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Theorem mpfind 19734
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 2845 . . . 4 (𝜑𝐴 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
42mpfrcl 19716 . . . . . . . . 9 (𝐴𝑄 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))
51, 4syl 17 . . . . . . . 8 (𝜑 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))
6 eqid 2756 . . . . . . . . 9 ((𝐼 evalSub 𝑆)‘𝑅) = ((𝐼 evalSub 𝑆)‘𝑅)
7 eqid 2756 . . . . . . . . 9 (𝐼 mPoly (𝑆s 𝑅)) = (𝐼 mPoly (𝑆s 𝑅))
8 eqid 2756 . . . . . . . . 9 (𝑆s 𝑅) = (𝑆s 𝑅)
9 eqid 2756 . . . . . . . . 9 (𝑆s (𝐵𝑚 𝐼)) = (𝑆s (𝐵𝑚 𝐼))
10 mpfind.cb . . . . . . . . 9 𝐵 = (Base‘𝑆)
116, 7, 8, 9, 10evlsrhm 19719 . . . . . . . 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 2756 . . . . . . . 8 (Base‘(𝐼 mPoly (𝑆s 𝑅))) = (Base‘(𝐼 mPoly (𝑆s 𝑅)))
14 eqid 2756 . . . . . . . 8 (Base‘(𝑆s (𝐵𝑚 𝐼))) = (Base‘(𝑆s (𝐵𝑚 𝐼)))
1513, 14rhmf 18924 . . . . . . 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 6202 . . . . . 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 6401 . . . . 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 6205 . . . . . . . 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 472 . . . . . 6 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → Fun ((𝐼 evalSub 𝑆)‘𝑅))
25 eqid 2756 . . . . . . 7 (Base‘(𝑆s 𝑅)) = (Base‘(𝑆s 𝑅))
26 eqid 2756 . . . . . . 7 (𝐼 mVar (𝑆s 𝑅)) = (𝐼 mVar (𝑆s 𝑅))
27 eqid 2756 . . . . . . 7 (+g‘(𝐼 mPoly (𝑆s 𝑅))) = (+g‘(𝐼 mPoly (𝑆s 𝑅)))
28 eqid 2756 . . . . . . 7 (.r‘(𝐼 mPoly (𝑆s 𝑅))) = (.r‘(𝐼 mPoly (𝑆s 𝑅)))
29 eqid 2756 . . . . . . 7 (algSc‘(𝐼 mPoly (𝑆s 𝑅))) = (algSc‘(𝐼 mPoly (𝑆s 𝑅)))
305simp1d 1137 . . . . . . . . . . . 12 (𝜑𝐼 ∈ V)
315simp2d 1138 . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ CRing)
325simp3d 1139 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ (SubRing‘𝑆))
338subrgcrng 18982 . . . . . . . . . . . . . 14 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑆s 𝑅) ∈ CRing)
3431, 32, 33syl2anc 696 . . . . . . . . . . . . 13 (𝜑 → (𝑆s 𝑅) ∈ CRing)
35 crngring 18754 . . . . . . . . . . . . 13 ((𝑆s 𝑅) ∈ CRing → (𝑆s 𝑅) ∈ Ring)
3634, 35syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑆s 𝑅) ∈ Ring)
377mplring 19650 . . . . . . . . . . . 12 ((𝐼 ∈ V ∧ (𝑆s 𝑅) ∈ Ring) → (𝐼 mPoly (𝑆s 𝑅)) ∈ Ring)
3830, 36, 37syl2anc 696 . . . . . . . . . . 11 (𝜑 → (𝐼 mPoly (𝑆s 𝑅)) ∈ Ring)
3938adantr 472 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (𝐼 mPoly (𝑆s 𝑅)) ∈ Ring)
40 simprl 811 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
41 elpreima 6496 . . . . . . . . . . . . . 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 472 . . . . . . . . . . . 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 477 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
46 simprr 813 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
47 elpreima 6496 . . . . . . . . . . . . . 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 472 . . . . . . . . . . . 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 477 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
5213, 27ringacl 18774 . . . . . . . . . 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 1477 . . . . . . . . 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 18923 . . . . . . . . . . . . . 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 472 . . . . . . . . . . . 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 2756 . . . . . . . . . . . . 13 (+g‘(𝑆s (𝐵𝑚 𝐼))) = (+g‘(𝑆s (𝐵𝑚 𝐼)))
