Theorem mpfind 19357

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	syl6eleq 2698	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
4	2	mpfrcl 19339	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑄 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))
5	1, 4	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))
6		eqid 2610	. . . . . . . . 9 ⊢ ((𝐼 evalSub 𝑆)‘𝑅) = ((𝐼 evalSub 𝑆)‘𝑅)
7		eqid 2610	. . . . . . . . 9 ⊢ (𝐼 mPoly (𝑆 ↾s 𝑅)) = (𝐼 mPoly (𝑆 ↾s 𝑅))
8		eqid 2610	. . . . . . . . 9 ⊢ (𝑆 ↾s 𝑅) = (𝑆 ↾s 𝑅)
9		eqid 2610	. . . . . . . . 9 ⊢ (𝑆 ↑s (𝐵 ↑𝑚 𝐼)) = (𝑆 ↑s (𝐵 ↑𝑚 𝐼))
10		mpfind.cb	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑆)
11	6, 7, 8, 9, 10	evlsrhm 19342	. . . . . . . 8 ⊢ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s (𝐵 ↑𝑚 𝐼))))
12	5, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s (𝐵 ↑𝑚 𝐼))))
13		eqid 2610	. . . . . . . 8 ⊢ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
14		eqid 2610	. . . . . . . 8 ⊢ (Base‘(𝑆 ↑s (𝐵 ↑𝑚 𝐼))) = (Base‘(𝑆 ↑s (𝐵 ↑𝑚 𝐼)))
15	13, 14	rhmf 18549	. . . . . . 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))))
16	12, 15	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))⟶(Base‘(𝑆 ↑s (𝐵 ↑𝑚 𝐼))))
17		ffn 5958	. . . . . 6 ⊢ (((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))⟶(Base‘(𝑆 ↑s (𝐵 ↑𝑚 𝐼))) → ((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
18	16, 17	syl 17	. . . . 5 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
19		fvelrnb 6153	. . . . 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))
20	18, 19	syl 17	. . . 4 ⊢ (𝜑 → (𝐴 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅) ↔ ∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴))
21	3, 20	mpbid 221	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴)
22		ffun 5961	. . . . . . . 8 ⊢ (((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))⟶(Base‘(𝑆 ↑s (𝐵 ↑𝑚 𝐼))) → Fun ((𝐼 evalSub 𝑆)‘𝑅))
23	16, 22	syl 17	. . . . . . 7 ⊢ (𝜑 → Fun ((𝐼 evalSub 𝑆)‘𝑅))
24	23	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → Fun ((𝐼 evalSub 𝑆)‘𝑅))
25		eqid 2610	. . . . . . 7 ⊢ (Base‘(𝑆 ↾s 𝑅)) = (Base‘(𝑆 ↾s 𝑅))
26		eqid 2610	. . . . . . 7 ⊢ (𝐼 mVar (𝑆 ↾s 𝑅)) = (𝐼 mVar (𝑆 ↾s 𝑅))
27		eqid 2610	. . . . . . 7 ⊢ (+g‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (+g‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
28		eqid 2610	. . . . . . 7 ⊢ (.r‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (.r‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
29		eqid 2610	. . . . . . 7 ⊢ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
30	5	simp1d 1066	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ V)
31	5	simp2d 1067	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ CRing)
32	5	simp3d 1068	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
33	8	subrgcrng 18607	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑆 ↾s 𝑅) ∈ CRing)
34	31, 32, 33	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ↾s 𝑅) ∈ CRing)
35		crngring 18381	. . . . . . . . . . . . 13 ⊢ ((𝑆 ↾s 𝑅) ∈ CRing → (𝑆 ↾s 𝑅) ∈ Ring)
36	34, 35	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 ↾s 𝑅) ∈ Ring)
37	7	mplring 19273	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ V ∧ (𝑆 ↾s 𝑅) ∈ Ring) → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ Ring)
38	30, 36, 37	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ Ring)
39	38	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ Ring)
40		simprl 790	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
41		elpreima 6245	. . . . . . . . . . . . . 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})))
42	18, 41	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})))
43	42	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})))
44	40, 43	mpbid 221	. . . . . . . . . . 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}))
45	44	simpld 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
46		simprr 792	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
47		elpreima 6245	. . . . . . . . . . . . . 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})))
48	18, 47	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})))
49	48	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})))
50	46, 49	mpbid 221	. . . . . . . . . . 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}))
51	50	simpld 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
52	13, 27	ringacl 18401	. . . . . . . . . 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))))
53	39, 45, 51, 52	syl3anc 1318	. . . . . . . . 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))))
54		rhmghm 18548	. . . . . . . . . . . . . 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))))
55	12, 54	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) GrpHom (𝑆 ↑s (𝐵 ↑𝑚 𝐼))))
56	55	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))))
