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Theorem mpfind 19455
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 + = (+g𝑆)
mpfind.ct · = (.r𝑆)
mpfind.cq 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)
mpfind.ad ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜁)
mpfind.mu ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜎)
mpfind.wa (𝑥 = ((𝐵𝑚 𝐼) × {𝑓}) → (𝜓𝜒))
mpfind.wb (𝑥 = (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑓)) → (𝜓𝜃))
mpfind.wc (𝑥 = 𝑓 → (𝜓𝜏))
mpfind.wd (𝑥 = 𝑔 → (𝜓𝜂))
mpfind.we (𝑥 = (𝑓𝑓 + 𝑔) → (𝜓𝜁))
mpfind.wf (𝑥 = (𝑓𝑓 · 𝑔) → (𝜓𝜎))
mpfind.wg (𝑥 = 𝐴 → (𝜓𝜌))
mpfind.co ((𝜑𝑓𝑅) → 𝜒)
mpfind.pr ((𝜑𝑓𝐼) → 𝜃)
mpfind.a (𝜑𝐴𝑄)
Assertion
Ref Expression
mpfind (𝜑𝜌)
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.
StepHypRef Expression
1 mpfind.a . . . . 5 (𝜑𝐴𝑄)
2 mpfind.cq . . . . 5 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)
31, 2syl6eleq 2708 . . . 4 (𝜑𝐴 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
42mpfrcl 19437 . . . . . . . . 9 (𝐴𝑄 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))
51, 4syl 17 . . . . . . . 8 (𝜑 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))
6 eqid 2621 . . . . . . . . 9 ((𝐼 evalSub 𝑆)‘𝑅) = ((𝐼 evalSub 𝑆)‘𝑅)
7 eqid 2621 . . . . . . . . 9 (𝐼 mPoly (𝑆s 𝑅)) = (𝐼 mPoly (𝑆s 𝑅))
8 eqid 2621 . . . . . . . . 9 (𝑆s 𝑅) = (𝑆s 𝑅)
9 eqid 2621 . . . . . . . . 9 (𝑆s (𝐵𝑚 𝐼)) = (𝑆s (𝐵𝑚 𝐼))
10 mpfind.cb . . . . . . . . 9 𝐵 = (Base‘𝑆)
116, 7, 8, 9, 10evlsrhm 19440 . . . . . . . 8 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆s 𝑅)) RingHom (𝑆s (𝐵𝑚 𝐼))))
125, 11syl 17 . . . . . . 7 (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆s 𝑅)) RingHom (𝑆s (𝐵𝑚 𝐼))))
13 eqid 2621 . . . . . . . 8 (Base‘(𝐼 mPoly (𝑆s 𝑅))) = (Base‘(𝐼 mPoly (𝑆s 𝑅)))
14 eqid 2621 . . . . . . . 8 (Base‘(𝑆s (𝐵𝑚 𝐼))) = (Base‘(𝑆s (𝐵𝑚 𝐼)))
1513, 14rhmf 18647 . . . . . . 7 (((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆s 𝑅)) RingHom (𝑆s (𝐵𝑚 𝐼))) → ((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆s 𝑅)))⟶(Base‘(𝑆s (𝐵𝑚 𝐼))))
1612, 15syl 17 . . . . . 6 (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆s 𝑅)))⟶(Base‘(𝑆s (𝐵𝑚 𝐼))))
17 ffn 6002 . . . . . 6 (((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆s 𝑅)))⟶(Base‘(𝑆s (𝐵𝑚 𝐼))) → ((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆s 𝑅))))
1816, 17syl 17 . . . . 5 (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆s 𝑅))))
19 fvelrnb 6200 . . . . 5 (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆s 𝑅))) → (𝐴 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅) ↔ ∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴))
2018, 19syl 17 . . . 4 (𝜑 → (𝐴 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅) ↔ ∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴))
213, 20mpbid 222 . . 3 (𝜑 → ∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴)
22 ffun 6005 . . . . . . . 8 (((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆s 𝑅)))⟶(Base‘(𝑆s (𝐵𝑚 𝐼))) → Fun ((𝐼 evalSub 𝑆)‘𝑅))
2316, 22syl 17 . . . . . . 7 (𝜑 → Fun ((𝐼 evalSub 𝑆)‘𝑅))
2423adantr 481 . . . . . 6 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → Fun ((𝐼 evalSub 𝑆)‘𝑅))
25 eqid 2621 . . . . . . 7 (Base‘(𝑆s 𝑅)) = (Base‘(𝑆s 𝑅))
26 eqid 2621 . . . . . . 7 (𝐼 mVar (𝑆s 𝑅)) = (𝐼 mVar (𝑆s 𝑅))
27 eqid 2621 . . . . . . 7 (+g‘(𝐼 mPoly (𝑆s 𝑅))) = (+g‘(𝐼 mPoly (𝑆s 𝑅)))
28 eqid 2621 . . . . . . 7 (.r‘(𝐼 mPoly (𝑆s 𝑅))) = (.r‘(𝐼 mPoly (𝑆s 𝑅)))
29 eqid 2621 . . . . . . 7 (algSc‘(𝐼 mPoly (𝑆s 𝑅))) = (algSc‘(𝐼 mPoly (𝑆s 𝑅)))
305simp1d 1071 . . . . . . . . . . . 12 (𝜑𝐼 ∈ V)
315simp2d 1072 . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ CRing)
325simp3d 1073 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ (SubRing‘𝑆))
338subrgcrng 18705 . . . . . . . . . . . . . 14 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑆s 𝑅) ∈ CRing)
3431, 32, 33syl2anc 692 . . . . . . . . . . . . 13 (𝜑 → (𝑆s 𝑅) ∈ CRing)