5813, 27, 57ghmlin 17862 . . . . . . . . . . . 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 1477 . . . . . . . . . . 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 472 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝑆 ∈ CRing)
61 ovexd 6839 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (𝐵𝑚 𝐼) ∈ V)
6216adantr 472 . . . . . . . . . . . . 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))))
6362, 45ffvelrnd 6519 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ (Base‘(𝑆s (𝐵𝑚 𝐼))))
6462, 51ffvelrnd 6519 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ (Base‘(𝑆s (𝐵𝑚 𝐼))))
65 mpfind.cp . . . . . . . . . . . 12 + = (+g𝑆)
669, 14, 60, 61, 63, 64, 65, 57pwsplusgval 16348 . . . . . . . . . . 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)))
6759, 66eqtrd 2790 . . . . . . . . . 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)))
68 simpl 474 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝜑)
6918adantr 472 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → ((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆s 𝑅))))
70 fnfvelrn 6515 . . . . . . . . . . . . . 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))
7169, 45, 70syl2anc 696 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
7271, 2syl6eleqr 2846 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄)
7323adantr 472 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → Fun ((𝐼 evalSub 𝑆)‘𝑅))
74 fvimacnvi 6490 . . . . . . . . . . . . 13 ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓})
7573, 40, 74syl2anc 696 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓})
7672, 75jca 555 . . . . . . . . . . 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}))
77 fnfvelrn 6515 . . . . . . . . . . . . . 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))
7869, 51, 77syl2anc 696 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
7978, 2syl6eleqr 2846 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄)
80 fvimacnvi 6490 . . . . . . . . . . . . 13 ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓})
8173, 46, 80syl2anc 696 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓})
8279, 81jca 555 . . . . . . . . . . 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}))
83 fvex 6358 . . . . . . . . . . . 12 (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ V
84 fvex 6358 . . . . . . . . . . . 12 (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ V
85 eleq1 2823 . . . . . . . . . . . . . . . 16 (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝑓𝑄 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄))
86 vex 3339 . . . . . . . . . . . . . . . . . 18 𝑓 ∈ V
87 mpfind.wc . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑓 → (𝜓𝜏))
8886, 87elab 3486 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ {𝑥𝜓} ↔ 𝜏)
89 eleq1 2823 . . . . . . . . . . . . . . . . 17 (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝑓 ∈ {𝑥𝜓} ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓}))
9088, 89syl5bbr 274 . . . . . . . . . . . . . . . 16 (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝜏 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓}))
