Theorem mpfind 22360

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 2870	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
4	2	mpfrcl 22330	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑄 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))
5	1, 4	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))
6		eqid 2760	. . . . . . . 8 ⊢ ((𝐼 evalSub 𝑆)‘𝑅) = ((𝐼 evalSub 𝑆)‘𝑅)
7		eqid 2760	. . . . . . . 8 ⊢ (𝐼 mPoly (𝑆 ↾s 𝑅)) = (𝐼 mPoly (𝑆 ↾s 𝑅))
8		eqid 2760	. . . . . . . 8 ⊢ (𝑆 ↾s 𝑅) = (𝑆 ↾s 𝑅)
9		eqid 2760	. . . . . . . 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 22333	. . . . . . 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 2760	. . . . . . . 8 ⊢ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
13		eqid 2760	. . . . . . . 8 ⊢ (Base‘(𝑆 ↑s (𝐵 ↑m 𝐼))) = (Base‘(𝑆 ↑s (𝐵 ↑m 𝐼)))
14	12, 13	rhmf 20651	. . . . . . 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 19	. . . . . 6 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))⟶(Base‘(𝑆 ↑s (𝐵 ↑m 𝐼))))
16	15	ffnd 6705	. . . . 5 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
17		fvelrnb 6940	. . . . 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 18	. . . 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 235	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴)
20	15	ffund 6709	. . . . . 6 ⊢ (𝜑 → Fun ((𝐼 evalSub 𝑆)‘𝑅))
21		eqid 2760	. . . . . . 7 ⊢ (Base‘(𝑆 ↾s 𝑅)) = (Base‘(𝑆 ↾s 𝑅))
22		eqid 2760	. . . . . . 7 ⊢ (𝐼 mVar (𝑆 ↾s 𝑅)) = (𝐼 mVar (𝑆 ↾s 𝑅))
23		eqid 2760	. . . . . . 7 ⊢ (+g‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
24		eqid 2760	. . . . . . 7 ⊢ (.r‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
25		eqid 2760	. . . . . . 7 ⊢ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
26	5	simp1d 1160	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ V)
27	5	simp2d 1161	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ CRing)
28	5	simp3d 1162	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
29	8	subrgcrng 20763	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑆 ↾s 𝑅) ∈ CRing)
30	27, 28, 29	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ↾s 𝑅) ∈ CRing)
31		crngring 20408	. . . . . . . . . . . . 13 ⊢ ((𝑆 ↾s 𝑅) ∈ CRing → (𝑆 ↾s 𝑅) ∈ Ring)
32	30, 31	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 ↾s 𝑅) ∈ Ring)
33	7, 26, 32	mplringd 22266	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ Ring)
34	33	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ Ring)
35		simprl 783	. . . . . . . . . . . 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 7052	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 235	. . . . . . . . . . 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 500	. . . . . . . . . 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 785	. . . . . . . . . . . 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 7052	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 235	. . . . . . . . . . 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 500	. . . . . . . . . 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 20443	. . . . . . . . . 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 1398	. . . . . . . . 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 20650	. . . . . . . . . . . . . 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 19	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) GrpHom (𝑆 ↑s (𝐵 ↑m 𝐼))))
51	50	adantr 486	. . . . . . . . . . . 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 2760	. . . . . . . . . . . . 13 ⊢ (+g‘(𝑆 ↑s (𝐵 ↑m 𝐼))) = (+g‘(𝑆 ↑s (𝐵 ↑m 𝐼)))
53	12, 23, 52	ghmlin 19371	. . . . . . . . . . . 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 1398	. . . . . . . . . . 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 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝑆 ∈ CRing)
56		ovexd 7450	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝐵 ↑m 𝐼) ∈ V)
57	15	adantr 486	. . . . . . . . . . . . 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 7080	. . . . . . . . . . . 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 7080	. . . . . . . . . . . 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 17598	. . . . . . . . . . 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 2795	. . . . . . . . . 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 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝜑)
64		fnfvelrn 7075	. . . . . . . . . . . . . 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 698	. . . . . . . . . . . . 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 2871	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄)
67		fvimacnvi 7046	. . . . . . . . . . . . 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 698	. . . . . . . . . . . 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 521	. . . . . . . . . . 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 7075	. . . . . . . . . . . . . 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 698	. . . . . . . . . . . . 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 2871	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄)
73		fvimacnvi 7046	. . . . . . . . . . . . 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 698	. . . . . . . . . . . 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 521	. . . . . . . . . . 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 6893	. . . . . . . . . . . 12 ⊢ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ V