35 crngring 18479 . . . . . . . . . . . . 13 ((𝑆s 𝑅) ∈ CRing → (𝑆s 𝑅) ∈ Ring)
3634, 35syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑆s 𝑅) ∈ Ring)
377mplring 19371 . . . . . . . . . . . 12 ((𝐼 ∈ V ∧ (𝑆s 𝑅) ∈ Ring) → (𝐼 mPoly (𝑆s 𝑅)) ∈ Ring)
3830, 36, 37syl2anc 692 . . . . . . . . . . 11 (𝜑 → (𝐼 mPoly (𝑆s 𝑅)) ∈ Ring)
3938adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (𝐼 mPoly (𝑆s 𝑅)) ∈ Ring)
40 simprl 793 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
41 elpreima 6293 . . . . . . . . . . . . . 14 (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆s 𝑅))) → (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ↔ (𝑖 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓})))
4218, 41syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ↔ (𝑖 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓})))
4342adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ↔ (𝑖 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓})))
4440, 43mpbid 222 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (𝑖 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓}))
4544simpld 475 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
46 simprr 795 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
47 elpreima 6293 . . . . . . . . . . . . . 14 (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆s 𝑅))) → (𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ↔ (𝑗 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓})))
4818, 47syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ↔ (𝑗 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓})))
4948adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ↔ (𝑗 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓})))
5046, 49mpbid 222 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (𝑗 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓}))
5150simpld 475 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
5213, 27ringacl 18499 . . . . . . . . . 10 (((𝐼 mPoly (𝑆s 𝑅)) ∈ Ring ∧ 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → (𝑖(+g‘(𝐼 mPoly (𝑆s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
5339, 45, 51, 52syl3anc 1323 . . . . . . . . 9 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (𝑖(+g‘(𝐼 mPoly (𝑆s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
54 rhmghm 18646 . . . . . . . . . . . . . 14 (((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆s 𝑅)) RingHom (𝑆s (𝐵𝑚 𝐼))) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆s 𝑅)) GrpHom (𝑆s (𝐵𝑚 𝐼))))
5512, 54syl 17 . . . . . . . . . . . . 13 (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆s 𝑅)) GrpHom (𝑆s (𝐵𝑚 𝐼))))
5655adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆s 𝑅)) GrpHom (𝑆s (𝐵𝑚 𝐼))))
57 eqid 2621 . . . . . . . . . . . . 13 (+g‘(𝑆s (𝐵𝑚 𝐼))) = (+g‘(𝑆s (𝐵𝑚 𝐼)))
5813, 27, 57ghmlin 17586 . . . . . . . . . . . 12 ((((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆s 𝑅)) GrpHom (𝑆s (𝐵𝑚 𝐼))) ∧ 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆s 𝑅)))𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖)(+g‘(𝑆s (𝐵𝑚 𝐼)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)))
5956, 45, 51, 58syl3anc 1323 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆s 𝑅)))𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖)(+g‘(𝑆s (𝐵𝑚 𝐼)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)))
6031adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝑆 ∈ CRing)
61 ovex 6632 . . . . . . . . . . . . 13 (𝐵𝑚 𝐼) ∈ V
6261a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (𝐵𝑚 𝐼) ∈ V)
6316adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → ((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆s 𝑅)))⟶(Base‘(𝑆s (𝐵𝑚 𝐼))))
6463, 45ffvelrnd 6316 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ (Base‘(𝑆s (𝐵𝑚 𝐼))))
6563, 51ffvelrnd 6316 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ (Base‘(𝑆s (𝐵𝑚 𝐼))))
66 mpfind.cp . . . . . . . . . . . 12 + = (+g𝑆)