9185, 90anbi12d 749 . . . . . . . . . . . . . . 15 (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → ((𝑓𝑄𝜏) ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓})))
92 eleq1 2823 . . . . . . . . . . . . . . . 16 (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝑔𝑄 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄))
93 vex 3339 . . . . . . . . . . . . . . . . . 18 𝑔 ∈ V
94 mpfind.wd . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑔 → (𝜓𝜂))
9593, 94elab 3486 . . . . . . . . . . . . . . . . 17 (𝑔 ∈ {𝑥𝜓} ↔ 𝜂)
96 eleq1 2823 . . . . . . . . . . . . . . . . 17 (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝑔 ∈ {𝑥𝜓} ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓}))
9795, 96syl5bbr 274 . . . . . . . . . . . . . . . 16 (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝜂 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓}))
9892, 97anbi12d 749 . . . . . . . . . . . . . . 15 (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → ((𝑔𝑄𝜂) ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓})))
9991, 98bi2anan9 953 . . . . . . . . . . . . . 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}))))
10099anbi2d 742 . . . . . . . . . . . . 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})))))
101 ovex 6837 . . . . . . . . . . . . . . 15 (𝑓𝑓 + 𝑔) ∈ V
102 mpfind.we . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑓 + 𝑔) → (𝜓𝜁))
103101, 102elab 3486 . . . . . . . . . . . . . 14 ((𝑓𝑓 + 𝑔) ∈ {𝑥𝜓} ↔ 𝜁)
104 oveq12 6818 . . . . . . . . . . . . . . 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)))
105104eleq1d 2820 . . . . . . . . . . . . . 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}))
106103, 105syl5bbr 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}))
107100, 106imbi12d 333 . . . . . . . . . . . 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})))
108 mpfind.ad . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜁)
10983, 84, 107, 108vtocl2 3397 . . . . . . . . . . 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})
11068, 76, 82, 109syl12anc 1475 . . . . . . . . . 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})
11167, 110eqeltrd 2835 . . . . . . . . 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})
112 elpreima 6496 . . . . . . . . . . 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})))
11318, 112syl 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})))
114113adantr 472 . . . . . . . . 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})))
11553, 111, 114mpbir2and 995 . . . . . . . 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}))
116115adantlr 753 . . . . . . 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}))
11713, 28ringcl 18757 . . . . . . . . . 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))))
11839, 45, 51, 117syl3anc 1477 . . . . . . . . 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))))
119 eqid 2756 . . . . . . . . . . . . . . 15 (mulGrp‘(𝐼 mPoly (𝑆s 𝑅))) = (mulGrp‘(𝐼 mPoly (𝑆s 𝑅)))
120 eqid 2756 . . . . . . . . . . . . . . 15 (mulGrp‘(𝑆s (𝐵𝑚 𝐼))) = (mulGrp‘(𝑆s (𝐵𝑚 𝐼)))
121119, 120rhmmhm 18920 . . . . . . . . . . . . . 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)))))
