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Theorem mpfind 21670

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 2844	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
4	2	mpfrcl 21648	. . . . . . . 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 2733	. . . . . . . 8 ⊢ ((𝐼 evalSub 𝑆)‘𝑅) = ((𝐼 evalSub 𝑆)‘𝑅)
7		eqid 2733	. . . . . . . 8 ⊢ (𝐼 mPoly (𝑆 ↾_s 𝑅)) = (𝐼 mPoly (𝑆 ↾_s 𝑅))
8		eqid 2733	. . . . . . . 8 ⊢ (𝑆 ↾_s 𝑅) = (𝑆 ↾_s 𝑅)
9		eqid 2733	. . . . . . . 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 21651	. . . . . . 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 2733	. . . . . . . 8 ⊢ (Base‘(𝐼 mPoly (𝑆 ↾_s 𝑅))) = (Base‘(𝐼 mPoly (𝑆 ↾_s 𝑅)))
13		eqid 2733	. . . . . . . 8 ⊢ (Base‘(𝑆 ↑_s (𝐵 ↑_m 𝐼))) = (Base‘(𝑆 ↑_s (𝐵 ↑_m 𝐼)))
14	12, 13	rhmf 20263	. . . . . . 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 6719	. . . . 5 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾_s 𝑅))))
17		fvelrnb 6953	. . . . 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 231	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾_s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴)
20	15	ffund 6722	. . . . . 6 ⊢ (𝜑 → Fun ((𝐼 evalSub 𝑆)‘𝑅))
21		eqid 2733	. . . . . . 7 ⊢ (Base‘(𝑆 ↾_s 𝑅)) = (Base‘(𝑆 ↾_s 𝑅))
22		eqid 2733	. . . . . . 7 ⊢ (𝐼 mVar (𝑆 ↾_s 𝑅)) = (𝐼 mVar (𝑆 ↾_s 𝑅))
23		eqid 2733	. . . . . . 7 ⊢ (+_g‘(𝐼 mPoly (𝑆 ↾_s 𝑅))) = (+_g‘(𝐼 mPoly (𝑆 ↾_s 𝑅)))
24		eqid 2733	. . . . . . 7 ⊢ (._r‘(𝐼 mPoly (𝑆 ↾_s 𝑅))) = (._r‘(𝐼 mPoly (𝑆 ↾_s 𝑅)))
25		eqid 2733	. . . . . . 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 20323	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑆 ↾_s 𝑅) ∈ CRing)
30	27, 28, 29	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ↾_s 𝑅) ∈ CRing)
31		crngring 20068	. . . . . . . . . . . . 13 ⊢ ((𝑆 ↾_s 𝑅) ∈ CRing → (𝑆 ↾_s 𝑅) ∈ Ring)
32	30, 31	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 ↾_s 𝑅) ∈ Ring)
33	7	mplring 21578	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ V ∧ (𝑆 ↾_s 𝑅) ∈ Ring) → (𝐼 mPoly (𝑆 ↾_s 𝑅)) ∈ Ring)
34	26, 32, 33	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼 mPoly (𝑆 ↾_s 𝑅)) ∈ Ring)
35	34	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝐼 mPoly (𝑆 ↾_s 𝑅)) ∈ Ring)
36		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}))
37		elpreima 7060	. . . . . . . . . . . . . 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})))
38	16, 37	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})))
39	38	adantr 482	. . . . . . . . . . . 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})))
40	36, 39	mpbid 231	. . . . . . . . . . 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}))
41	40	simpld 496	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾_s 𝑅))))
42		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}))
43		elpreima 7060	. . . . . . . . . . . . . 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})))
44	16, 43	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})))
45	44	adantr 482	. . . . . . . . . . . 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})))
46	42, 45	mpbid 231	. . . . . . . . . . 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}))
47	46	simpld 496	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾_s 𝑅))))
48	12, 23	ringacl 20095	. . . . . . . . . 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))))
49	35, 41, 47, 48	syl3anc 1372	. . . . . . . . 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))))
50		rhmghm 20262	. . . . . . . . . . . . . 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))))
51	5, 11, 50	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾_s 𝑅)) GrpHom (𝑆 ↑_s (𝐵 ↑_m 𝐼))))
52	51	adantr 482	. . . . . . . . . . . 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))))
53		eqid 2733	. . . . . . . . . . . . 13 ⊢ (+_g‘(𝑆 ↑_s (𝐵 ↑_m 𝐼))) = (+_g‘(𝑆 ↑_s (𝐵 ↑_m 𝐼)))
54	12, 23, 53	ghmlin 19097	. . . . . . . . . . . 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)))
55	52, 41, 47, 54	syl3anc 1372	. . . . . . . . . . 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)))
56	27	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝑆 ∈ CRing)
57		ovexd 7444	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝐵 ↑_m 𝐼) ∈ V)
58	15	adantr 482	. . . . . . . . . . . . 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))))
59	58, 41	ffvelcdmd 7088	. . . . . . . . . . . 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))))
60	58, 47	ffvelcdmd 7088	. . . . . . . . . . . 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))))
61		mpfind.cp	. . . . . . . . . . . 12 ⊢ + = (+_g‘𝑆)
62	9, 13, 56, 57, 59, 60, 61, 53	pwsplusgval 17436	. . . . . . . . . . 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)))