679, 14, 60, 62, 64, 65, 66, 57pwsplusgval 16071 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖)(+g‘(𝑆s (𝐵𝑚 𝐼)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘𝑓 + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)))
6859, 67eqtrd 2655 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆s 𝑅)))𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘𝑓 + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)))
69 simpl 473 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝜑)
7018adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → ((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆s 𝑅))))
71 fnfvelrn 6312 . . . . . . . . . . . . . 14 ((((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
7270, 45, 71syl2anc 692 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
7372, 2syl6eleqr 2709 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄)
7423adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → Fun ((𝐼 evalSub 𝑆)‘𝑅))
75 fvimacnvi 6287 . . . . . . . . . . . . 13 ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓})
7674, 40, 75syl2anc 692 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓})
7773, 76jca 554 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓}))
78 fnfvelrn 6312 . . . . . . . . . . . . . 14 ((((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
7970, 51, 78syl2anc 692 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
8079, 2syl6eleqr 2709 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄)
81 fvimacnvi 6287 . . . . . . . . . . . . 13 ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓})
8274, 46, 81syl2anc 692 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓})
8380, 82jca 554 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓}))
84 fvex 6158 . . . . . . . . . . . 12 (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ V
85 fvex 6158 . . . . . . . . . . . 12 (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ V
86 eleq1 2686 . . . . . . . . . . . . . . . 16 (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝑓𝑄 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄))
87 vex 3189 . . . . . . . . . . . . . . . . . 18 𝑓 ∈ V
88 mpfind.wc . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑓 → (𝜓𝜏))
8987, 88elab 3333 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ {𝑥𝜓} ↔ 𝜏)
90 eleq1 2686 . . . . . . . . . . . . . . . . 17 (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝑓 ∈ {𝑥𝜓} ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓}))
9189, 90syl5bbr 274 . . . . . . . . . . . . . . . 16 (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝜏 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓}))
9286, 91anbi12d 746 . . . . . . . . . . . . . . 15 (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → ((𝑓𝑄𝜏) ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓})))
93 eleq1 2686 . . . . . . . . . . . . . . . 16 (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝑔𝑄 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄))
94 vex 3189 . . . . . . . . . . . . . . . . . 18 𝑔 ∈ V
95 mpfind.wd . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑔 → (𝜓𝜂))
9694, 95elab 3333 . . . . . . . . . . . . . . . . 17 (𝑔 ∈ {𝑥𝜓} ↔ 𝜂)
97 eleq1 2686 . . . . . . . . . . . . . . . . 17 (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝑔 ∈ {𝑥𝜓} ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓}))
9896, 97syl5bbr 274 . . . . . . . . . . . . . . . 16 (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝜂 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓}))
9993, 98anbi12d 746 . . . . . . . . . . . . . . 15 (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → ((𝑔𝑄𝜂) ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓})))
10092, 99bi2anan9 916 . . . . . . . . . . . . . 14 ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → (((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂)) ↔ (((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓}) ∧ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓}))))