12212, 121syl 17 . . . . . . . . . . . . 13 (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((mulGrp‘(𝐼 mPoly (𝑆s 𝑅))) MndHom (mulGrp‘(𝑆s (𝐵𝑚 𝐼)))))
123122adantr 472 . . . . . . . . . . . 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)))))
124119, 13mgpbas 18691 . . . . . . . . . . . . 13 (Base‘(𝐼 mPoly (𝑆s 𝑅))) = (Base‘(mulGrp‘(𝐼 mPoly (𝑆s 𝑅))))
125119, 28mgpplusg 18689 . . . . . . . . . . . . 13 (.r‘(𝐼 mPoly (𝑆s 𝑅))) = (+g‘(mulGrp‘(𝐼 mPoly (𝑆s 𝑅))))
126 eqid 2756 . . . . . . . . . . . . . 14 (.r‘(𝑆s (𝐵𝑚 𝐼))) = (.r‘(𝑆s (𝐵𝑚 𝐼)))
127120, 126mgpplusg 18689 . . . . . . . . . . . . 13 (.r‘(𝑆s (𝐵𝑚 𝐼))) = (+g‘(mulGrp‘(𝑆s (𝐵𝑚 𝐼))))
128124, 125, 127mhmlin 17539 . . . . . . . . . . . 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)))
129123, 45, 51, 128syl3anc 1477 . . . . . . . . . . 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)))
130 mpfind.ct . . . . . . . . . . . 12 · = (.r𝑆)
1319, 14, 60, 61, 63, 64, 130, 126pwsmulrval 16349 . . . . . . . . . . 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)))
132129, 131eqtrd 2790 . . . . . . . . . 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)))
133 ovex 6837 . . . . . . . . . . . . . . 15 (𝑓𝑓 · 𝑔) ∈ V
134 mpfind.wf . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑓 · 𝑔) → (𝜓𝜎))
135133, 134elab 3486 . . . . . . . . . . . . . 14 ((𝑓𝑓 · 𝑔) ∈ {𝑥𝜓} ↔ 𝜎)
136 oveq12 6818 . . . . . . . . . . . . . . 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)))
137136eleq1d 2820 . . . . . . . . . . . . . 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}))
138135, 137syl5bbr 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}))
139100, 138imbi12d 333 . . . . . . . . . . . 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})))
140 mpfind.mu . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜎)
14183, 84, 139, 140vtocl2 3397 . . . . . . . . . . 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})
14268, 76, 82, 141syl12anc 1475 . . . . . . . . . 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})
143132, 142eqeltrd 2835 . . . . . . . . 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})
144 elpreima 6496 . . . . . . . . . . 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})))
14518, 144syl 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})))
146145adantr 472 . . . . . . . . 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})))
147118, 143, 146mpbir2and 995 . . . . . . . 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}))
148147adantlr 753 . . . . . . 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}))
1497mplassa 19652 . . . . . . . . . . . . . 14 ((𝐼 ∈ V ∧ (𝑆s 𝑅) ∈ CRing) → (𝐼 mPoly (𝑆s 𝑅)) ∈ AssAlg)
15030, 34, 149syl2anc 696 . . . . . . . . . . . . 13 (𝜑 → (𝐼 mPoly (𝑆s 𝑅)) ∈ AssAlg)
151 eqid 2756 . . . . . . . . . . . . . 14 (Scalar‘(𝐼 mPoly (𝑆s 𝑅))) = (Scalar‘(𝐼 mPoly (𝑆s 𝑅)))
15229, 151asclrhm 19540 . . . . . . . . . . . . 13 ((𝐼 mPoly (𝑆s 𝑅)) ∈ AssAlg → (algSc‘(𝐼 mPoly (𝑆s 𝑅))) ∈ ((Scalar‘(𝐼 mPoly (𝑆s 𝑅))) RingHom (𝐼 mPoly (𝑆s 𝑅))))
153150, 152syl 17 . . . . . . . . . . . 12 (𝜑 → (algSc‘(𝐼 mPoly (𝑆s 𝑅))) ∈ ((Scalar‘(𝐼 mPoly (𝑆s 𝑅))) RingHom (𝐼 mPoly (𝑆s 𝑅))))
154 eqid 2756 . . . . . . . . . . . . 13 (Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅)))) = (Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅))))
155154, 13rhmf 18924 . . . . . . . . . . . 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))))