77		fvex 6893	. . . . . . . . . . . 12 ⊢ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ V
78		eleq1 2848	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝑓 ∈ 𝑄 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄))
79		vex 3454	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
80		mpfind.wc	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏))
81	79, 80	elab 3633	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ 𝜏)
82		eleq1 2848	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}))
83	81, 82	bitr3id 288	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝜏 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}))
84	78, 83	anbi12d 644	. . . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝑔 ∈ 𝑄 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄))
86		vex 3454	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑔 ∈ V
87		mpfind.wd	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂))
88	86, 87	elab 3633	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ 𝜂)
89		eleq1 2848	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))
90	88, 89	bitr3id 288	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝜂 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))
91	85, 90	anbi12d 644	. . . . . . . . . . . . . . 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 650	. . . . . . . . . . . . . 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 642	. . . . . . . . . . . . 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 7448	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∘f + 𝑔) ∈ V
95		mpfind.we	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 ∘f + 𝑔) → (𝜓 ↔ 𝜁))
96	94, 95	elab 3633	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜁)
97		oveq12 7424	. . . . . . . . . . . . . . 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 2845	. . . . . . . . . . . . . 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 288	. . . . . . . . . . . . 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 347	. . . . . . . . . . . 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 3526	. . . . . . . . . . 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 850	. . . . . . . . . 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 2860	. . . . . . . . 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 7052	. . . . . . . . . . 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 18	. . . . . . . . . 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 486	. . . . . . . . 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 726	. . . . . . . 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 728	. . . . . . 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 20413	. . . . . . . . . 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 1398	. . . . . . . . 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 2760	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
113		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘(𝑆 ↑s (𝐵 ↑m 𝐼))) = (mulGrp‘(𝑆 ↑s (𝐵 ↑m 𝐼)))
114	112, 113	rhmmhm 20646	. . . . . . . . . . . . . 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 19	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))) MndHom (mulGrp‘(𝑆 ↑s (𝐵 ↑m 𝐼)))))
116	115	adantr 486	. . . . . . . . . . . 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 20301	. . . . . . . . . . . . 13 ⊢ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (Base‘(mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
118	112, 24	mgpplusg 20300	. . . . . . . . . . . . 13 ⊢ (.r‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (+g‘(mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
119		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (.r‘(𝑆 ↑s (𝐵 ↑m 𝐼))) = (.r‘(𝑆 ↑s (𝐵 ↑m 𝐼)))
120	113, 119	mgpplusg 20300	. . . . . . . . . . . . 13 ⊢ (.r‘(𝑆 ↑s (𝐵 ↑m 𝐼))) = (+g‘(mulGrp‘(𝑆 ↑s (𝐵 ↑m 𝐼))))
121	117, 118, 120	mhmlin 18924	. . . . . . . . . . . 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 1398	. . . . . . . . . . 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 17599	. . . . . . . . . . 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 2795	. . . . . . . . . 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 7448	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∘f · 𝑔) ∈ V
127		mpfind.wf	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 ∘f · 𝑔) → (𝜓 ↔ 𝜎))
128	126, 127	elab 3633	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜎)
129		oveq12 7424	. . . . . . . . . . . . . . 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 2845	. . . . . . . . . . . . . 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 288	. . . . . . . . . . . . 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 347	. . . . . . . . . . . 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 3526	. . . . . . . . . . 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 850	. . . . . . . . . 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 2860	. . . . . . . . 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 7052	. . . . . . . . . . 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 18	. . . . . . . . . 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 486	. . . . . . . . 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 726	. . . . . . . 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 728	. . . . . . 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 22265	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ V ∧ (𝑆 ↾s 𝑅) ∈ CRing) → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ AssAlg)
