Theorem mpfind 22154

Description: Prove a property of polynomials by "structural" induction, under a simplified model of structure which loses the sum of products structure. (Contributed by Mario Carneiro, 19-Mar-2015.)

Hypotheses

Ref	Expression
mpfind.cb	⊢ 𝐵 = (Base‘𝑆)
mpfind.cp	⊢ + = (+g‘𝑆)
mpfind.ct	⊢ · = (.r‘𝑆)
mpfind.cq	⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)
mpfind.ad	⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜁)
mpfind.mu	⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜎)
mpfind.wa	⊢ (𝑥 = ((𝐵 ↑m 𝐼) × {𝑓}) → (𝜓 ↔ 𝜒))
mpfind.wb	⊢ (𝑥 = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) → (𝜓 ↔ 𝜃))
mpfind.wc	⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏))
mpfind.wd	⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂))
mpfind.we	⊢ (𝑥 = (𝑓 ∘f + 𝑔) → (𝜓 ↔ 𝜁))
mpfind.wf	⊢ (𝑥 = (𝑓 ∘f · 𝑔) → (𝜓 ↔ 𝜎))
mpfind.wg	⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜌))
mpfind.co	⊢ ((𝜑 ∧ 𝑓 ∈ 𝑅) → 𝜒)
mpfind.pr	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐼) → 𝜃)
mpfind.a	⊢ (𝜑 → 𝐴 ∈ 𝑄)

Assertion

Ref	Expression
mpfind	⊢ (𝜑 → 𝜌)

Distinct variable groups:   𝜒,𝑥   𝜂,𝑥   𝜑,𝑓,𝑔   𝜓,𝑓,𝑔   𝜌,𝑥   𝜎,𝑥   𝜏,𝑥   𝜃,𝑥   𝜁,𝑥   𝑥,𝐴   𝐵,𝑓,𝑔,𝑥   𝑓,𝐼,𝑔,𝑥   + ,𝑓,𝑔,𝑥   𝑄,𝑓,𝑔   𝑅,𝑓,𝑔   𝑆,𝑓,𝑔   · ,𝑓,𝑔,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑓,𝑔)   𝜃(𝑓,𝑔)   𝜏(𝑓,𝑔)   𝜂(𝑓,𝑔)   𝜁(𝑓,𝑔)   𝜎(𝑓,𝑔)   𝜌(𝑓,𝑔)   𝐴(𝑓,𝑔)   𝑄(𝑥)   𝑅(𝑥)   𝑆(𝑥)

Proof of Theorem mpfind

Dummy variables 𝑖 𝑗 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpfind.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑄)
2		mpfind.cq	. . . . 5 ⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)
3	1, 2	eleqtrdi 2854	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
4	2	mpfrcl 22132	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑄 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))
5	1, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))
6		eqid 2740	. . . . . . . 8 ⊢ ((𝐼 evalSub 𝑆)‘𝑅) = ((𝐼 evalSub 𝑆)‘𝑅)
7		eqid 2740	. . . . . . . 8 ⊢ (𝐼 mPoly (𝑆 ↾s 𝑅)) = (𝐼 mPoly (𝑆 ↾s 𝑅))
8		eqid 2740	. . . . . . . 8 ⊢ (𝑆 ↾s 𝑅) = (𝑆 ↾s 𝑅)
9		eqid 2740	. . . . . . . 8 ⊢ (𝑆 ↑s (𝐵 ↑m 𝐼)) = (𝑆 ↑s (𝐵 ↑m 𝐼))
10		mpfind.cb	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑆)
11	6, 7, 8, 9, 10	evlsrhm 22135	. . . . . . 7 ⊢ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s (𝐵 ↑m 𝐼))))
12		eqid 2740	. . . . . . . 8 ⊢ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
13		eqid 2740	. . . . . . . 8 ⊢ (Base‘(𝑆 ↑s (𝐵 ↑m 𝐼))) = (Base‘(𝑆 ↑s (𝐵 ↑m 𝐼)))
14	12, 13	rhmf 20511	. . . . . . 7 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s (𝐵 ↑m 𝐼))) → ((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))⟶(Base‘(𝑆 ↑s (𝐵 ↑m 𝐼))))
15	5, 11, 14	3syl 18	. . . . . 6 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))⟶(Base‘(𝑆 ↑s (𝐵 ↑m 𝐼))))
16	15	ffnd 6748	. . . . 5 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
17		fvelrnb 6982	. . . . 5 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) → (𝐴 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅) ↔ ∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴))
18	16, 17	syl 17	. . . 4 ⊢ (𝜑 → (𝐴 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅) ↔ ∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴))
19	3, 18	mpbid 232	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴)
20	15	ffund 6751	. . . . . 6 ⊢ (𝜑 → Fun ((𝐼 evalSub 𝑆)‘𝑅))
21		eqid 2740	. . . . . . 7 ⊢ (Base‘(𝑆 ↾s 𝑅)) = (Base‘(𝑆 ↾s 𝑅))
22		eqid 2740	. . . . . . 7 ⊢ (𝐼 mVar (𝑆 ↾s 𝑅)) = (𝐼 mVar (𝑆 ↾s 𝑅))
23		eqid 2740	. . . . . . 7 ⊢ (+g‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
24		eqid 2740	. . . . . . 7 ⊢ (.r‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
25		eqid 2740	. . . . . . 7 ⊢ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
26	5	simp1d 1142	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ V)
27	5	simp2d 1143	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ CRing)
28	5	simp3d 1144	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
29	8	subrgcrng 20603	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑆 ↾s 𝑅) ∈ CRing)
30	27, 28, 29	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ↾s 𝑅) ∈ CRing)
31		crngring 20272	. . . . . . . . . . . . 13 ⊢ ((𝑆 ↾s 𝑅) ∈ CRing → (𝑆 ↾s 𝑅) ∈ Ring)
32	30, 31	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 ↾s 𝑅) ∈ Ring)
33	7, 26, 32	mplringd 22066	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ Ring)
34	33	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ Ring)
35		simprl 770	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
36		elpreima 7091	. . . . . . . . . . . . . 14 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) → (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓})))