63	55, 62	eqtrd 2773	. . . . . . . . . 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)))
64		simpl 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝜑)
65		fnfvelrn 7083	. . . . . . . . . . . . . 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))
66	16, 41, 65	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))
67	66, 2	eleqtrrdi 2845	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄)
68		fvimacnvi 7054	. . . . . . . . . . . . 13 ⊢ ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑖 ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓})
69	20, 36, 68	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})
70	67, 69	jca 513	. . . . . . . . . . 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}))
71		fnfvelrn 7083	. . . . . . . . . . . . . 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))
72	16, 47, 71	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))
73	72, 2	eleqtrrdi 2845	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄)
74		fvimacnvi 7054	. . . . . . . . . . . . 13 ⊢ ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑗 ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓})
75	20, 42, 74	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})
76	73, 75	jca 513	. . . . . . . . . . 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}))
77		fvex 6905	. . . . . . . . . . . 12 ⊢ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ V
78		fvex 6905	. . . . . . . . . . . 12 ⊢ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ V
79		eleq1 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝑓 ∈ 𝑄 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄))
80		vex 3479	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
81		mpfind.wc	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏))
82	80, 81	elab 3669	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ 𝜏)
83		eleq1 2822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}))
84	82, 83	bitr3id 285	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝜏 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}))
85	79, 84	anbi12d 632	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → ((𝑓 ∈ 𝑄 ∧ 𝜏) ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓})))
86		eleq1 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝑔 ∈ 𝑄 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄))
87		vex 3479	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑔 ∈ V
88		mpfind.wd	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂))
89	87, 88	elab 3669	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ 𝜂)
90		eleq1 2822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))
91	89, 90	bitr3id 285	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝜂 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))
92	86, 91	anbi12d 632	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → ((𝑔 ∈ 𝑄 ∧ 𝜂) ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓})))
93	85, 92	bi2anan9 638	. . . . . . . . . . . . . 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}))))
94	93	anbi2d 630	. . . . . . . . . . . . 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})))))
95		ovex 7442	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∘_f + 𝑔) ∈ V
96		mpfind.we	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 ∘_f + 𝑔) → (𝜓 ↔ 𝜁))
97	95, 96	elab 3669	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘_f + 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜁)
98		oveq12 7418	. . . . . . . . . . . . . . 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)))
99	98	eleq1d 2819	. . . . . . . . . . . . . 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}))
100	97, 99	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}))
101	94, 100	imbi12d 345	. . . . . . . . . . . 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})))
102		mpfind.ad	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜁)
103	77, 78, 101, 102	vtocl2 3552	. . . . . . . . . . 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})
104	64, 70, 76, 103	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})
105	63, 104	eqeltrd 2834	. . . . . . . . 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})
106		elpreima 7060	. . . . . . . . . . 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})))
107	16, 106	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})))
108	107	adantr 482	. . . . . . . . 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})))
109	49, 105, 108	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}))
110	109	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}))
111	12, 24	ringcl 20073	. . . . . . . . . 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))))
112	35, 41, 47, 111	syl3anc 1372	. . . . . . . . 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))))
