20713  df-idom 20714  df-lmod 20898  df-lss 20968  df-lsp 21008  df-cnfld 21394  df-assa 21874  df-asp 21875  df-ascl 21876  df-psr 21930  df-mvr 21931  df-mpl 21932  df-opsr 21934  df-evls 22096  df-evl 22097  df-selv 22139  df-psr1 22211  df-vr1 22212  df-ply1 22213  df-coe1 22214  df-mdeg 26084  df-deg1 26085

This theorem is referenced by:  mplidom  33769