37	16, 36	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓})))
38	37	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓})))
39	35, 38	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}))
40	39	simpld 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
41		simprr 772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
42		elpreima 7091	. . . . . . . . . . . . . 14 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) → (𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓})))
43	16, 42	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓})))
44	43	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓})))
45	41, 44	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))
46	45	simpld 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
47	12, 23	ringacl 20301	. . . . . . . . . 10 ⊢ (((𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ Ring ∧ 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
48	34, 40, 46, 47	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
49		rhmghm 20510	. . . . . . . . . . . . . 14 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s (𝐵 ↑m 𝐼))) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) GrpHom (𝑆 ↑s (𝐵 ↑m 𝐼))))
50	5, 11, 49	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) GrpHom (𝑆 ↑s (𝐵 ↑m 𝐼))))
51	50	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) GrpHom (𝑆 ↑s (𝐵 ↑m 𝐼))))
52		eqid 2740	. . . . . . . . . . . . 13 ⊢ (+g‘(𝑆 ↑s (𝐵 ↑m 𝐼))) = (+g‘(𝑆 ↑s (𝐵 ↑m 𝐼)))
53	12, 23, 52	ghmlin 19261	. . . . . . . . . . . 12 ⊢ ((((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) GrpHom (𝑆 ↑s (𝐵 ↑m 𝐼))) ∧ 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖)(+g‘(𝑆 ↑s (𝐵 ↑m 𝐼)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)))
54	51, 40, 46, 53	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖)(+g‘(𝑆 ↑s (𝐵 ↑m 𝐼)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)))
55	27	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝑆 ∈ CRing)
56		ovexd 7483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝐵 ↑m 𝐼) ∈ V)
57	15	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))⟶(Base‘(𝑆 ↑s (𝐵 ↑m 𝐼))))
58	57, 40	ffvelcdmd 7119	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ (Base‘(𝑆 ↑s (𝐵 ↑m 𝐼))))
59	57, 46	ffvelcdmd 7119	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ (Base‘(𝑆 ↑s (𝐵 ↑m 𝐼))))
60		mpfind.cp	. . . . . . . . . . . 12 ⊢ + = (+g‘𝑆)
61	9, 13, 55, 56, 58, 59, 60, 52	pwsplusgval 17550	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖)(+g‘(𝑆 ↑s (𝐵 ↑m 𝐼)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)))
62	54, 61	eqtrd 2780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)))
63		simpl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝜑)
64		fnfvelrn 7114	. . . . . . . . . . . . . 14 ⊢ ((((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
65	16, 40, 64	syl2an2r 684	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
66	65, 2	eleqtrrdi 2855	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄)
67		fvimacnvi 7085	. . . . . . . . . . . . 13 ⊢ ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓})
68	20, 35, 67	syl2an2r 684	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓})
69	66, 68	jca 511	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}))
70		fnfvelrn 7114	. . . . . . . . . . . . . 14 ⊢ ((((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
71	16, 46, 70	syl2an2r 684	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
72	71, 2	eleqtrrdi 2855	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄)
73		fvimacnvi 7085	. . . . . . . . . . . . 13 ⊢ ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓})
74	20, 41, 73	syl2an2r 684	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓})
75	72, 74	jca 511	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))
76		fvex 6933	. . . . . . . . . . . 12 ⊢ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ V
77		fvex 6933	. . . . . . . . . . . 12 ⊢ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ V
78		eleq1 2832	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝑓 ∈ 𝑄 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄))
79		vex 3492	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
80		mpfind.wc	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏))
81	79, 80	elab 3694	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ 𝜏)
82		eleq1 2832	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}))
83	81, 82	bitr3id 285	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝜏 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}))
84	78, 83	anbi12d 631	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → ((𝑓 ∈ 𝑄 ∧ 𝜏) ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓})))
