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Theorem mpfind 21517
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, 2eleqtrdi 2848 . . . 4 (𝜑𝐴 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
42mpfrcl 21495 . . . . . . . 8 (𝐴𝑄 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))
51, 4syl 17 . . . . . . 7 (𝜑 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))
6 eqid 2736 . . . . . . . 8 ((𝐼 evalSub 𝑆)‘𝑅) = ((𝐼 evalSub 𝑆)‘𝑅)
7 eqid 2736 . . . . . . . 8 (𝐼 mPoly (𝑆s 𝑅)) = (𝐼 mPoly (𝑆s 𝑅))
8 eqid 2736 . . . . . . . 8 (𝑆s 𝑅) = (𝑆s 𝑅)
9 eqid 2736 . . . . . . . 8 (𝑆s (𝐵m 𝐼)) = (𝑆s (𝐵m 𝐼))
10 mpfind.cb . . . . . . . 8 𝐵 = (Base‘𝑆)
116, 7, 8, 9, 10evlsrhm 21498 . . . . . . 7 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆s 𝑅)) RingHom (𝑆s (𝐵m 𝐼))))
12 eqid 2736 . . . . . . . 8 (Base‘(𝐼 mPoly (𝑆s 𝑅))) = (Base‘(𝐼 mPoly (𝑆s 𝑅)))
13 eqid 2736 . . . . . . . 8 (Base‘(𝑆s (𝐵m 𝐼))) = (Base‘(𝑆s (𝐵m 𝐼)))
1412, 13rhmf 20158 . . . . . . 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))))
155, 11, 143syl 18 . . . . . 6 (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆s 𝑅)))⟶(Base‘(𝑆s (𝐵m 𝐼))))
1615ffnd 6669 . . . . 5 (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆s 𝑅))))
17 fvelrnb 6903 . . . . 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))
1816, 17syl 17 . . . 4 (𝜑 → (𝐴 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅) ↔ ∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴))
193, 18mpbid 231 . . 3 (𝜑 → ∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴)
2015ffund 6672 . . . . . 6 (𝜑 → Fun ((𝐼 evalSub 𝑆)‘𝑅))
21 eqid 2736 . . . . . . 7 (Base‘(𝑆s 𝑅)) = (Base‘(𝑆s 𝑅))
22 eqid 2736 . . . . . . 7 (𝐼 mVar (𝑆s 𝑅)) = (𝐼 mVar (𝑆s 𝑅))
23 eqid 2736 . . . . . . 7 (+g‘(𝐼 mPoly (𝑆s 𝑅))) = (+g‘(𝐼 mPoly (𝑆s 𝑅)))
24 eqid 2736 . . . . . . 7 (.r‘(𝐼 mPoly (𝑆s 𝑅))) = (.r‘(𝐼 mPoly (𝑆s 𝑅)))
25 eqid 2736 . . . . . . 7 (algSc‘(𝐼 mPoly (𝑆s 𝑅))) = (algSc‘(𝐼 mPoly (𝑆s 𝑅)))
265simp1d 1142 . . . . . . . . . . . 12 (𝜑𝐼 ∈ V)
275simp2d 1143 . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ CRing)
285simp3d 1144 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ (SubRing‘𝑆))
298subrgcrng 20226 . . . . . . . . . . . . . 14 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑆s 𝑅) ∈ CRing)
3027, 28, 29syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → (𝑆s 𝑅) ∈ CRing)
31 crngring 19976 . . . . . . . . . . . . 13 ((𝑆s 𝑅) ∈ CRing → (𝑆s 𝑅) ∈ Ring)
3230, 31syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑆s 𝑅) ∈ Ring)
337mplring 21424 . . . . . . . . . . . 12 ((𝐼 ∈ V ∧ (𝑆s 𝑅) ∈ Ring) → (𝐼 mPoly (𝑆s 𝑅)) ∈ Ring)
3426, 32, 33syl2anc 584 . . . . . . . . . . 11 (𝜑 → (𝐼 mPoly (𝑆s 𝑅)) ∈ Ring)
3534adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (𝐼 mPoly (𝑆s 𝑅)) ∈ Ring)
36 simprl 769 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
37 elpreima 7008 . . . . . . . . . . . . . 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})))
3816, 37syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ↔ (𝑖 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓})))
3938adantr 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})))
4036, 39mpbid 231 . . . . . . . . . . 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}))
4140simpld 495 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
42 simprr 771 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