101100anbi2d 739 . . . . . . . . . . . . 13 ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) ↔ (𝜑 ∧ (((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓}) ∧ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓})))))
102 ovex 6632 . . . . . . . . . . . . . . 15 (𝑓𝑓 + 𝑔) ∈ V
103 mpfind.we . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑓 + 𝑔) → (𝜓𝜁))
104102, 103elab 3333 . . . . . . . . . . . . . 14 ((𝑓𝑓 + 𝑔) ∈ {𝑥𝜓} ↔ 𝜁)
105 oveq12 6613 . . . . . . . . . . . . . . 15 ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → (𝑓𝑓 + 𝑔) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘𝑓 + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)))
106105eleq1d 2683 . . . . . . . . . . . . . 14 ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → ((𝑓𝑓 + 𝑔) ∈ {𝑥𝜓} ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘𝑓 + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥𝜓}))
107104, 106syl5bbr 274 . . . . . . . . . . . . 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)) ∈ {𝑥𝜓}))
108101, 107imbi12d 334 . . . . . . . . . . . 12 ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → (((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜁) ↔ ((𝜑 ∧ (((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓}) ∧ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘𝑓 + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥𝜓})))
109 mpfind.ad . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜁)
11084, 85, 108, 109vtocl2 3247 . . . . . . . . . . 11 ((𝜑 ∧ (((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓}) ∧ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘𝑓 + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥𝜓})
11169, 77, 83, 110syl12anc 1321 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘𝑓 + (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥𝜓})
11268, 111eqeltrd 2698 . . . . . . . . 9 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆s 𝑅)))𝑗)) ∈ {𝑥𝜓})
113 elpreima 6293 . . . . . . . . . . 11 (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆s 𝑅))) → ((𝑖(+g‘(𝐼 mPoly (𝑆s 𝑅)))𝑗) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ↔ ((𝑖(+g‘(𝐼 mPoly (𝑆s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆s 𝑅)))𝑗)) ∈ {𝑥𝜓})))
11418, 113syl 17 . . . . . . . . . 10 (𝜑 → ((𝑖(+g‘(𝐼 mPoly (𝑆s 𝑅)))𝑗) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ↔ ((𝑖(+g‘(𝐼 mPoly (𝑆s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆s 𝑅)))𝑗)) ∈ {𝑥𝜓})))
115114adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → ((𝑖(+g‘(𝐼 mPoly (𝑆s 𝑅)))𝑗) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ↔ ((𝑖(+g‘(𝐼 mPoly (𝑆s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(+g‘(𝐼 mPoly (𝑆s 𝑅)))𝑗)) ∈ {𝑥𝜓})))
11653, 112, 115mpbir2and 956 . . . . . . . 8 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (𝑖(+g‘(𝐼 mPoly (𝑆s 𝑅)))𝑗) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
117116adantlr 750 . . . . . . 7 (((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (𝑖(+g‘(𝐼 mPoly (𝑆s 𝑅)))𝑗) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
11813, 28ringcl 18482 . . . . . . . . . 10 (((𝐼 mPoly (𝑆s 𝑅)) ∈ Ring ∧ 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → (𝑖(.r‘(𝐼 mPoly (𝑆s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
11939, 45, 51, 118syl3anc 1323 . . . . . . . . 9 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (𝑖(.r‘(𝐼 mPoly (𝑆s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
120 eqid 2621 . . . . . . . . . . . . . . 15 (mulGrp‘(𝐼 mPoly (𝑆s 𝑅))) = (mulGrp‘(𝐼 mPoly (𝑆s 𝑅)))
121 eqid 2621 . . . . . . . . . . . . . . 15 (mulGrp‘(𝑆s (𝐵𝑚 𝐼))) = (mulGrp‘(𝑆s (𝐵𝑚 𝐼)))
122120, 121rhmmhm 18643 . . . . . . . . . . . . . 14 (((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆s 𝑅)) RingHom (𝑆s (𝐵𝑚 𝐼))) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((mulGrp‘(𝐼 mPoly (𝑆s 𝑅))) MndHom (mulGrp‘(𝑆s (𝐵𝑚 𝐼)))))