113		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘(𝐼 mPoly (𝑆 ↾_s 𝑅))) = (mulGrp‘(𝐼 mPoly (𝑆 ↾_s 𝑅)))
114		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘(𝑆 ↑_s (𝐵 ↑_m 𝐼))) = (mulGrp‘(𝑆 ↑_s (𝐵 ↑_m 𝐼)))
115	113, 114	rhmmhm 20258	. . . . . . . . . . . . . 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)))))
116	5, 11, 115	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((mulGrp‘(𝐼 mPoly (𝑆 ↾_s 𝑅))) MndHom (mulGrp‘(𝑆 ↑_s (𝐵 ↑_m 𝐼)))))
117	116	adantr 482	. . . . . . . . . . . 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)))))
118	113, 12	mgpbas 19993	. . . . . . . . . . . . 13 ⊢ (Base‘(𝐼 mPoly (𝑆 ↾_s 𝑅))) = (Base‘(mulGrp‘(𝐼 mPoly (𝑆 ↾_s 𝑅))))
119	113, 24	mgpplusg 19991	. . . . . . . . . . . . 13 ⊢ (._r‘(𝐼 mPoly (𝑆 ↾_s 𝑅))) = (+_g‘(mulGrp‘(𝐼 mPoly (𝑆 ↾_s 𝑅))))
120		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (._r‘(𝑆 ↑_s (𝐵 ↑_m 𝐼))) = (._r‘(𝑆 ↑_s (𝐵 ↑_m 𝐼)))
121	114, 120	mgpplusg 19991	. . . . . . . . . . . . 13 ⊢ (._r‘(𝑆 ↑_s (𝐵 ↑_m 𝐼))) = (+_g‘(mulGrp‘(𝑆 ↑_s (𝐵 ↑_m 𝐼))))
122	118, 119, 121	mhmlin 18679	. . . . . . . . . . . 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)))
123	117, 41, 47, 122	syl3anc 1372	. . . . . . . . . . 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)))
124		mpfind.ct	. . . . . . . . . . . 12 ⊢ · = (._r‘𝑆)
125	9, 13, 56, 57, 59, 60, 124, 120	pwsmulrval 17437	. . . . . . . . . . 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)))
126	123, 125	eqtrd 2773	. . . . . . . . . 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)))
127		ovex 7442	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∘_f · 𝑔) ∈ V
128		mpfind.wf	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 ∘_f · 𝑔) → (𝜓 ↔ 𝜎))
129	127, 128	elab 3669	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘_f · 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜎)
130		oveq12 7418	. . . . . . . . . . . . . . 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)))
131	130	eleq1d 2819	. . . . . . . . . . . . . 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}))
132	129, 131	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}))
133	94, 132	imbi12d 345	. . . . . . . . . . . 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})))
134		mpfind.mu	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜎)
135	77, 78, 133, 134	vtocl2 3552	. . . . . . . . . . 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})
136	64, 70, 76, 135	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})
137	126, 136	eqeltrd 2834	. . . . . . . . 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})
138		elpreima 7060	. . . . . . . . . . 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})))
139	16, 138	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})))
140	139	adantr 482	. . . . . . . . 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})))
141	112, 137, 140	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}))
142	141	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}))
143	7	mplassa 21581	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ V ∧ (𝑆 ↾_s 𝑅) ∈ CRing) → (𝐼 mPoly (𝑆 ↾_s 𝑅)) ∈ AssAlg)
144	26, 30, 143	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐼 mPoly (𝑆 ↾_s 𝑅)) ∈ AssAlg)
145		eqid 2733	. . . . . . . . . . . . 13 ⊢ (Scalar‘(𝐼 mPoly (𝑆 ↾_s 𝑅))) = (Scalar‘(𝐼 mPoly (𝑆 ↾_s 𝑅)))
146	25, 145	asclrhm 21444	. . . . . . . . . . . 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))))
147		eqid 2733	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾_s 𝑅)))) = (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾_s 𝑅))))
148	147, 12	rhmf 20263	. . . . . . . . . . . 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))))
149	144, 146, 148	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (algSc‘(𝐼 mPoly (𝑆 ↾_s 𝑅))):(Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾_s 𝑅))))⟶(Base‘(𝐼 mPoly (𝑆 ↾_s 𝑅))))
150	149	adantr 482	. . . . . . . . . 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))))
151	7, 26, 30	mplsca 21572	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ↾_s 𝑅) = (Scalar‘(𝐼 mPoly (𝑆 ↾_s 𝑅))))
152	151	fveq2d 6896	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘(𝑆 ↾_s 𝑅)) = (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾_s 𝑅)))))
153	152	eleq2d 2820	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑖 ∈ (Base‘(𝑆 ↾_s 𝑅)) ↔ 𝑖 ∈ (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾_s 𝑅))))))
154	153	biimpa 478	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾_s 𝑅))) → 𝑖 ∈ (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾_s 𝑅)))))
155	150, 154	ffvelcdmd 7088	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾_s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆 ↾_s 𝑅)))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾_s 𝑅))))