43 elpreima 7008 . . . . . . . . . . . . . 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})))
4416, 43syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ↔ (𝑗 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))) ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓})))
4544adantr 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})))
4642, 45mpbid 231 . . . . . . . . . . 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}))
4746simpld 495 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
4812, 23ringacl 19999 . . . . . . . . . 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))))
4935, 41, 47, 48syl3anc 1371 . . . . . . . . 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))))
50 rhmghm 20157 . . . . . . . . . . . . . 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))))
515, 11, 503syl 18 . . . . . . . . . . . . 13 (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆s 𝑅)) GrpHom (𝑆s (𝐵m 𝐼))))
5251adantr 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))))
53 eqid 2736 . . . . . . . . . . . . 13 (+g‘(𝑆s (𝐵m 𝐼))) = (+g‘(𝑆s (𝐵m 𝐼)))
5412, 23, 53ghmlin 19013 . . . . . . . . . . . 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)))
5552, 41, 47, 54syl3anc 1371 . . . . . . . . . . 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)))
5627adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝑆 ∈ CRing)
57 ovexd 7392 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (𝐵m 𝐼) ∈ V)
5815adantr 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))))
5958, 41ffvelcdmd 7036 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ (Base‘(𝑆s (𝐵m 𝐼))))
6058, 47ffvelcdmd 7036 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ (Base‘(𝑆s (𝐵m 𝐼))))
61 mpfind.cp . . . . . . . . . . . 12 + = (+g𝑆)
629, 13, 56, 57, 59, 60, 61, 53pwsplusgval 17372 . . . . . . . . . . 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)))
6355, 62eqtrd 2776 . . . . . . . . . 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)))
64 simpl 483 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝜑)
65 fnfvelrn 7031 . . . . . . . . . . . . . 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))
6616, 41, 65syl2an2r 683 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
6766, 2eleqtrrdi 2849 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄)
68 fvimacnvi 7002 . . . . . . . . . . . . 13 ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓})
6920, 36, 68syl2an2r 683 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓})
7067, 69jca 512 . . . . . . . . . . 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}))
71 fnfvelrn 7031 . . . . . . . . . . . . . 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))
7216, 47, 71syl2an2r 683 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
7372, 2eleqtrrdi 2849 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄)
74 fvimacnvi 7002 . . . . . . . . . . . . 13 ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓})
7520, 42, 74syl2an2r 683 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓})
7673, 75jca 512 . . . . . . . . . . 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}))
77 fvex 6855 . . . . . . . . . . . 12 (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ V
78 fvex 6855 . . . . . . . . . . . 12 (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ V