85		eleq1 2832	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝑔 ∈ 𝑄 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄))
86		vex 3492	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑔 ∈ V
87		mpfind.wd	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂))
88	86, 87	elab 3694	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ 𝜂)
89		eleq1 2832	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))
90	88, 89	bitr3id 285	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝜂 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))
91	85, 90	anbi12d 631	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → ((𝑔 ∈ 𝑄 ∧ 𝜂) ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓})))
92	84, 91	bi2anan9 637	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → (((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂)) ↔ (((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}) ∧ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))))
93	92	anbi2d 629	. . . . . . . . . . . . 13 ⊢ ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) ↔ (𝜑 ∧ (((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}) ∧ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓})))))
94		ovex 7481	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∘f + 𝑔) ∈ V
95		mpfind.we	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 ∘f + 𝑔) → (𝜓 ↔ 𝜁))
96	94, 95	elab 3694	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜁)
97		oveq12 7457	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → (𝑓 ∘f + 𝑔) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)))
98	97	eleq1d 2829	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → ((𝑓 ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥 ∣ 𝜓}))
99	96, 98	bitr3id 285	. . . . . . . . . . . . 13 ⊢ ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → (𝜁 ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥 ∣ 𝜓}))
100	93, 99	imbi12d 344	. . . . . . . . . . . 12 ⊢ ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → (((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜁) ↔ ((𝜑 ∧ (((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}) ∧ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥 ∣ 𝜓})))
101		mpfind.ad	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜁)
102	76, 77, 100, 101	vtocl2 3578	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}) ∧ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥 ∣ 𝜓})
103	63, 69, 75, 102	syl12anc 836	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥 ∣ 𝜓})
104	62, 103	eqeltrd 2844	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) ∈ {𝑥 ∣ 𝜓})
105		elpreima 7091	. . . . . . . . . . 11 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) → ((𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ ((𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) ∈ {𝑥 ∣ 𝜓})))
106	16, 105	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ ((𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) ∈ {𝑥 ∣ 𝜓})))
107	106	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ ((𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) ∈ {𝑥 ∣ 𝜓})))
108	48, 104, 107	mpbir2and 712	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
109	108	adantlr 714	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝑖(+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
110	12, 24	ringcl 20277	. . . . . . . . . 10 ⊢ (((𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ Ring ∧ 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
111	34, 40, 46, 110	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
112		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
113		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘(𝑆 ↑s (𝐵 ↑m 𝐼))) = (mulGrp‘(𝑆 ↑s (𝐵 ↑m 𝐼)))
114	112, 113	rhmmhm 20505	. . . . . . . . . . . . . 14 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s (𝐵 ↑m 𝐼))) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))) MndHom (mulGrp‘(𝑆 ↑s (𝐵 ↑m 𝐼)))))
115	5, 11, 114	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))) MndHom (mulGrp‘(𝑆 ↑s (𝐵 ↑m 𝐼)))))
116	115	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))) MndHom (mulGrp‘(𝑆 ↑s (𝐵 ↑m 𝐼)))))
117	112, 12	mgpbas 20167	. . . . . . . . . . . . 13 ⊢ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (Base‘(mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
118	112, 24	mgpplusg 20165	. . . . . . . . . . . . 13 ⊢ (.r‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (+g‘(mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
119		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (.r‘(𝑆 ↑s (𝐵 ↑m 𝐼))) = (.r‘(𝑆 ↑s (𝐵 ↑m 𝐼)))