143	26, 30, 142	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ AssAlg)
144		eqid 2760	. . . . . . . . . . . . 13 ⊢ (Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
145	25, 144	asclrhm 22134	. . . . . . . . . . . 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 2760	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) = (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
147	146, 12	rhmf 20651	. . . . . . . . . . . 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 19	. . . . . . . . . . 11 ⊢ (𝜑 → (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))):(Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))⟶(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
149	148	adantr 486	. . . . . . . . . 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 22256	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ↾s 𝑅) = (Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
151	150	fveq2d 6884	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘(𝑆 ↾s 𝑅)) = (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))))
152	151	eleq2d 2846	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑖 ∈ (Base‘(𝑆 ↾s 𝑅)) ↔ 𝑖 ∈ (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))))
153	152	biimpa 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → 𝑖 ∈ (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))))
154	149, 153	ffvelcdmd 7080	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
155	26	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → 𝐼 ∈ V)
156	27	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → 𝑆 ∈ CRing)
157	28	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → 𝑅 ∈ (SubRing‘𝑆))
158	10	subrgss 20760	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵)
159	8, 10	ressbas2 17352	. . . . . . . . . . . . . 14 ⊢ (𝑅 ⊆ 𝐵 → 𝑅 = (Base‘(𝑆 ↾s 𝑅)))
160	28, 158, 159	3syl 19	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 = (Base‘(𝑆 ↾s 𝑅)))
161	160	eleq2d 2846	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑖 ∈ 𝑅 ↔ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))))
162	161	biimpar 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → 𝑖 ∈ 𝑅)
163	6, 7, 8, 10, 25, 155, 156, 157, 162	evlssca 22339	. . . . . . . . . 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 3154	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑓 ∈ 𝑅 𝜒)
166		ovex 7448	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ↑m 𝐼) ∈ V
167		vsnex 5400	. . . . . . . . . . . . . . . . 17 ⊢ {𝑓} ∈ V
168	166, 167	xpex 7754	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ↑m 𝐼) × {𝑓}) ∈ V
169		mpfind.wa	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ((𝐵 ↑m 𝐼) × {𝑓}) → (𝜓 ↔ 𝜒))
170	168, 169	elab 3633	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ↑m 𝐼) × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)
171		sneq 4594	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝑖 → {𝑓} = {𝑖})
172	171	xpeq2d 5685	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑖 → ((𝐵 ↑m 𝐼) × {𝑓}) = ((𝐵 ↑m 𝐼) × {𝑖}))
173	172	eleq1d 2845	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑖 → (((𝐵 ↑m 𝐼) × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ ((𝐵 ↑m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓}))
174	170, 173	bitr3id 288	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑖 → (𝜒 ↔ ((𝐵 ↑m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓}))
175	174	cbvralvw 3240	. . . . . . . . . . . . 13 ⊢ (∀𝑓 ∈ 𝑅 𝜒 ↔ ∀𝑖 ∈ 𝑅 ((𝐵 ↑m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓})
176	165, 175	sylib 221	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑖 ∈ 𝑅 ((𝐵 ↑m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓})
177	176	r19.21bi 3254	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑅) → ((𝐵 ↑m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓})
178	162, 177	syldan 603	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → ((𝐵 ↑m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓})
179	163, 178	eqeltrd 2860	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖)) ∈ {𝑥 ∣ 𝜓})
180		elpreima 7052	. . . . . . . . . . 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 18	. . . . . . . . . 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 486	. . . . . . . . 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 726	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
184	183	adantlr 728	. . . . . . 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 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ V)
186	32	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑆 ↾s 𝑅) ∈ Ring)
187		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
188	7, 22, 12, 185, 186, 187	mvrcl 22235	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
189	27	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ CRing)
190	28	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑅 ∈ (SubRing‘𝑆))