12312, 122syl 17 . . . . . . . . . . . . 13 (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((mulGrp‘(𝐼 mPoly (𝑆s 𝑅))) MndHom (mulGrp‘(𝑆s (𝐵𝑚 𝐼)))))
124123adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((mulGrp‘(𝐼 mPoly (𝑆s 𝑅))) MndHom (mulGrp‘(𝑆s (𝐵𝑚 𝐼)))))
125120, 13mgpbas 18416 . . . . . . . . . . . . 13 (Base‘(𝐼 mPoly (𝑆s 𝑅))) = (Base‘(mulGrp‘(𝐼 mPoly (𝑆s 𝑅))))
126120, 28mgpplusg 18414 . . . . . . . . . . . . 13 (.r‘(𝐼 mPoly (𝑆s 𝑅))) = (+g‘(mulGrp‘(𝐼 mPoly (𝑆s 𝑅))))
127 eqid 2621 . . . . . . . . . . . . . 14 (.r‘(𝑆s (𝐵𝑚 𝐼))) = (.r‘(𝑆s (𝐵𝑚 𝐼)))
128121, 127mgpplusg 18414 . . . . . . . . . . . . 13 (.r‘(𝑆s (𝐵𝑚 𝐼))) = (+g‘(mulGrp‘(𝑆s (𝐵𝑚 𝐼))))
129125, 126, 128mhmlin 17263 . . . . . . . . . . . 12 ((((𝐼 evalSub 𝑆)‘𝑅) ∈ ((mulGrp‘(𝐼 mPoly (𝑆s 𝑅))) MndHom (mulGrp‘(𝑆s (𝐵𝑚 𝐼)))) ∧ 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆s 𝑅)))𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖)(.r‘(𝑆s (𝐵𝑚 𝐼)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)))
130124, 45, 51, 129syl3anc 1323 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆s 𝑅)))𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖)(.r‘(𝑆s (𝐵𝑚 𝐼)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)))
131 mpfind.ct . . . . . . . . . . . 12 · = (.r𝑆)
1329, 14, 60, 62, 64, 65, 131, 127pwsmulrval 16072 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖)(.r‘(𝑆s (𝐵𝑚 𝐼)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘𝑓 · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)))
133130, 132eqtrd 2655 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆s 𝑅)))𝑗)) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘𝑓 · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)))
134 ovex 6632 . . . . . . . . . . . . . . 15 (𝑓𝑓 · 𝑔) ∈ V
135 mpfind.wf . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑓 · 𝑔) → (𝜓𝜎))
136134, 135elab 3333 . . . . . . . . . . . . . 14 ((𝑓𝑓 · 𝑔) ∈ {𝑥𝜓} ↔ 𝜎)
137 oveq12 6613 . . . . . . . . . . . . . . 15 ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → (𝑓𝑓 · 𝑔) = ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘𝑓 · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)))
138137eleq1d 2683 . . . . . . . . . . . . . 14 ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → ((𝑓𝑓 · 𝑔) ∈ {𝑥𝜓} ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘𝑓 · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥𝜓}))
139136, 138syl5bbr 274 . . . . . . . . . . . . 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)) ∈ {𝑥𝜓}))
140101, 139imbi12d 334 . . . . . . . . . . . 12 ((𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∧ 𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) → (((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜎) ↔ ((𝜑 ∧ (((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓}) ∧ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘𝑓 · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥𝜓})))
141 mpfind.mu . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜎)
14284, 85, 140, 141vtocl2 3247 . . . . . . . . . . 11 ((𝜑 ∧ (((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓}) ∧ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘𝑓 · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥𝜓})
14369, 77, 83, 142syl12anc 1321 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∘𝑓 · (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗)) ∈ {𝑥𝜓})
144133, 143eqeltrd 2698 . . . . . . . . 9 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆s 𝑅)))𝑗)) ∈ {𝑥𝜓})
145 elpreima 6293 . . . . . . . . . . 11 (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆s 𝑅))) → ((𝑖(.r‘(𝐼 mPoly (𝑆s 𝑅)))𝑗) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ↔ ((𝑖(.r‘(𝐼 mPoly (𝑆s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆s 𝑅)))𝑗)) ∈ {𝑥𝜓})))