156153, 155syl 17 . . . . . . . . . . 11 (𝜑 → (algSc‘(𝐼 mPoly (𝑆s 𝑅))):(Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅))))⟶(Base‘(𝐼 mPoly (𝑆s 𝑅))))
157156adantr 472 . . . . . . . . . 10 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → (algSc‘(𝐼 mPoly (𝑆s 𝑅))):(Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅))))⟶(Base‘(𝐼 mPoly (𝑆s 𝑅))))
1587, 30, 34mplsca 19643 . . . . . . . . . . . . 13 (𝜑 → (𝑆s 𝑅) = (Scalar‘(𝐼 mPoly (𝑆s 𝑅))))
159158fveq2d 6352 . . . . . . . . . . . 12 (𝜑 → (Base‘(𝑆s 𝑅)) = (Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅)))))
160159eleq2d 2821 . . . . . . . . . . 11 (𝜑 → (𝑖 ∈ (Base‘(𝑆s 𝑅)) ↔ 𝑖 ∈ (Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅))))))
161160biimpa 502 . . . . . . . . . 10 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → 𝑖 ∈ (Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅)))))
162157, 161ffvelrnd 6519 . . . . . . . . 9 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
16330adantr 472 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → 𝐼 ∈ V)
16431adantr 472 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → 𝑆 ∈ CRing)
16532adantr 472 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → 𝑅 ∈ (SubRing‘𝑆))
16610subrgss 18979 . . . . . . . . . . . . . . 15 (𝑅 ∈ (SubRing‘𝑆) → 𝑅𝐵)
16732, 166syl 17 . . . . . . . . . . . . . 14 (𝜑𝑅𝐵)
1688, 10ressbas2 16129 . . . . . . . . . . . . . 14 (𝑅𝐵𝑅 = (Base‘(𝑆s 𝑅)))
169167, 168syl 17 . . . . . . . . . . . . 13 (𝜑𝑅 = (Base‘(𝑆s 𝑅)))
170169eleq2d 2821 . . . . . . . . . . . 12 (𝜑 → (𝑖𝑅𝑖 ∈ (Base‘(𝑆s 𝑅))))
171170biimpar 503 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → 𝑖𝑅)
1726, 7, 8, 10, 29, 163, 164, 165, 171evlssca 19720 . . . . . . . . . 10 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖)) = ((𝐵𝑚 𝐼) × {𝑖}))
173 mpfind.co . . . . . . . . . . . . . 14 ((𝜑𝑓𝑅) → 𝜒)
174173ralrimiva 3100 . . . . . . . . . . . . 13 (𝜑 → ∀𝑓𝑅 𝜒)
175 ovex 6837 . . . . . . . . . . . . . . . . 17 (𝐵𝑚 𝐼) ∈ V
176 snex 5053 . . . . . . . . . . . . . . . . 17 {𝑓} ∈ V
177175, 176xpex 7123 . . . . . . . . . . . . . . . 16 ((𝐵𝑚 𝐼) × {𝑓}) ∈ V
178 mpfind.wa . . . . . . . . . . . . . . . 16 (𝑥 = ((𝐵𝑚 𝐼) × {𝑓}) → (𝜓𝜒))
179177, 178elab 3486 . . . . . . . . . . . . . . 15 (((𝐵𝑚 𝐼) × {𝑓}) ∈ {𝑥𝜓} ↔ 𝜒)
180 sneq 4327 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝑖 → {𝑓} = {𝑖})
181180xpeq2d 5292 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑖 → ((𝐵𝑚 𝐼) × {𝑓}) = ((𝐵𝑚 𝐼) × {𝑖}))
182181eleq1d 2820 . . . . . . . . . . . . . . 15 (𝑓 = 𝑖 → (((𝐵𝑚 𝐼) × {𝑓}) ∈ {𝑥𝜓} ↔ ((𝐵𝑚 𝐼) × {𝑖}) ∈ {𝑥𝜓}))
183179, 182syl5bbr 274 . . . . . . . . . . . . . 14 (𝑓 = 𝑖 → (𝜒 ↔ ((𝐵𝑚 𝐼) × {𝑖}) ∈ {𝑥𝜓}))
184183cbvralv 3306 . . . . . . . . . . . . 13 (∀𝑓𝑅 𝜒 ↔ ∀𝑖𝑅 ((𝐵𝑚 𝐼) × {𝑖}) ∈ {𝑥𝜓})
185174, 184sylib 208 . . . . . . . . . . . 12 (𝜑 → ∀𝑖𝑅 ((𝐵𝑚 𝐼) × {𝑖}) ∈ {𝑥𝜓})
186185r19.21bi 3066 . . . . . . . . . . 11 ((𝜑𝑖𝑅) → ((𝐵𝑚 𝐼) × {𝑖}) ∈ {𝑥𝜓})
187171, 186syldan 488 . . . . . . . . . 10 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → ((𝐵𝑚 𝐼) × {𝑖}) ∈ {𝑥𝜓})
188172, 187eqeltrd 2835 . . . . . . . . 9 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖)) ∈ {𝑥𝜓})
189 elpreima 6496 . . . . . . . . . . 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 472 . . . . . . . . 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})))