79 eleq1 2825 . . . . . . . . . . . . . . . 16 (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝑓𝑄 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄))
80 vex 3449 . . . . . . . . . . . . . . . . . 18 𝑓 ∈ V
81 mpfind.wc . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑓 → (𝜓𝜏))
8280, 81elab 3630 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ {𝑥𝜓} ↔ 𝜏)
83 eleq1 2825 . . . . . . . . . . . . . . . . 17 (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝑓 ∈ {𝑥𝜓} ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓}))
8482, 83bitr3id 284 . . . . . . . . . . . . . . . 16 (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝜏 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓}))
8579, 84anbi12d 631 . . . . . . . . . . . . . . 15 (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → ((𝑓𝑄𝜏) ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓})))
86 eleq1 2825 . . . . . . . . . . . . . . . 16 (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝑔𝑄 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄))
87 vex 3449 . . . . . . . . . . . . . . . . . 18 𝑔 ∈ V
88 mpfind.wd . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑔 → (𝜓𝜂))
8987, 88elab 3630 . . . . . . . . . . . . . . . . 17 (𝑔 ∈ {𝑥𝜓} ↔ 𝜂)
90 eleq1 2825 . . . . . . . . . . . . . . . . 17 (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝑔 ∈ {𝑥𝜓} ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓}))
9189, 90bitr3id 284 . . . . . . . . . . . . . . . 16 (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝜂 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓}))
9286, 91anbi12d 631 . . . . . . . . . . . . . . 15 (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → ((𝑔𝑄𝜂) ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓})))
9385, 92bi2anan9 637 . . . . . . . . . . . . . 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}))))
9493anbi2d 629 . . . . . . . . . . . . 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})))))
95 ovex 7390 . . . . . . . . . . . . . . 15 (𝑓f + 𝑔) ∈ V
96 mpfind.we . . . . . . . . . . . . . . 15 (𝑥 = (𝑓f + 𝑔) → (𝜓𝜁))
9795, 96elab 3630 . . . . . . . . . . . . . 14 ((𝑓f + 𝑔) ∈ {𝑥𝜓} ↔ 𝜁)
98 oveq12 7366 . . . . . . . . . . . . . . 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)))
9998eleq1d 2822 . . . . . . . . . . . . . 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}))
10097, 99bitr3id 284 . . . . . . . . . . . . 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}))
10194, 100imbi12d 344 . . . . . . . . . . . 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})))
102 mpfind.ad . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜁)
10377, 78, 101, 102vtocl2 3520 . . . . . . . . . . 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})
10464, 70, 76, 103syl12anc 835 . . . . . . . . . 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})
10563, 104eqeltrd 2838 . . . . . . . . 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})
106 elpreima 7008 . . . . . . . . . . 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})))
10716, 106syl 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})))
108107adantr 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})))
10949, 105, 108mpbir2and 711 . . . . . . . 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}))
110109adantlr 713 . . . . . . 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}))