156	26	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾_s 𝑅))) → 𝐼 ∈ V)
157	27	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾_s 𝑅))) → 𝑆 ∈ CRing)
158	28	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾_s 𝑅))) → 𝑅 ∈ (SubRing‘𝑆))
159	10	subrgss 20320	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵)
160	8, 10	ressbas2 17182	. . . . . . . . . . . . . 14 ⊢ (𝑅 ⊆ 𝐵 → 𝑅 = (Base‘(𝑆 ↾_s 𝑅)))
161	28, 159, 160	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 = (Base‘(𝑆 ↾_s 𝑅)))
162	161	eleq2d 2820	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑖 ∈ 𝑅 ↔ 𝑖 ∈ (Base‘(𝑆 ↾_s 𝑅))))
163	162	biimpar 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾_s 𝑅))) → 𝑖 ∈ 𝑅)
164	6, 7, 8, 10, 25, 156, 157, 158, 163	evlssca 21652	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾_s 𝑅))) → (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆 ↾_s 𝑅)))‘𝑖)) = ((𝐵 ↑_m 𝐼) × {𝑖}))
165		mpfind.co	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑅) → 𝜒)
166	165	ralrimiva 3147	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑓 ∈ 𝑅 𝜒)
167		ovex 7442	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ↑_m 𝐼) ∈ V
168		vsnex 5430	. . . . . . . . . . . . . . . . 17 ⊢ {𝑓} ∈ V
169	167, 168	xpex 7740	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ↑_m 𝐼) × {𝑓}) ∈ V
170		mpfind.wa	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ((𝐵 ↑_m 𝐼) × {𝑓}) → (𝜓 ↔ 𝜒))
171	169, 170	elab 3669	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ↑_m 𝐼) × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)
172		sneq 4639	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝑖 → {𝑓} = {𝑖})
173	172	xpeq2d 5707	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑖 → ((𝐵 ↑_m 𝐼) × {𝑓}) = ((𝐵 ↑_m 𝐼) × {𝑖}))
174	173	eleq1d 2819	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑖 → (((𝐵 ↑_m 𝐼) × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ ((𝐵 ↑_m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓}))
175	171, 174	bitr3id 285	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑖 → (𝜒 ↔ ((𝐵 ↑_m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓}))
176	175	cbvralvw 3235	. . . . . . . . . . . . 13 ⊢ (∀𝑓 ∈ 𝑅 𝜒 ↔ ∀𝑖 ∈ 𝑅 ((𝐵 ↑_m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓})
177	166, 176	sylib 217	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑖 ∈ 𝑅 ((𝐵 ↑_m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓})
178	177	r19.21bi 3249	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑅) → ((𝐵 ↑_m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓})
179	163, 178	syldan 592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾_s 𝑅))) → ((𝐵 ↑_m 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓})
180	164, 179	eqeltrd 2834	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾_s 𝑅))) → (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆 ↾_s 𝑅)))‘𝑖)) ∈ {𝑥 ∣ 𝜓})
181		elpreima 7060	. . . . . . . . . . 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})))
182	16, 181	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})))
183	182	adantr 482	. . . . . . . . 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})))
184	155, 180, 183	mpbir2and 712	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾_s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆 ↾_s 𝑅)))‘𝑖) ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
185	184	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}))
186	26	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ V)
187	32	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑆 ↾_s 𝑅) ∈ Ring)
188		simpr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
189	7, 22, 12, 186, 187, 188	mvrcl 21551	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐼 mVar (𝑆 ↾_s 𝑅))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾_s 𝑅))))
190	27	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ CRing)
191	28	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑅 ∈ (SubRing‘𝑆))
192	6, 22, 8, 10, 186, 190, 191, 188	evlsvar 21653	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾_s 𝑅))‘𝑖)) = (𝑔 ∈ (𝐵 ↑_m 𝐼) ↦ (𝑔‘𝑖)))
193		mpfind.pr	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐼) → 𝜃)
194	167	mptex 7225	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (𝐵 ↑_m 𝐼) ↦ (𝑔‘𝑓)) ∈ V
195		mpfind.wb	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑔 ∈ (𝐵 ↑_m 𝐼) ↦ (𝑔‘𝑓)) → (𝜓 ↔ 𝜃))
196	194, 195	elab 3669	. . . . . . . . . . . . . 14 ⊢ ((𝑔 ∈ (𝐵 ↑_m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓} ↔ 𝜃)
197	193, 196	sylibr 233	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐼) → (𝑔 ∈ (𝐵 ↑_m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓})