192162, 188, 191mpbir2and 995 . . . . . . . 8 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
193192adantlr 753 . . . . . . 7 (((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) ∧ 𝑖 ∈ (Base‘(𝑆s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
19430adantr 472 . . . . . . . . . 10 ((𝜑𝑖𝐼) → 𝐼 ∈ V)
19536adantr 472 . . . . . . . . . 10 ((𝜑𝑖𝐼) → (𝑆s 𝑅) ∈ Ring)
196 simpr 479 . . . . . . . . . 10 ((𝜑𝑖𝐼) → 𝑖𝐼)
1977, 26, 13, 194, 195, 196mvrcl 19647 . . . . . . . . 9 ((𝜑𝑖𝐼) → ((𝐼 mVar (𝑆s 𝑅))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
19831adantr 472 . . . . . . . . . . 11 ((𝜑𝑖𝐼) → 𝑆 ∈ CRing)
19932adantr 472 . . . . . . . . . . 11 ((𝜑𝑖𝐼) → 𝑅 ∈ (SubRing‘𝑆))
2006, 26, 8, 10, 194, 198, 199, 196evlsvar 19721 . . . . . . . . . 10 ((𝜑𝑖𝐼) → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆s 𝑅))‘𝑖)) = (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑖)))
201 mpfind.pr . . . . . . . . . . . . . 14 ((𝜑𝑓𝐼) → 𝜃)
202175mptex 6646 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑓)) ∈ V
203 mpfind.wb . . . . . . . . . . . . . . 15 (𝑥 = (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑓)) → (𝜓𝜃))
204202, 203elab 3486 . . . . . . . . . . . . . 14 ((𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑓)) ∈ {𝑥𝜓} ↔ 𝜃)
205201, 204sylibr 224 . . . . . . . . . . . . 13 ((𝜑𝑓𝐼) → (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑓)) ∈ {𝑥𝜓})
206205ralrimiva 3100 . . . . . . . . . . . 12 (𝜑 → ∀𝑓𝐼 (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑓)) ∈ {𝑥𝜓})
207 fveq2 6348 . . . . . . . . . . . . . . 15 (𝑓 = 𝑖 → (𝑔𝑓) = (𝑔𝑖))
208207mpteq2dv 4893 . . . . . . . . . . . . . 14 (𝑓 = 𝑖 → (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑓)) = (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑖)))
209208eleq1d 2820 . . . . . . . . . . . . 13 (𝑓 = 𝑖 → ((𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑓)) ∈ {𝑥𝜓} ↔ (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑖)) ∈ {𝑥𝜓}))
210209cbvralv 3306 . . . . . . . . . . . 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 3066 . . . . . . . . . 10 ((𝜑𝑖𝐼) → (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑖)) ∈ {𝑥𝜓})
213200, 212eqeltrd 2835 . . . . . . . . 9 ((𝜑𝑖𝐼) → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆s 𝑅))‘𝑖)) ∈ {𝑥𝜓})
214 elpreima 6496 . . . . . . . . . . 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 472 . . . . . . . . 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 995 . . . . . . . 8 ((𝜑𝑖𝐼) → ((𝐼 mVar (𝑆s 𝑅))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
218217adantlr 753 . . . . . . 7 (((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) ∧ 𝑖𝐼) → ((𝐼 mVar (𝑆s 𝑅))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
219 simpr 479 . . . . . . 7 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
22030adantr 472 . . . . . . 7 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → 𝐼 ∈ V)
22134adantr 472 . . . . . . 7 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → (𝑆s 𝑅) ∈ CRing)
22225, 26, 7, 27, 28, 29, 13, 116, 148, 193, 218, 219, 220, 221mplind 19700 . . . . . 6 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → 𝑦 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
223 fvimacnvi 6490 . . . . . 6 ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑦 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥𝜓})