11112, 24ringcl 19981 . . . . . . . . . 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))))
11235, 41, 47, 111syl3anc 1371 . . . . . . . . 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))))
113 eqid 2736 . . . . . . . . . . . . . . 15 (mulGrp‘(𝐼 mPoly (𝑆s 𝑅))) = (mulGrp‘(𝐼 mPoly (𝑆s 𝑅)))
114 eqid 2736 . . . . . . . . . . . . . . 15 (mulGrp‘(𝑆s (𝐵m 𝐼))) = (mulGrp‘(𝑆s (𝐵m 𝐼)))
115113, 114rhmmhm 20153 . . . . . . . . . . . . . 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)))))
1165, 11, 1153syl 18 . . . . . . . . . . . . 13 (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((mulGrp‘(𝐼 mPoly (𝑆s 𝑅))) MndHom (mulGrp‘(𝑆s (𝐵m 𝐼)))))
117116adantr 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)))))
118113, 12mgpbas 19902 . . . . . . . . . . . . 13 (Base‘(𝐼 mPoly (𝑆s 𝑅))) = (Base‘(mulGrp‘(𝐼 mPoly (𝑆s 𝑅))))
119113, 24mgpplusg 19900 . . . . . . . . . . . . 13 (.r‘(𝐼 mPoly (𝑆s 𝑅))) = (+g‘(mulGrp‘(𝐼 mPoly (𝑆s 𝑅))))
120 eqid 2736 . . . . . . . . . . . . . 14 (.r‘(𝑆s (𝐵m 𝐼))) = (.r‘(𝑆s (𝐵m 𝐼)))
121114, 120mgpplusg 19900 . . . . . . . . . . . . 13 (.r‘(𝑆s (𝐵m 𝐼))) = (+g‘(mulGrp‘(𝑆s (𝐵m 𝐼))))
122118, 119, 121mhmlin 18609 . . . . . . . . . . . 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)))
123117, 41, 47, 122syl3anc 1371 . . . . . . . . . . 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)))
124 mpfind.ct . . . . . . . . . . . 12 · = (.r𝑆)
1259, 13, 56, 57, 59, 60, 124, 120pwsmulrval 17373 . . . . . . . . . . 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)))
126123, 125eqtrd 2776 . . . . . . . . . 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)))
127 ovex 7390 . . . . . . . . . . . . . . 15 (𝑓f · 𝑔) ∈ V
128 mpfind.wf . . . . . . . . . . . . . . 15 (𝑥 = (𝑓f · 𝑔) → (𝜓𝜎))
129127, 128elab 3630 . . . . . . . . . . . . . 14 ((𝑓f · 𝑔) ∈ {𝑥𝜓} ↔ 𝜎)
130 oveq12 7366 . . . . . . . . . . . . . . 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)))
131130eleq1d 2822 . . . . . . . . . . . . . 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}))
132129, 131bitr3id 284 . . . . . . . . . . . . 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}))
13394, 132imbi12d 344 . . . . . . . . . . . 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})))
134 mpfind.mu . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜎)
13577, 78, 133, 134vtocl2 3520 . . . . . . . . . . 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})
13664, 70, 76, 135syl12anc 835 . . . . . . . . . 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})
137126, 136eqeltrd 2838 . . . . . . . . 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})
138 elpreima 7008 . . . . . . . . . . 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})))
13916, 138syl 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})))
140139adantr 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})))
141112, 137, 140mpbir2and 711 . . . . . . . 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}))
142141adantlr 713 . . . . . . 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}))
1437mplassa 21427 . . . . . . . . . . . . 13 ((𝐼 ∈ V ∧ (𝑆s 𝑅) ∈ CRing) → (𝐼 mPoly (𝑆s 𝑅)) ∈ AssAlg)
14426, 30, 143syl2anc 584 . . . . . . . . . . . 12 (𝜑 → (𝐼 mPoly (𝑆s 𝑅)) ∈ AssAlg)
145 eqid 2736 . . . . . . . . . . . . 13 (Scalar‘(𝐼 mPoly (𝑆s 𝑅))) = (Scalar‘(𝐼 mPoly (𝑆s 𝑅)))