191	6, 22, 8, 10, 185, 189, 190, 187	evlsvar 22340	. . . . . . . . . 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 7224	. . . . . . . . . . . . . . 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 3633	. . . . . . . . . . . . . 14 ⊢ ((𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓} ↔ 𝜃)
196	192, 195	sylibr 237	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐼) → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓})
197	196	ralrimiva 3154	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑓 ∈ 𝐼 (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓})
198		fveq2 6880	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑖 → (𝑔‘𝑓) = (𝑔‘𝑖))
199	198	mpteq2dv 5199	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑖 → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑖)))
200	199	eleq1d 2845	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑖 → ((𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓} ↔ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓}))
201	200	cbvralvw 3240	. . . . . . . . . . . 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 221	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓})
203	202	r19.21bi 3254	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓})
204	191, 203	eqeltrd 2860	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖)) ∈ {𝑥 ∣ 𝜓})
205		elpreima 7052	. . . . . . . . . . 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 18	. . . . . . . . . 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 486	. . . . . . . . 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 726	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
209	208	adantlr 728	. . . . . . 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 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
211	26	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → 𝐼 ∈ V)
212	30	adantr 486	. . . . . . 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 22315	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → 𝑦 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
214		fvimacnvi 7046	. . . . . 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 698	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥 ∣ 𝜓})
216		eleq1 2848	. . . . 5 ⊢ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴 → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥 ∣ 𝜓} ↔ 𝐴 ∈ {𝑥 ∣ 𝜓}))
217	215, 216	syl5ibcom 248	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴 → 𝐴 ∈ {𝑥 ∣ 𝜓}))
218	217	rexlimdva 3163	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴 → 𝐴 ∈ {𝑥 ∣ 𝜓}))
219	19, 218	mpd 16	. 2 ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜓})
220		mpfind.wg	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜌))
221	220	elabg 3630	. . 3 ⊢ (𝐴 ∈ 𝑄 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜌))
222	1, 221	syl 18	. 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜌))
223	219, 222	mpbid 235	1 ⊢ (𝜑 → 𝜌)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3899  {csn 4584   ↦ cmpt 5186   × cxp 5653  ^◡ccnv 5654  ran crn 5656   “ cima 5658  Fun wfun 6528   Fn wfn 6529  ⟶wf 6530  ‘cfv 6534  (class class class)co 7415   ∘_f cof 7678   ↑_m cmap 8830  Basecbs 17323   ↾_s cress 17344  +_gcplusg 17364  ._rcmulr 17365  Scalarcsca 17367   ↑_s cpws 17553   MndHom cmhm 18912   GrpHom cghm 19363  mulGrpcmgp 20296  Ringcrg 20395  CRingccrg 20396   RingHom crh 20635  SubRingcsubrg 20757  AssAlgcasa 22094  algSccascl 22096   mVar cmvr 22149   mPoly cmpl 22150   evalSub ces 22317

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 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7738  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223

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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  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 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-isom 6543  df-riota 7372  df-ov 7418  df-oprab 7419  df-mpo 7420  df-of 7680  df-ofr 7681  df-om 7865  df-1st 7988  df-2nd 7989  df-supp 8161  df-frecs 8282  df-wrecs 8313  df-recs 8362  df-rdg 8401  df-1o 8459  df-2o 8460  df-er 8700  df-map 8832  df-pm 8833  df-ixp 8909  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-fsupp 9336  df-sup 9416  df-oi 9486  df-card 9966  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11489  df-neg 11490  df-nn 12280  df-2 12349  df-3 12350  df-4 12351  df-5 12352  df-6 12353  df-7 12354  df-8 12355  df-9 12356  df-n0 12551  df-z 12638  df-dec 12759  df-uz 12910  df-fz 13584  df-fzo 13732  df-seq 14088  df-hash 14417  df-struct 17261  df-sets 17278  df-slot 17296  df-ndx 17308  df-base 17324  df-ress 17345  df-plusg 17377  df-mulr 17378  df-sca 17380  df-vsca 17381  df-ip 17382  df-tset 17383  df-ple 17384  df-ds 17386  df-hom 17388  df-cco 17389  df-0g 17548  df-gsum 17549  df-prds 17554  df-pws 17556  df-mre 17692  df-mrc 17693  df-acs 17695  df-mgm 18752  df-sgrp 18844  df-mnd 18860  df-mhm 18914  df-submnd 18915  df-grp 19083  df-minusg 19084  df-sbg 19085  df-mulg 19214  df-subg 19269  df-ghm 19364  df-cntz 19467  df-cmn 19932  df-abl 19933  df-mgp 20297  df-rng 20311  df-ur 20344  df-srg 20349  df-ring 20397  df-cring 20398  df-rhm 20638  df-subrng 20734  df-subrg 20758  df-lmod 21073  df-lss 21143  df-lsp 21183  df-assa 22097  df-asp 22098  df-ascl 22099  df-psr 22153  df-mvr 22154  df-mpl 22155  df-evls 22319

This theorem is used by:  pf1ind  22609  mzpmfp  43606