120	113, 119	mgpplusg 20165	. . . . . . . . . . . . 13 ⊢ (.r‘(𝑆 ↑s (𝐵 ↑m 𝐼))) = (+g‘(mulGrp‘(𝑆 ↑s (𝐵 ↑m 𝐼))))
121	117, 118, 120	mhmlin 18828	. . . . . . . . . . . 12 ⊢ ((((𝐼 evalSub 𝑆)‘𝑅) ∈ ((mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))) MndHom (mulGrp‘(𝑆 ↑s (𝐵 ↑m 𝐼)))) ∧ 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖)(.r‘(𝑆 ↑s (𝐵 ↑m 𝐼)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)))
122	116, 40, 46, 121	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖)(.r‘(𝑆 ↑s (𝐵 ↑m 𝐼)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)))
123		mpfind.ct	. . . . . . . . . . . 12 ⊢ · = (.r‘𝑆)
124	9, 13, 55, 56, 58, 59, 123, 119	pwsmulrval 17551	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖)(.r‘(𝑆 ↑s (𝐵 ↑m 𝐼)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)))
125	122, 124	eqtrd 2780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)))
126		ovex 7481	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∘f · 𝑔) ∈ V
127		mpfind.wf	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 ∘f · 𝑔) → (𝜓 ↔ 𝜎))
128	126, 127	elab 3694	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜎)
129		oveq12 7457	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → (𝑓 ∘f · 𝑔) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)))
130	129	eleq1d 2829	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → ((𝑓 ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥 ∣ 𝜓}))
131	128, 130	bitr3id 285	. . . . . . . . . . . . 13 ⊢ ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → (𝜎 ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥 ∣ 𝜓}))
132	93, 131	imbi12d 344	. . . . . . . . . . . 12 ⊢ ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → (((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜎) ↔ ((𝜑 ∧ (((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}) ∧ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥 ∣ 𝜓})))
133		mpfind.mu	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜎)
134	76, 77, 132, 133	vtocl2 3578	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}) ∧ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥 ∣ 𝜓})
135	63, 69, 75, 134	syl12anc 836	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘f · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥 ∣ 𝜓})
136	125, 135	eqeltrd 2844	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) ∈ {𝑥 ∣ 𝜓})
137		elpreima 7091	. . . . . . . . . . 11 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) → ((𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ ((𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) ∈ {𝑥 ∣ 𝜓})))
138	16, 137	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ ((𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) ∈ {𝑥 ∣ 𝜓})))
139	138	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ ((𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗)) ∈ {𝑥 ∣ 𝜓})))
140	111, 136, 139	mpbir2and 712	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
141	140	adantlr 714	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝑖(.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))𝑗) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
142	7	mplassa 22065	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ V ∧ (𝑆 ↾s 𝑅) ∈ CRing) → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ AssAlg)
143	26, 30, 142	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ AssAlg)
144		eqid 2740	. . . . . . . . . . . . 13 ⊢ (Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
145	25, 144	asclrhm 21933	. . . . . . . . . . . 12 ⊢ ((𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ AssAlg → (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∈ ((Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))) RingHom (𝐼 mPoly (𝑆 ↾s 𝑅))))
146		eqid 2740	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) = (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
147	146, 12	rhmf 20511	. . . . . . . . . . . 12 ⊢ ((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∈ ((Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))) RingHom (𝐼 mPoly (𝑆 ↾s 𝑅))) → (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))):(Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))⟶(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
148	143, 145, 147	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))):(Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))⟶(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
149	148	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))):(Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))⟶(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
150	7, 26, 30	mplsca 22056	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ↾s 𝑅) = (Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