57		eqid 2610	. . . . . . . . . . . . 13 ⊢ (+g‘(𝑆 ↑s (𝐵 ↑𝑚 𝐼))) = (+g‘(𝑆 ↑s (𝐵 ↑𝑚 𝐼)))
58	13, 27, 57	ghmlin 17488	. . . . . . . . . . . 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)))
59	56, 45, 51, 58	syl3anc 1318	. . . . . . . . . . 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)))
60	31	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝑆 ∈ CRing)
61		ovex 6577	. . . . . . . . . . . . 13 ⊢ (𝐵 ↑𝑚 𝐼) ∈ V
62	61	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (𝐵 ↑𝑚 𝐼) ∈ V)
63	16	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))))
64	63, 45	ffvelrnd 6268	. . . . . . . . . . . 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))))
65	63, 51	ffvelrnd 6268	. . . . . . . . . . . 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))))
66		mpfind.cp	. . . . . . . . . . . 12 ⊢ + = (+g‘𝑆)
67	9, 14, 60, 62, 64, 65, 66, 57	pwsplusgval 15973	. . . . . . . . . . 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)))
68	59, 67	eqtrd 2644	. . . . . . . . . 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)))
69		simpl 472	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → 𝜑)
70	18	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → ((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
71		fnfvelrn 6264	. . . . . . . . . . . . . 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))
72	70, 45, 71	syl2anc 691	. . . . . . . . . . . . 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))
73	72, 2	syl6eleqr 2699	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄)
74	23	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → Fun ((𝐼 evalSub 𝑆)‘𝑅))
75		fvimacnvi 6239	. . . . . . . . . . . . 13 ⊢ ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓})
76	74, 40, 75	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓})
77	73, 76	jca 553	. . . . . . . . . . 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}))
78		fnfvelrn 6264	. . . . . . . . . . . . . 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))
79	70, 51, 78	syl2anc 691	. . . . . . . . . . . . 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))
80	79, 2	syl6eleqr 2699	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄)
81		fvimacnvi 6239	. . . . . . . . . . . . 13 ⊢ ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓})
82	74, 46, 81	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}) ∧ 𝑗 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓})
83	80, 82	jca 553	. . . . . . . . . . 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}))
84		fvex 6113	. . . . . . . . . . . 12 ⊢ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ V
85		fvex 6113	. . . . . . . . . . . 12 ⊢ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ V
86		eleq1 2676	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝑓 ∈ 𝑄 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄))
87		vex 3176	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
88		mpfind.wc	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏))
89	87, 88	elab 3319	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ 𝜏)
90		eleq1 2676	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}))
91	89, 90	syl5bbr 273	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝜏 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓}))
92	86, 91	anbi12d 743	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → ((𝑓 ∈ 𝑄 ∧ 𝜏) ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥 ∣ 𝜓})))
93		eleq1 2676	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝑔 ∈ 𝑄 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄))
94		vex 3176	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑔 ∈ V
95		mpfind.wd	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂))
96	94, 95	elab 3319	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ 𝜂)
97		eleq1 2676	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))
98	96, 97	syl5bbr 273	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝜂 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓}))
99	93, 98	anbi12d 743	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → ((𝑔 ∈ 𝑄 ∧ 𝜂) ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥 ∣ 𝜓})))
100	92, 99	bi2anan9 913	. . . . . . . . . . . . . 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}))))
101	100	anbi2d 736	. . . . . . . . . . . . 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})))))
102		ovex 6577	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∘𝑓 + 𝑔) ∈ V
103		mpfind.we	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 ∘𝑓 + 𝑔) → (𝜓 ↔ 𝜁))
104	102, 103	elab 3319	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘𝑓 + 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜁)
105		oveq12 6558	. . . . . . . . . . . . . . 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)))