198	197	ralrimiva 3147	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑓 ∈ 𝐼 (𝑔 ∈ (𝐵 ↑_m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓})
199		fveq2 6892	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑖 → (𝑔‘𝑓) = (𝑔‘𝑖))
200	199	mpteq2dv 5251	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑖 → (𝑔 ∈ (𝐵 ↑_m 𝐼) ↦ (𝑔‘𝑓)) = (𝑔 ∈ (𝐵 ↑_m 𝐼) ↦ (𝑔‘𝑖)))
201	200	eleq1d 2819	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑖 → ((𝑔 ∈ (𝐵 ↑_m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓} ↔ (𝑔 ∈ (𝐵 ↑_m 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓}))
202	201	cbvralvw 3235	. . . . . . . . . . . 12 ⊢ (∀𝑓 ∈ 𝐼 (𝑔 ∈ (𝐵 ↑_m 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓} ↔ ∀𝑖 ∈ 𝐼 (𝑔 ∈ (𝐵 ↑_m 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓})
203	198, 202	sylib 217	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 (𝑔 ∈ (𝐵 ↑_m 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓})
204	203	r19.21bi 3249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑔 ∈ (𝐵 ↑_m 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓})
205	192, 204	eqeltrd 2834	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾_s 𝑅))‘𝑖)) ∈ {𝑥 ∣ 𝜓})
206		elpreima 7060	. . . . . . . . . . 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})))
207	16, 206	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})))
208	207	adantr 482	. . . . . . . . 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})))
209	189, 205, 208	mpbir2and 712	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐼 mVar (𝑆 ↾_s 𝑅))‘𝑖) ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
210	209	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}))
211		simpr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾_s 𝑅)))) → 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾_s 𝑅))))
212	26	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾_s 𝑅)))) → 𝐼 ∈ V)
213	30	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾_s 𝑅)))) → (𝑆 ↾_s 𝑅) ∈ CRing)
214	21, 22, 7, 23, 24, 25, 12, 110, 142, 185, 210, 211, 212, 213	mplind 21631	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾_s 𝑅)))) → 𝑦 ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
215		fvimacnvi 7054	. . . . . 6 ⊢ ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑦 ∈ (^◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥 ∣ 𝜓})
216	20, 214, 215	syl2an2r 684	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾_s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥 ∣ 𝜓})
217		eleq1 2822	. . . . 5 ⊢ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴 → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥 ∣ 𝜓} ↔ 𝐴 ∈ {𝑥 ∣ 𝜓}))
218	216, 217	syl5ibcom 244	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾_s 𝑅)))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴 → 𝐴 ∈ {𝑥 ∣ 𝜓}))
219	218	rexlimdva 3156	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾_s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴 → 𝐴 ∈ {𝑥 ∣ 𝜓}))
220	19, 219	mpd 15	. 2 ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜓})
221		mpfind.wg	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜌))
222	221	elabg 3667	. . 3 ⊢ (𝐴 ∈ 𝑄 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜌))
223	1, 222	syl 17	. 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜌))
224	220, 223	mpbid 231	1 ⊢ (𝜑 → 𝜌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 {cab 2710 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ⊆ wss 3949 {csn 4629 ↦ cmpt 5232 × cxp 5675 ^◡ccnv 5676 ran crn 5678 “ cima 5680 Fun wfun 6538 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ∘_f cof 7668 ↑_m cmap 8820 Basecbs 17144 ↾_s cress 17173 +_gcplusg 17197 ._rcmulr 17198 Scalarcsca 17200 ↑_s cpws 17392 MndHom cmhm 18669 GrpHom cghm 19089 mulGrpcmgp 19987 Ringcrg 20056 CRingccrg 20057 RingHom crh 20248 SubRingcsubrg 20315 AssAlgcasa 21405 algSccascl 21407 mVar cmvr 21458 mPoly cmpl 21459 evalSub ces 21633

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-ofr 7671 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-sup 9437 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-fz 13485 df-fzo 13628 df-seq 13967 df-hash 14291 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-hom 17221 df-cco 17222 df-0g 17387 df-gsum 17388 df-prds 17393 df-pws 17395 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-submnd 18672 df-grp 18822 df-minusg 18823 df-sbg 18824 df-mulg 18951 df-subg 19003 df-ghm 19090 df-cntz 19181 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-srg 20010 df-ring 20058 df-cring 20059 df-rnghom 20251 df-subrg 20317 df-lmod 20473 df-lss 20543 df-lsp 20583 df-assa 21408 df-asp 21409 df-ascl 21410 df-psr 21462 df-mvr 21463 df-mpl 21464 df-evls 21635

This theorem is referenced by: pf1ind 21874 mzpmfp 41533

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