151	150	fveq2d 6924	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘(𝑆 ↾s 𝑅)) = (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))))
152	151	eleq2d 2830	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑖 ∈ (Base‘(𝑆 ↾s 𝑅)) ↔ 𝑖 ∈ (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))))
153	152	biimpa 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → 𝑖 ∈ (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))))
154	149, 153	ffvelcdmd 7119	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
155	26	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → 𝐼 ∈ V)
156	27	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → 𝑆 ∈ CRing)
157	28	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → 𝑅 ∈ (SubRing‘𝑆))
158	10	subrgss 20600	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵)
159	8, 10	ressbas2 17296	. . . . . . . . . . . . . 14 ⊢ (𝑅 ⊆ 𝐵 → 𝑅 = (Base‘(𝑆 ↾s 𝑅)))
160	28, 158, 159	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 = (Base‘(𝑆 ↾s 𝑅)))
161	160	eleq2d 2830	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑖 ∈ 𝑅 ↔ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))))
162	161	biimpar 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → 𝑖 ∈ 𝑅)
163	6, 7, 8, 10, 25, 155, 156, 157, 162	evlssca 22136	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖)) = ((𝐵 ↑m 𝐼) × {𝑖}))
164		mpfind.co	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑅) → 𝜒)
165	164	ralrimiva 3152	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑓 ∈ 𝑅 𝜒)
166		ovex 7481	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ↑m 𝐼) ∈ V
167		vsnex 5449	. . . . . . . . . . . . . . . . 17 ⊢ {𝑓} ∈ V
168	166, 167	xpex 7788	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ↑m 𝐼) × {𝑓}) ∈ V
169		mpfind.wa	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ((𝐵 ↑m 𝐼) × {𝑓}) → (𝜓 ↔ 𝜒))
170	168, 169	elab 3694	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ↑m 𝐼) × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)
171		sneq 4658	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝑖 → {𝑓} = {𝑖})
172	171	xpeq2d 5730	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑖 → ((𝐵 ↑m 𝐼) × {𝑓}) = ((𝐵 ↑m 𝐼) × {𝑖}))
173	172	eleq1d 2829	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑖 → (((𝐵 ↑m 𝐼) × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ ((𝐵 ↑m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓}))
174	170, 173	bitr3id 285	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑖 → (𝜒 ↔ ((𝐵 ↑m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓}))
175	174	cbvralvw 3243	. . . . . . . . . . . . 13 ⊢ (∀𝑓 ∈ 𝑅 𝜒 ↔ ∀𝑖 ∈ 𝑅 ((𝐵 ↑m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓})
176	165, 175	sylib 218	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑖 ∈ 𝑅 ((𝐵 ↑m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓})
177	176	r19.21bi 3257	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑅) → ((𝐵 ↑m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓})
178	162, 177	syldan 590	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → ((𝐵 ↑m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓})
179	163, 178	eqeltrd 2844	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖)) ∈ {𝑥 ∣ 𝜓})
180		elpreima 7091	. . . . . . . . . . 11 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) → (((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖)) ∈ {𝑥 ∣ 𝜓})))
181	16, 180	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖)) ∈ {𝑥 ∣ 𝜓})))
182	181	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → (((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖)) ∈ {𝑥 ∣ 𝜓})))
183	154, 179, 182	mpbir2and 712	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
184	183	adantlr 714	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
185	26	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ V)
186	32	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑆 ↾s 𝑅) ∈ Ring)
187		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
188	7, 22, 12, 185, 186, 187	mvrcl 22035	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
189	27	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ CRing)
190	28	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑅 ∈ (SubRing‘𝑆))
191	6, 22, 8, 10, 185, 189, 190, 187	evlsvar 22137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑖)))
192		mpfind.pr	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐼) → 𝜃)
193	166	mptex 7260	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) ∈ V
194		mpfind.wb	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) → (𝜓 ↔ 𝜃))
195	193, 194	elab 3694	. . . . . . . . . . . . . 14 ⊢ ((𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓} ↔ 𝜃)