106	105	eleq1d 2672	. . . . . . . . . . . . . 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}))
107	104, 106	syl5bbr 273	. . . . . . . . . . . . 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}))
108	101, 107	imbi12d 333	. . . . . . . . . . . 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})))
109		mpfind.ad	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜁)
110	84, 85, 108, 109	vtocl2 3234	. . . . . . . . . . 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})
111	69, 77, 83, 110	syl12anc 1316	. . . . . . . . . 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})
112	68, 111	eqeltrd 2688	. . . . . . . . 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})
113		elpreima 6245	. . . . . . . . . . 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})))
114	18, 113	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})))
115	114	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})))
116	53, 112, 115	mpbir2and 959	. . . . . . . 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}))
117	116	adantlr 747	. . . . . . 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}))
118	13, 28	ringcl 18384	. . . . . . . . . 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))))
119	39, 45, 51, 118	syl3anc 1318	. . . . . . . . 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))))
120		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
121		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘(𝑆 ↑s (𝐵 ↑𝑚 𝐼))) = (mulGrp‘(𝑆 ↑s (𝐵 ↑𝑚 𝐼)))
122	120, 121	rhmmhm 18545	. . . . . . . . . . . . . 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)))))
123	12, 122	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))) MndHom (mulGrp‘(𝑆 ↑s (𝐵 ↑𝑚 𝐼)))))
124	123	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)))))
125	120, 13	mgpbas 18318	. . . . . . . . . . . . 13 ⊢ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (Base‘(mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
126	120, 28	mgpplusg 18316	. . . . . . . . . . . . 13 ⊢ (.r‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (+g‘(mulGrp‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
127		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (.r‘(𝑆 ↑s (𝐵 ↑𝑚 𝐼))) = (.r‘(𝑆 ↑s (𝐵 ↑𝑚 𝐼)))
128	121, 127	mgpplusg 18316	. . . . . . . . . . . . 13 ⊢ (.r‘(𝑆 ↑s (𝐵 ↑𝑚 𝐼))) = (+g‘(mulGrp‘(𝑆 ↑s (𝐵 ↑𝑚 𝐼))))
129	125, 126, 128	mhmlin 17165	. . . . . . . . . . . 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)))
130	124, 45, 51, 129	syl3anc 1318	. . . . . . . . . . 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)))
131		mpfind.ct	. . . . . . . . . . . 12 ⊢ · = (.r‘𝑆)
132	9, 14, 60, 62, 64, 65, 131, 127	pwsmulrval 15974	. . . . . . . . . . 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)))
133	130, 132	eqtrd 2644	. . . . . . . . . 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)))
134		ovex 6577	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∘𝑓 · 𝑔) ∈ V
135		mpfind.wf	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 ∘𝑓 · 𝑔) → (𝜓 ↔ 𝜎))
136	134, 135	elab 3319	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘𝑓 · 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜎)
137		oveq12 6558	. . . . . . . . . . . . . . 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)))
138	137	eleq1d 2672	. . . . . . . . . . . . . 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}))
139	136, 138	syl5bbr 273	. . . . . . . . . . . . 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}))
140	101, 139	imbi12d 333	. . . . . . . . . . . 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})))
141		mpfind.mu	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜎)
142	84, 85, 140, 141	vtocl2 3234	. . . . . . . . . . 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})
143	69, 77, 83, 142	syl12anc 1316	. . . . . . . . . 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})
144	133, 143	eqeltrd 2688	. . . . . . . . 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})
145		elpreima 6245	. . . . . . . . . . 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})))
146	18, 145	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})))
147	146	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})))
148	119, 144, 147	mpbir2and 959	. . . . . . . 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}))
149	148	adantlr 747	. . . . . . 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}))
150	7	mplassa 19275	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ V ∧ (𝑆 ↾s 𝑅) ∈ CRing) → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ AssAlg)
151	30, 34, 150	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ AssAlg)
152		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
153	29, 152	asclrhm 19163	. . . . . . . . . . . . 13 ⊢ ((𝐼 mPoly (𝑆 ↾s 𝑅)) ∈ AssAlg → (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∈ ((Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))) RingHom (𝐼 mPoly (𝑆 ↾s 𝑅))))