22424, 222, 223syl2anc 696 . . . . 5 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥𝜓})
225 eleq1 2823 . . . . 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 3165 . . 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 3487 . . 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 383  w3a 1072   = wceq 1628  wcel 2135  {cab 2742  wral 3046  wrex 3047  Vcvv 3336  wss 3711  {csn 4317  cmpt 4877   × cxp 5260  ccnv 5261  ran crn 5263  cima 5265  Fun wfun 6039   Fn wfn 6040  wf 6041  cfv 6045  (class class class)co 6809  𝑓 cof 7056  𝑚 cmap 8019  Basecbs 16055  s cress 16056  +gcplusg 16139  .rcmulr 16140  Scalarcsca 16142  s cpws 16305   MndHom cmhm 17530   GrpHom cghm 17854  mulGrpcmgp 18685  Ringcrg 18743  CRingccrg 18744   RingHom crh 18910  SubRingcsubrg 18974  AssAlgcasa 19507  algSccascl 19509   mVar cmvr 19550   mPoly cmpl 19551   evalSub ces 19702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110  ax-inf2 8707  ax-cnex 10180  ax-resscn 10181  ax-1cn 10182  ax-icn 10183  ax-addcl 10184  ax-addrcl 10185  ax-mulcl 10186  ax-mulrcl 10187  ax-mulcom 10188  ax-addass 10189  ax-mulass 10190  ax-distr 10191  ax-i2m1 10192  ax-1ne0 10193  ax-1rid 10194  ax-rnegex 10195  ax-rrecex 10196  ax-cnre 10197  ax-pre-lttri 10198  ax-pre-lttrn 10199  ax-pre-ltadd 10200  ax-pre-mulgt0 10201
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-nel 3032  df-ral 3051  df-rex 3052  df-reu 3053  df-rmo 3054  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-int 4624  df-iun 4670  df-iin 4671  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-se 5222  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-isom 6054  df-riota 6770  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-of 7058  df-ofr 7059  df-om 7227  df-1st 7329  df-2nd 7330  df-supp 7460  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-1o 7725  df-2o 7726  df-oadd 7729  df-er 7907  df-map 8021  df-pm 8022  df-ixp 8071  df-en 8118  df-dom 8119  df-sdom 8120  df-fin 8121  df-fsupp 8437  df-sup 8509  df-oi 8576  df-card 8951  df-pnf 10264  df-mnf 10265  df-xr 10266  df-ltxr 10267  df-le 10268  df-sub 10456  df-neg 10457  df-nn 11209  df-2 11267  df-3 11268  df-4 11269  df-5 11270  df-6 11271  df-7 11272  df-8 11273  df-9 11274  df-n0 11481  df-z 11566  df-dec 11682  df-uz 11876  df-fz 12516  df-fzo 12656  df-seq 12992  df-hash 13308  df-struct 16057  df-ndx 16058  df-slot 16059  df-base 16061  df-sets 16062  df-ress 16063  df-plusg 16152  df-mulr 16153  df-sca 16155  df-vsca 16156  df-ip 16157  df-tset 16158  df-ple 16159  df-ds 16162  df-hom 16164  df-cco 16165  df-0g 16300  df-gsum 16301  df-prds 16306  df-pws 16308  df-mre 16444  df-mrc 16445  df-acs 16447  df-mgm 17439  df-sgrp 17481  df-mnd 17492  df-mhm 17532  df-submnd 17533  df-grp 17622  df-minusg 17623  df-sbg 17624  df-mulg 17738  df-subg 17788  df-ghm 17855  df-cntz 17946  df-cmn 18391  df-abl 18392  df-mgp 18686  df-ur 18698  df-srg 18702  df-ring 18745  df-cring 18746  df-rnghom 18913  df-subrg 18976  df-lmod 19063  df-lss 19131  df-lsp 19170  df-assa 19510  df-asp 19511  df-ascl 19512  df-psr 19554  df-mvr 19555  df-mpl 19556  df-evls 19704
This theorem is referenced by:  pf1ind  19917  mzpmfp  37808
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