196	192, 195	sylibr 234	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐼) → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓})
197	196	ralrimiva 3152	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑓 ∈ 𝐼 (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓})
198		fveq2 6920	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑖 → (𝑔‘𝑓) = (𝑔‘𝑖))
199	198	mpteq2dv 5268	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑖 → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑖)))
200	199	eleq1d 2829	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑖 → ((𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓} ↔ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓}))
201	200	cbvralvw 3243	. . . . . . . . . . . 12 ⊢ (∀𝑓 ∈ 𝐼 (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓} ↔ ∀𝑖 ∈ 𝐼 (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓})
202	197, 201	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓})
203	202	r19.21bi 3257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓})
204	191, 203	eqeltrd 2844	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖)) ∈ {𝑥 ∣ 𝜓})
205		elpreima 7091	. . . . . . . . . . 11 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) → (((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖)) ∈ {𝑥 ∣ 𝜓})))
206	16, 205	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖)) ∈ {𝑥 ∣ 𝜓})))
207	206	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ↔ (((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖)) ∈ {𝑥 ∣ 𝜓})))
208	188, 204, 207	mpbir2and 712	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
209	208	adantlr 714	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) ∧ 𝑖 ∈ 𝐼) → ((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
210		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
211	26	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → 𝐼 ∈ V)
212	30	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (𝑆 ↾s 𝑅) ∈ CRing)
213	21, 22, 7, 23, 24, 25, 12, 109, 141, 184, 209, 210, 211, 212	mplind 22117	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → 𝑦 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
214		fvimacnvi 7085	. . . . . 6 ⊢ ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑦 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥 ∣ 𝜓})
215	20, 213, 214	syl2an2r 684	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥 ∣ 𝜓})
216		eleq1 2832	. . . . 5 ⊢ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴 → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥 ∣ 𝜓} ↔ 𝐴 ∈ {𝑥 ∣ 𝜓}))
217	215, 216	syl5ibcom 245	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴 → 𝐴 ∈ {𝑥 ∣ 𝜓}))
218	217	rexlimdva 3161	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴 → 𝐴 ∈ {𝑥 ∣ 𝜓}))
219	19, 218	mpd 15	. 2 ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜓})
220		mpfind.wg	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜌))
221	220	elabg 3690	. . 3 ⊢ (𝐴 ∈ 𝑄 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜌))
222	1, 221	syl 17	. 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜌))
223	219, 222	mpbid 232	1 ⊢ (𝜑 → 𝜌)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ⊆ wss 3976  {csn 4648   ↦ cmpt 5249   × cxp 5698  ^◡ccnv 5699  ran crn 5701   “ cima 5703  Fun wfun 6567   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∘_f cof 7712   ↑_m cmap 8884  Basecbs 17258   ↾_s cress 17287  +_gcplusg 17311  ._rcmulr 17312  Scalarcsca 17314   ↑_s cpws 17506   MndHom cmhm 18816   GrpHom cghm 19252  mulGrpcmgp 20161  Ringcrg 20260  CRingccrg 20261   RingHom crh 20495  SubRingcsubrg 20595  AssAlgcasa 21893  algSccascl 21895   mVar cmvr 21948   mPoly cmpl 21949   evalSub ces 22119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-ofr 7715  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-pm 8887  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-sup 9511  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-fzo 13712  df-seq 14053  df-hash 14380  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-hom 17335  df-cco 17336  df-0g 17501  df-gsum 17502  df-prds 17507  df-pws 17509  df-mre 17644  df-mrc 17645  df-acs 17647  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-mhm 18818  df-submnd 18819  df-grp 18976  df-minusg 18977  df-sbg 18978  df-mulg 19108  df-subg 19163  df-ghm 19253  df-cntz 19357  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-srg 20214  df-ring 20262  df-cring 20263  df-rhm 20498  df-subrng 20572  df-subrg 20597  df-lmod 20882  df-lss 20953  df-lsp 20993  df-assa 21896  df-asp 21897  df-ascl 21898  df-psr 21952  df-mvr 21953  df-mpl 21954  df-evls 22121

This theorem is referenced by:  pf1ind  22380  mzpmfp  42703