154	151, 153	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∈ ((Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))) RingHom (𝐼 mPoly (𝑆 ↾s 𝑅))))
155		eqid 2610	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) = (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
156	155, 13	rhmf 18549	. . . . . . . . . . . 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))))
157	154, 156	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))):(Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))⟶(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
158	157	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))))
159	7, 30, 34	mplsca 19266	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ↾s 𝑅) = (Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
160	159	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘(𝑆 ↾s 𝑅)) = (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))))
161	160	eleq2d 2673	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑖 ∈ (Base‘(𝑆 ↾s 𝑅)) ↔ 𝑖 ∈ (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅))))))
162	161	biimpa 500	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → 𝑖 ∈ (Base‘(Scalar‘(𝐼 mPoly (𝑆 ↾s 𝑅)))))
163	158, 162	ffvelrnd 6268	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
164	30	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → 𝐼 ∈ V)
165	31	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → 𝑆 ∈ CRing)
166	32	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → 𝑅 ∈ (SubRing‘𝑆))
167	10	subrgss 18604	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵)
168	32, 167	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ⊆ 𝐵)
169	8, 10	ressbas2 15758	. . . . . . . . . . . . . 14 ⊢ (𝑅 ⊆ 𝐵 → 𝑅 = (Base‘(𝑆 ↾s 𝑅)))
170	168, 169	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 = (Base‘(𝑆 ↾s 𝑅)))
171	170	eleq2d 2673	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑖 ∈ 𝑅 ↔ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))))
172	171	biimpar 501	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → 𝑖 ∈ 𝑅)
173	6, 7, 8, 10, 29, 164, 165, 166, 172	evlssca 19343	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖)) = ((𝐵 ↑𝑚 𝐼) × {𝑖}))
174		mpfind.co	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑅) → 𝜒)
175	174	ralrimiva 2949	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑓 ∈ 𝑅 𝜒)
176		snex 4835	. . . . . . . . . . . . . . . . 17 ⊢ {𝑓} ∈ V
177	61, 176	xpex 6860	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ↑𝑚 𝐼) × {𝑓}) ∈ V
178		mpfind.wa	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ((𝐵 ↑𝑚 𝐼) × {𝑓}) → (𝜓 ↔ 𝜒))
179	177, 178	elab 3319	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ↑𝑚 𝐼) × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)
180		sneq 4135	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝑖 → {𝑓} = {𝑖})
181	180	xpeq2d 5063	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑖 → ((𝐵 ↑𝑚 𝐼) × {𝑓}) = ((𝐵 ↑𝑚 𝐼) × {𝑖}))
182	181	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑖 → (((𝐵 ↑𝑚 𝐼) × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ ((𝐵 ↑𝑚 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓}))
183	179, 182	syl5bbr 273	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑖 → (𝜒 ↔ ((𝐵 ↑𝑚 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓}))
184	183	cbvralv 3147	. . . . . . . . . . . . 13 ⊢ (∀𝑓 ∈ 𝑅 𝜒 ↔ ∀𝑖 ∈ 𝑅 ((𝐵 ↑𝑚 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓})
185	175, 184	sylib 207	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑖 ∈ 𝑅 ((𝐵 ↑𝑚 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓})
186	185	r19.21bi 2916	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑅) → ((𝐵 ↑𝑚 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓})
187	172, 186	syldan 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → ((𝐵 ↑𝑚 𝐼) × {𝑖}) ∈ {𝑥 ∣ 𝜓})
188	173, 187	eqeltrd 2688	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖)) ∈ {𝑥 ∣ 𝜓})
189		elpreima 6245	. . . . . . . . . . 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})))
190	18, 189	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})))
191	190	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})))
192	163, 188, 191	mpbir2and 959	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (Base‘(𝑆 ↾s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
193	192	adantlr 747	. . . . . . 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}))
194	30	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐼 ∈ V)
195	36	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑆 ↾s 𝑅) ∈ Ring)
196		simpr 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
197	7, 26, 13, 194, 195, 196	mvrcl 19270	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