14618, 145syl 17 . . . . . . . . . 10 (𝜑 → ((𝑖(.r‘(𝐼 mPoly (𝑆s 𝑅)))𝑗) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ↔ ((𝑖(.r‘(𝐼 mPoly (𝑆s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆s 𝑅)))𝑗)) ∈ {𝑥𝜓})))
147146adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → ((𝑖(.r‘(𝐼 mPoly (𝑆s 𝑅)))𝑗) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ↔ ((𝑖(.r‘(𝐼 mPoly (𝑆s 𝑅)))𝑗) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘(𝑖(.r‘(𝐼 mPoly (𝑆s 𝑅)))𝑗)) ∈ {𝑥𝜓})))
148119, 144, 147mpbir2and 956 . . . . . . . 8 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (𝑖(.r‘(𝐼 mPoly (𝑆s 𝑅)))𝑗) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
149148adantlr 750 . . . . . . 7 (((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (𝑖(.r‘(𝐼 mPoly (𝑆s 𝑅)))𝑗) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
1507mplassa 19373 . . . . . . . . . . . . . 14 ((𝐼 ∈ V ∧ (𝑆s 𝑅) ∈ CRing) → (𝐼 mPoly (𝑆s 𝑅)) ∈ AssAlg)
15130, 34, 150syl2anc 692 . . . . . . . . . . . . 13 (𝜑 → (𝐼 mPoly (𝑆s 𝑅)) ∈ AssAlg)
152 eqid 2621 . . . . . . . . . . . . . 14 (Scalar‘(𝐼 mPoly (𝑆s 𝑅))) = (Scalar‘(𝐼 mPoly (𝑆s 𝑅)))
15329, 152asclrhm 19261 . . . . . . . . . . . . 13 ((𝐼 mPoly (𝑆s 𝑅)) ∈ AssAlg → (algSc‘(𝐼 mPoly (𝑆s 𝑅))) ∈ ((Scalar‘(𝐼 mPoly (𝑆s 𝑅))) RingHom (𝐼 mPoly (𝑆s 𝑅))))
154151, 153syl 17 . . . . . . . . . . . 12 (𝜑 → (algSc‘(𝐼 mPoly (𝑆s 𝑅))) ∈ ((Scalar‘(𝐼 mPoly (𝑆s 𝑅))) RingHom (𝐼 mPoly (𝑆s 𝑅))))
155 eqid 2621 . . . . . . . . . . . . 13 (Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅)))) = (Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅))))
156155, 13rhmf 18647 . . . . . . . . . . . 12 ((algSc‘(𝐼 mPoly (𝑆s 𝑅))) ∈ ((Scalar‘(𝐼 mPoly (𝑆s 𝑅))) RingHom (𝐼 mPoly (𝑆s 𝑅))) → (algSc‘(𝐼 mPoly (𝑆s 𝑅))):(Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅))))⟶(Base‘(𝐼 mPoly (𝑆s 𝑅))))
157154, 156syl 17 . . . . . . . . . . 11 (𝜑 → (algSc‘(𝐼 mPoly (𝑆s 𝑅))):(Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅))))⟶(Base‘(𝐼 mPoly (𝑆s 𝑅))))
158157adantr 481 . . . . . . . . . 10 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → (algSc‘(𝐼 mPoly (𝑆s 𝑅))):(Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅))))⟶(Base‘(𝐼 mPoly (𝑆s 𝑅))))
1597, 30, 34mplsca 19364 . . . . . . . . . . . . 13 (𝜑 → (𝑆s 𝑅) = (Scalar‘(𝐼 mPoly (𝑆s 𝑅))))
160159fveq2d 6152 . . . . . . . . . . . 12 (𝜑 → (Base‘(𝑆s 𝑅)) = (Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅)))))
161160eleq2d 2684 . . . . . . . . . . 11 (𝜑 → (𝑖 ∈ (Base‘(𝑆s 𝑅)) ↔ 𝑖 ∈ (Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅))))))
162161biimpa 501 . . . . . . . . . 10 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → 𝑖 ∈ (Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅)))))
163158, 162ffvelrnd 6316 . . . . . . . . 9 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
16430adantr 481 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → 𝐼 ∈ V)
16531adantr 481 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → 𝑆 ∈ CRing)
16632adantr 481 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → 𝑅 ∈ (SubRing‘𝑆))
16710subrgss 18702 . . . . . . . . . . . . . . 15 (𝑅 ∈ (SubRing‘𝑆) → 𝑅𝐵)
16832, 167syl 17 . . . . . . . . . . . . . 14 (𝜑𝑅𝐵)
1698, 10ressbas2 15852 . . . . . . . . . . . . . 14 (𝑅𝐵𝑅 = (Base‘(𝑆s 𝑅)))
170168, 169syl 17 . . . . . . . . . . . . 13 (𝜑𝑅 = (Base‘(𝑆s 𝑅)))
171170eleq2d 2684 . . . . . . . . . . . 12 (𝜑 → (𝑖𝑅𝑖 ∈ (Base‘(𝑆s 𝑅))))
172171biimpar 502 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → 𝑖𝑅)
1736, 7, 8, 10, 29, 164, 165, 166, 172evlssca 19441 . . . . . . . . . 10 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖)) = ((𝐵𝑚 𝐼) × {𝑖}))
174 mpfind.co . . . . . . . . . . . . . 14 ((𝜑𝑓𝑅) → 𝜒)
175174ralrimiva 2960 . . . . . . . . . . . . 13 (𝜑 → ∀𝑓𝑅 𝜒)
176 snex 4869 . . . . . . . . . . . . . . . . 17 {𝑓} ∈ V
17761, 176xpex 6915 . . . . . . . . . . . . . . . 16 ((𝐵𝑚 𝐼) × {𝑓}) ∈ V
178 mpfind.wa . . . . . . . . . . . . . . . 16 (𝑥 = ((𝐵𝑚 𝐼) × {𝑓}) → (𝜓𝜒))
179177, 178elab 3333 . . . . . . . . . . . . . . 15 (((𝐵𝑚 𝐼) × {𝑓}) ∈ {𝑥𝜓} ↔ 𝜒)
