Theorem mpfind 22093

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 2846	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
4	2	mpfrcl 22063	. . . . . . . 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 2736	. . . . . . . 8 ⊢ ((𝐼 evalSub 𝑆)‘𝑅) = ((𝐼 evalSub 𝑆)‘𝑅)
7		eqid 2736	. . . . . . . 8 ⊢ (𝐼 mPoly (𝑆 ↾s 𝑅)) = (𝐼 mPoly (𝑆 ↾s 𝑅))
8		eqid 2736	. . . . . . . 8 ⊢ (𝑆 ↾s 𝑅) = (𝑆 ↾s 𝑅)
9		eqid 2736	. . . . . . . 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 22066	. . . . . . 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 2736	. . . . . . . 8 ⊢ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
13		eqid 2736	. . . . . . . 8 ⊢ (Base‘(𝑆 ↑s (𝐵 ↑m 𝐼))) = (Base‘(𝑆 ↑s (𝐵 ↑m 𝐼)))
14	12, 13	rhmf 20464	. . . . . . 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 6669	. . . . 5 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
17		fvelrnb 6900	. . . . 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 6672	. . . . . 6 ⊢ (𝜑 → Fun ((𝐼 evalSub 𝑆)‘𝑅))
21		eqid 2736	. . . . . . 7 ⊢ (Base‘(𝑆 ↾s 𝑅)) = (Base‘(𝑆 ↾s 𝑅))
22		eqid 2736	. . . . . . 7 ⊢ (𝐼 mVar (𝑆 ↾s 𝑅)) = (𝐼 mVar (𝑆 ↾s 𝑅))
23		eqid 2736	. . . . . . 7 ⊢ (+g‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
24		eqid 2736	. . . . . . 7 ⊢ (.r‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
25		eqid 2736	. . . . . . 7 ⊢ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
26	5	simp1d 1143	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ V)
27	5	simp2d 1144	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ CRing)
28	5	simp3d 1145	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
29	8	subrgcrng 20552	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑆 ↾s 𝑅) ∈ CRing)
30	27, 28, 29	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ↾s 𝑅) ∈ CRing)
31		crngring 20226	. . . . . . . . . . . . 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 22001	. . . . . . . . . . 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 771	. . . . . . . . . . . 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 7010	. . . . . . . . . . . . . 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 773	. . . . . . . . . . . 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 7010	. . . . . . . . . . . . . 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 20259	. . . . . . . . . 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 1374	. . . . . . . . 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 20463	. . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ (+g‘(𝑆 ↑s (𝐵 ↑m 𝐼))) = (+g‘(𝑆 ↑s (𝐵 ↑m 𝐼)))
53	12, 23, 52	ghmlin 19196	. . . . . . . . . . . 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 1374	. . . . . . . . . . 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 7402	. . . . . . . . . . . 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 7037	. . . . . . . . . . . 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 7037	. . . . . . . . . . . 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 17454	. . . . . . . . . . 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 2771	. . . . . . . . . 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 7032	. . . . . . . . . . . . . 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 686	. . . . . . . . . . . . 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 2847	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄)
67		fvimacnvi 7004	. . . . . . . . . . . . 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 686	. . . . . . . . . . . 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 7032	. . . . . . . . . . . . . 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 686	. . . . . . . . . . . . 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 2847	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄)
73		fvimacnvi 7004	. . . . . . . . . . . . 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 686	. . . . . . . . . . . 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 6853	. . . . . . . . . . . 12 ⊢ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ V
77		fvex 6853	. . . . . . . . . . . 12 ⊢ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ V
78		eleq1 2824	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝑓 ∈ 𝑄 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄))
79		vex 3433	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
80		mpfind.wc	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏))
81	79, 80	elab 3622	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ 𝜏)
82		eleq1 2824	. . . . . . . . . . . . . . . . 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 633	. . . . . . . . . . . . . . 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 2824	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝑔 ∈ 𝑄 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄))
86		vex 3433	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑔 ∈ V
87		mpfind.wd	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂))
88	86, 87	elab 3622	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ 𝜂)
89		eleq1 2824	. . . . . . . . . . . . . . . . 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 633	. . . . . . . . . . . . . . 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 639	. . . . . . . . . . . . . 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 631	. . . . . . . . . . . . 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 7400	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∘f + 𝑔) ∈ V
95		mpfind.we	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 ∘f + 𝑔) → (𝜓 ↔ 𝜁))
96	94, 95	elab 3622	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜁)
97		oveq12 7376	. . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . 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 3510	. . . . . . . . . . 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 837	. . . . . . . . . 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 2836	. . . . . . . . 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 7010	. . . . . . . . . . 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 714	. . . . . . . 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 716	. . . . . . 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 20231	. . . . . . . . . 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 1374	. . . . . . . . 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 2736	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
113		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘(𝑆 ↑s (𝐵 ↑m 𝐼))) = (mulGrp‘(𝑆 ↑s (𝐵 ↑m 𝐼)))
114	112, 113	rhmmhm 20459	. . . . . . . . . . . . . 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 20126	. . . . . . . . . . . . 13 ⊢ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (Base‘(mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
118	112, 24	mgpplusg 20125	. . . . . . . . . . . . 13 ⊢ (.r‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (+g‘(mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
119		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (.r‘(𝑆 ↑s (𝐵 ↑m 𝐼))) = (.r‘(𝑆 ↑s (𝐵 ↑m 𝐼)))
120	113, 119	mgpplusg 20125	. . . . . . . . . . . . 13 ⊢ (.r‘(𝑆 ↑s (𝐵 ↑m 𝐼))) = (+g‘(mulGrp‘(𝑆 ↑s (𝐵 ↑m 𝐼))))