198	31	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ CRing)
199	32	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑅 ∈ (SubRing‘𝑆))
200	6, 26, 8, 10, 194, 198, 199, 196	evlsvar 19344	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖)) = (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑖)))
201		mpfind.pr	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐼) → 𝜃)
202	61	mptex 6390	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑓)) ∈ V
203		mpfind.wb	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑓)) → (𝜓 ↔ 𝜃))
204	202, 203	elab 3319	. . . . . . . . . . . . . 14 ⊢ ((𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓} ↔ 𝜃)
205	201, 204	sylibr 223	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐼) → (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓})
206	205	ralrimiva 2949	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑓 ∈ 𝐼 (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓})
207		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑖 → (𝑔‘𝑓) = (𝑔‘𝑖))
208	207	mpteq2dv 4673	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑖 → (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑓)) = (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑖)))
209	208	eleq1d 2672	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑖 → ((𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓} ↔ (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓}))
210	209	cbvralv 3147	. . . . . . . . . . . 12 ⊢ (∀𝑓 ∈ 𝐼 (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∣ 𝜓} ↔ ∀𝑖 ∈ 𝐼 (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓})
211	206, 210	sylib 207	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓})
212	211	r19.21bi 2916	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑖)) ∈ {𝑥 ∣ 𝜓})
213	200, 212	eqeltrd 2688	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖)) ∈ {𝑥 ∣ 𝜓})
214		elpreima 6245	. . . . . . . . . . 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})))
215	18, 214	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})))
216	215	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})))
217	197, 213, 216	mpbir2and 959	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
218	217	adantlr 747	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) ∧ 𝑖 ∈ 𝐼) → ((𝐼 mVar (𝑆 ↾s 𝑅))‘𝑖) ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
219		simpr 476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
220	30	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → 𝐼 ∈ V)
221	34	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (𝑆 ↾s 𝑅) ∈ CRing)
222	25, 26, 7, 27, 28, 29, 13, 117, 149, 193, 218, 219, 220, 221	mplind 19323	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → 𝑦 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓}))
223		fvimacnvi 6239	. . . . . 6 ⊢ ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑦 ∈ (◡((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥 ∣ 𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥 ∣ 𝜓})
224	24, 222, 223	syl2anc 691	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥 ∣ 𝜓})
225		eleq1 2676	. . . . 5 ⊢ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴 → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥 ∣ 𝜓} ↔ 𝐴 ∈ {𝑥 ∣ 𝜓}))
226	224, 225	syl5ibcom 234	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴 → 𝐴 ∈ {𝑥 ∣ 𝜓}))
227	226	rexlimdva 3013	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴 → 𝐴 ∈ {𝑥 ∣ 𝜓}))
228	21, 227	mpd 15	. 2 ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜓})
229		mpfind.wg	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜌))
230	229	elabg 3320	. . 3 ⊢ (𝐴 ∈ 𝑄 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜌))
231	1, 230	syl 17	. 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜌))
232	228, 231	mpbid 221	1 ⊢ (𝜑 → 𝜌)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  {csn 4125   ↦ cmpt 4643   × cxp 5036  ^◡ccnv 5037  ran crn 5039   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793   ↑_𝑚 cmap 7744  Basecbs 15695   ↾_s cress 15696  +_gcplusg 15768  ._rcmulr 15769  Scalarcsca 15771   ↑_s cpws 15930   MndHom cmhm 17156   GrpHom cghm 17480  mulGrpcmgp 18312  Ringcrg 18370  CRingccrg 18371   RingHom crh 18535  SubRingcsubrg 18599  AssAlgcasa 19130  algSccascl 19132   mVar cmvr 19173   mPoly cmpl 19174   evalSub ces 19325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-hom 15793  df-cco 15794  df-0g 15925  df-gsum 15926  df-prds 15931  df-pws 15933  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-srg 18329  df-ring 18372  df-cring 18373  df-rnghom 18538  df-subrg 18601  df-lmod 18688  df-lss 18754  df-lsp 18793  df-assa 19133  df-asp 19134  df-ascl 19135  df-psr 19177  df-mvr 19178  df-mpl 19179  df-evls 19327

This theorem is referenced by:  pf1ind  19540  mzpmfp  36328