180 sneq 4158 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝑖 → {𝑓} = {𝑖})
181180xpeq2d 5099 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑖 → ((𝐵𝑚 𝐼) × {𝑓}) = ((𝐵𝑚 𝐼) × {𝑖}))
182181eleq1d 2683 . . . . . . . . . . . . . . 15 (𝑓 = 𝑖 → (((𝐵𝑚 𝐼) × {𝑓}) ∈ {𝑥𝜓} ↔ ((𝐵𝑚 𝐼) × {𝑖}) ∈ {𝑥𝜓}))
183179, 182syl5bbr 274 . . . . . . . . . . . . . 14 (𝑓 = 𝑖 → (𝜒 ↔ ((𝐵𝑚 𝐼) × {𝑖}) ∈ {𝑥𝜓}))
184183cbvralv 3159 . . . . . . . . . . . . 13 (∀𝑓𝑅 𝜒 ↔ ∀𝑖𝑅 ((𝐵𝑚 𝐼) × {𝑖}) ∈ {𝑥𝜓})
185175, 184sylib 208 . . . . . . . . . . . 12 (𝜑 → ∀𝑖𝑅 ((𝐵𝑚 𝐼) × {𝑖}) ∈ {𝑥𝜓})
186185r19.21bi 2927 . . . . . . . . . . 11 ((𝜑𝑖𝑅) → ((𝐵𝑚 𝐼) × {𝑖}) ∈ {𝑥𝜓})
187172, 186syldan 487 . . . . . . . . . 10 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → ((𝐵𝑚 𝐼) × {𝑖}) ∈ {𝑥𝜓})
188173, 187eqeltrd 2698 . . . . . . . . 9 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖)) ∈ {𝑥𝜓})
189 elpreima 6293 . . . . . . . . . . 11 (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆s 𝑅))) → (((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ↔ (((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖)) ∈ {𝑥𝜓})))
19018, 189syl 17 . . . . . . . . . 10 (𝜑 → (((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ↔ (((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖)) ∈ {𝑥𝜓})))
191190adantr 481 . . . . . . . . 9 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → (((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ↔ (((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖)) ∈ {𝑥𝜓})))
192163, 188, 191mpbir2and 956 . . . . . . . 8 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
193192adantlr 750 . . . . . . 7 (((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) ∧ 𝑖 ∈ (Base‘(𝑆s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
19430adantr 481 . . . . . . . . . 10 ((𝜑𝑖𝐼) → 𝐼 ∈ V)
19536adantr 481 . . . . . . . . . 10 ((𝜑𝑖𝐼) → (𝑆s 𝑅) ∈ Ring)
196 simpr 477 . . . . . . . . . 10 ((𝜑𝑖𝐼) → 𝑖𝐼)
1977, 26, 13, 194, 195, 196mvrcl 19368 . . . . . . . . 9 ((𝜑𝑖𝐼) → ((𝐼 mVar (𝑆s 𝑅))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
19831adantr 481 . . . . . . . . . . 11 ((𝜑𝑖𝐼) → 𝑆 ∈ CRing)
19932adantr 481 . . . . . . . . . . 11 ((𝜑𝑖𝐼) → 𝑅 ∈ (SubRing‘𝑆))
2006, 26, 8, 10, 194, 198, 199, 196evlsvar 19442 . . . . . . . . . 10 ((𝜑𝑖𝐼) → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆s 𝑅))‘𝑖)) = (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑖)))
201 mpfind.pr . . . . . . . . . . . . . 14 ((𝜑𝑓𝐼) → 𝜃)
20261mptex 6440 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑓)) ∈ V
203 mpfind.wb . . . . . . . . . . . . . . 15 (𝑥 = (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑓)) → (𝜓𝜃))
204202, 203elab 3333 . . . . . . . . . . . . . 14 ((𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑓)) ∈ {𝑥𝜓} ↔ 𝜃)
205201, 204sylibr 224 . . . . . . . . . . . . 13 ((𝜑𝑓𝐼) → (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑓)) ∈ {𝑥𝜓})
206205ralrimiva 2960 . . . . . . . . . . . 12 (𝜑 → ∀𝑓𝐼 (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑓)) ∈ {𝑥𝜓})
207 fveq2 6148 . . . . . . . . . . . . . . 15 (𝑓 = 𝑖 → (𝑔𝑓) = (𝑔𝑖))
208207mpteq2dv 4705 . . . . . . . . . . . . . 14 (𝑓 = 𝑖 → (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑓)) = (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑖)))
209208eleq1d 2683 . . . . . . . . . . . . 13 (𝑓 = 𝑖 → ((𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑓)) ∈ {𝑥𝜓} ↔ (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑖)) ∈ {𝑥𝜓}))
210209cbvralv 3159 . . . . . . . . . . . 12 (∀𝑓𝐼 (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑓)) ∈ {𝑥𝜓} ↔ ∀𝑖𝐼 (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑖)) ∈ {𝑥𝜓})
211206, 210sylib 208 . . . . . . . . . . 11 (𝜑 → ∀𝑖𝐼 (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑖)) ∈ {𝑥𝜓})
212211r19.21bi 2927 . . . . . . . . . 10 ((𝜑𝑖𝐼) → (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑖)) ∈ {𝑥𝜓})