121	117, 118, 120	mhmlin 18761	. . . . . . . . . . . 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 1374	. . . . . . . . . . 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 17455	. . . . . . . . . . 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 2771	. . . . . . . . . 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 7400	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∘f · 𝑔) ∈ V
127		mpfind.wf	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 ∘f · 𝑔) → (𝜓 ↔ 𝜎))
128	126, 127	elab 3622	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜎)
129		oveq12 7376	. . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . 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 3510	. . . . . . . . . . 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 837	. . . . . . . . . 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 2836	. . . . . . . . 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 7010	. . . . . . . . . . 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 714	. . . . . . . 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 716	. . . . . . 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 22000	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ V ∧ (𝑆 ↾s 𝑅) ∈ CRing) → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ AssAlg)
143	26, 30, 142	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ AssAlg)
144		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
145	25, 144	asclrhm 21870	. . . . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) = (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
147	146, 12	rhmf 20464	. . . . . . . . . . . 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 21991	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ↾s 𝑅) = (Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
151	150	fveq2d 6844	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘(𝑆 ↾s 𝑅)) = (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))))
152	151	eleq2d 2822	. . . . . . . . . . 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 7037	. . . . . . . . 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 20549	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵)
159	8, 10	ressbas2 17208	. . . . . . . . . . . . . 14 ⊢ (𝑅 ⊆ 𝐵 → 𝑅 = (Base‘(𝑆 ↾s 𝑅)))
160	28, 158, 159	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 = (Base‘(𝑆 ↾s 𝑅)))
161	160	eleq2d 2822	. . . . . . . . . . . 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 22072	. . . . . . . . . 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 3129	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑓 ∈ 𝑅 𝜒)
166		ovex 7400	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ↑m 𝐼) ∈ V
167		vsnex 5377	. . . . . . . . . . . . . . . . 17 ⊢ {𝑓} ∈ V
168	166, 167	xpex 7707	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ↑m 𝐼) × {𝑓}) ∈ V
169		mpfind.wa	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ((𝐵 ↑m 𝐼) × {𝑓}) → (𝜓 ↔ 𝜒))
170	168, 169	elab 3622	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ↑m 𝐼) × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)
171		sneq 4577	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝑖 → {𝑓} = {𝑖})
172	171	xpeq2d 5661	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑖 → ((𝐵 ↑m 𝐼) × {𝑓}) = ((𝐵 ↑m 𝐼) × {𝑖}))
173	172	eleq1d 2821	. . . . . . . . . . . . . . 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 3215	. . . . . . . . . . . . 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 3229	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑅) → ((𝐵 ↑m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓})
178	162, 177	syldan 592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → ((𝐵 ↑m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓})
179	163, 178	eqeltrd 2836	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖)) ∈ {𝑥 ∣ 𝜓})
180		elpreima 7010	. . . . . . . . . . 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 714	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
184	183	adantlr 716	. . . . . . 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 21970	. . . . . . . . 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 22073	. . . . . . . . . 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 7178	. . . . . . . . . . . . . . 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 3622	. . . . . . . . . . . . . 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 3129	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑓 ∈ 𝐼 (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓})
198		fveq2 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑖 → (𝑔‘𝑓) = (𝑔‘𝑖))
199	198	mpteq2dv 5179	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑖 → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑖)))
200	199	eleq1d 2821	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑖 → ((𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓} ↔ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓}))
201	200	cbvralvw 3215	. . . . . . . . . . . 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 3229	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓})
204	191, 203	eqeltrd 2836	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖)) ∈ {𝑥 ∣ 𝜓})
205		elpreima 7010	. . . . . . . . . . 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 714	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
209	208	adantlr 716	. . . . . . 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 22048	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → 𝑦 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
214		fvimacnvi 7004	. . . . . 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 686	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥 ∣ 𝜓})
216		eleq1 2824	. . . . 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 3138	. . 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 3619	. . 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 1542   ∈ wcel 2114  {cab 2714  ∀wral 3051  ∃wrex 3061  Vcvv 3429   ⊆ wss 3889  {csn 4567   ↦ cmpt 5166   × cxp 5629  ^◡ccnv 5630  ran crn 5632   “ cima 5634  Fun wfun 6492   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ∘_f cof 7629   ↑_m cmap 8773  Basecbs 17179   ↾_s cress 17200  +_gcplusg 17220  ._rcmulr 17221  Scalarcsca 17223   ↑_s cpws 17409   MndHom cmhm 18749   GrpHom cghm 19187  mulGrpcmgp 20121  Ringcrg 20214  CRingccrg 20215   RingHom crh 20449  SubRingcsubrg 20546  AssAlgcasa 21830  algSccascl 21832   mVar cmvr 21885   mPoly cmpl 21886   evalSub ces 22050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-ofr 7632  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-pm 8776  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-sup 9355  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-fzo 13609  df-seq 13964  df-hash 14293  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-mre 17548  df-mrc 17549  df-acs 17551  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-grp 18912  df-minusg 18913  df-sbg 18914  df-mulg 19044  df-subg 19099  df-ghm 19188  df-cntz 19292  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-srg 20168  df-ring 20216  df-cring 20217  df-rhm 20452  df-subrng 20523  df-subrg 20547  df-lmod 20857  df-lss 20927  df-lsp 20967  df-assa 21833  df-asp 21834  df-ascl 21835  df-psr 21889  df-mvr 21890  df-mpl 21891  df-evls 22052

This theorem is referenced by:  pf1ind  22320  mzpmfp  43179