213200, 212eqeltrd 2698 . . . . . . . . 9 ((𝜑𝑖𝐼) → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆s 𝑅))‘𝑖)) ∈ {𝑥𝜓})
214 elpreima 6293 . . . . . . . . . . 11 (((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆s 𝑅))) → (((𝐼 mVar (𝑆s 𝑅))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ↔ (((𝐼 mVar (𝑆s 𝑅))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆s 𝑅))‘𝑖)) ∈ {𝑥𝜓})))
21518, 214syl 17 . . . . . . . . . 10 (𝜑 → (((𝐼 mVar (𝑆s 𝑅))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ↔ (((𝐼 mVar (𝑆s 𝑅))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆s 𝑅))‘𝑖)) ∈ {𝑥𝜓})))
216215adantr 481 . . . . . . . . 9 ((𝜑𝑖𝐼) → (((𝐼 mVar (𝑆s 𝑅))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ↔ (((𝐼 mVar (𝑆s 𝑅))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆s 𝑅))‘𝑖)) ∈ {𝑥𝜓})))
217197, 213, 216mpbir2and 956 . . . . . . . 8 ((𝜑𝑖𝐼) → ((𝐼 mVar (𝑆s 𝑅))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
218217adantlr 750 . . . . . . 7 (((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) ∧ 𝑖𝐼) → ((𝐼 mVar (𝑆s 𝑅))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
219 simpr 477 . . . . . . 7 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
22030adantr 481 . . . . . . 7 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → 𝐼 ∈ V)
22134adantr 481 . . . . . . 7 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → (𝑆s 𝑅) ∈ CRing)
22225, 26, 7, 27, 28, 29, 13, 117, 149, 193, 218, 219, 220, 221mplind 19421 . . . . . 6 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → 𝑦 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
223 fvimacnvi 6287 . . . . . 6 ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑦 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥𝜓})
22424, 222, 223syl2anc 692 . . . . 5 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥𝜓})
225 eleq1 2686 . . . . 5 ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴 → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥𝜓} ↔ 𝐴 ∈ {𝑥𝜓}))
226224, 225syl5ibcom 235 . . . 4 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴𝐴 ∈ {𝑥𝜓}))
227226rexlimdva 3024 . . 3 (𝜑 → (∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴𝐴 ∈ {𝑥𝜓}))
22821, 227mpd 15 . 2 (𝜑𝐴 ∈ {𝑥𝜓})
229 mpfind.wg . . . 4 (𝑥 = 𝐴 → (𝜓𝜌))
230229elabg 3334 . . 3 (𝐴𝑄 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜌))
2311, 230syl 17 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜌))
232228, 231mpbid 222 1 (𝜑𝜌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  Vcvv 3186  wss 3555  {csn 4148  cmpt 4673   × cxp 5072  ccnv 5073  ran crn 5075  cima 5077  Fun wfun 5841   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  𝑓 cof 6848  𝑚 cmap 7802  Basecbs 15781  s cress 15782  +gcplusg 15862  .rcmulr 15863  Scalarcsca 15865  s cpws 16028   MndHom cmhm 17254   GrpHom cghm 17578  mulGrpcmgp 18410  Ringcrg 18468  CRingccrg 18469   RingHom crh 18633  SubRingcsubrg 18697  AssAlgcasa 19228  algSccascl 19230   mVar cmvr 19271   mPoly cmpl 19272   evalSub ces 19423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-ofr 6851  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-sup 8292  df-oi 8359  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-fzo 12407  df-seq 12742  df-hash 13058  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-hom 15887  df-cco 15888  df-0g 16023  df-gsum 16024  df-prds 16029  df-pws 16031  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-submnd 17257  df-grp 17346  df-minusg 17347  df-sbg 17348  df-mulg 17462  df-subg 17512  df-ghm 17579  df-cntz 17671  df-cmn 18116  df-abl 18117  df-mgp 18411  df-ur 18423  df-srg 18427  df-ring 18470  df-cring 18471  df-rnghom 18636  df-subrg 18699  df-lmod 18786  df-lss 18852  df-lsp 18891  df-assa 19231  df-asp 19232  df-ascl 19233  df-psr 19275  df-mvr 19276  df-mpl 19277  df-evls 19425
This theorem is referenced by:  pf1ind  19638  mzpmfp  36787
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