14625, 145asclrhm 21293 . . . . . . . . . . . 12 ((𝐼 mPoly (𝑆s 𝑅)) ∈ AssAlg → (algSc‘(𝐼 mPoly (𝑆s 𝑅))) ∈ ((Scalar‘(𝐼 mPoly (𝑆s 𝑅))) RingHom (𝐼 mPoly (𝑆s 𝑅))))
147 eqid 2736 . . . . . . . . . . . . 13 (Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅)))) = (Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅))))
148147, 12rhmf 20158 . . . . . . . . . . . 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))))
149144, 146, 1483syl 18 . . . . . . . . . . 11 (𝜑 → (algSc‘(𝐼 mPoly (𝑆s 𝑅))):(Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅))))⟶(Base‘(𝐼 mPoly (𝑆s 𝑅))))
150149adantr 481 . . . . . . . . . 10 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → (algSc‘(𝐼 mPoly (𝑆s 𝑅))):(Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅))))⟶(Base‘(𝐼 mPoly (𝑆s 𝑅))))
1517, 26, 30mplsca 21417 . . . . . . . . . . . . 13 (𝜑 → (𝑆s 𝑅) = (Scalar‘(𝐼 mPoly (𝑆s 𝑅))))
152151fveq2d 6846 . . . . . . . . . . . 12 (𝜑 → (Base‘(𝑆s 𝑅)) = (Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅)))))
153152eleq2d 2823 . . . . . . . . . . 11 (𝜑 → (𝑖 ∈ (Base‘(𝑆s 𝑅)) ↔ 𝑖 ∈ (Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅))))))
154153biimpa 477 . . . . . . . . . 10 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → 𝑖 ∈ (Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅)))))
155150, 154ffvelcdmd 7036 . . . . . . . . 9 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
15626adantr 481 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → 𝐼 ∈ V)
15727adantr 481 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → 𝑆 ∈ CRing)
15828adantr 481 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → 𝑅 ∈ (SubRing‘𝑆))
15910subrgss 20223 . . . . . . . . . . . . . 14 (𝑅 ∈ (SubRing‘𝑆) → 𝑅𝐵)
1608, 10ressbas2 17120 . . . . . . . . . . . . . 14 (𝑅𝐵𝑅 = (Base‘(𝑆s 𝑅)))
16128, 159, 1603syl 18 . . . . . . . . . . . . 13 (𝜑𝑅 = (Base‘(𝑆s 𝑅)))
162161eleq2d 2823 . . . . . . . . . . . 12 (𝜑 → (𝑖𝑅𝑖 ∈ (Base‘(𝑆s 𝑅))))
163162biimpar 478 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → 𝑖𝑅)
1646, 7, 8, 10, 25, 156, 157, 158, 163evlssca 21499 . . . . . . . . . 10 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖)) = ((𝐵m 𝐼) × {𝑖}))
165 mpfind.co . . . . . . . . . . . . . 14 ((𝜑𝑓𝑅) → 𝜒)
166165ralrimiva 3143 . . . . . . . . . . . . 13 (𝜑 → ∀𝑓𝑅 𝜒)
167 ovex 7390 . . . . . . . . . . . . . . . . 17 (𝐵m 𝐼) ∈ V
168 vsnex 5386 . . . . . . . . . . . . . . . . 17 {𝑓} ∈ V
169167, 168xpex 7687 . . . . . . . . . . . . . . . 16 ((𝐵m 𝐼) × {𝑓}) ∈ V
170 mpfind.wa . . . . . . . . . . . . . . . 16 (𝑥 = ((𝐵m 𝐼) × {𝑓}) → (𝜓𝜒))
171169, 170elab 3630 . . . . . . . . . . . . . . 15 (((𝐵m 𝐼) × {𝑓}) ∈ {𝑥𝜓} ↔ 𝜒)
172 sneq 4596 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝑖 → {𝑓} = {𝑖})
173172xpeq2d 5663 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑖 → ((𝐵m 𝐼) × {𝑓}) = ((𝐵m 𝐼) × {𝑖}))
174173eleq1d 2822 . . . . . . . . . . . . . . 15 (𝑓 = 𝑖 → (((𝐵m 𝐼) × {𝑓}) ∈ {𝑥𝜓} ↔ ((𝐵m 𝐼) × {𝑖}) ∈ {𝑥𝜓}))
175171, 174bitr3id 284 . . . . . . . . . . . . . 14 (𝑓 = 𝑖 → (𝜒 ↔ ((𝐵m 𝐼) × {𝑖}) ∈ {𝑥𝜓}))
176175cbvralvw 3225 . . . . . . . . . . . . 13 (∀𝑓𝑅 𝜒 ↔ ∀𝑖𝑅 ((𝐵m 𝐼) × {𝑖}) ∈ {𝑥𝜓})
177166, 176sylib 217 . . . . . . . . . . . 12 (𝜑 → ∀𝑖𝑅 ((𝐵m 𝐼) × {𝑖}) ∈ {𝑥𝜓})
178177r19.21bi 3234 . . . . . . . . . . 11 ((𝜑𝑖𝑅) → ((𝐵m 𝐼) × {𝑖}) ∈ {𝑥𝜓})
179163, 178syldan 591 . . . . . . . . . 10 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → ((𝐵m 𝐼) × {𝑖}) ∈ {𝑥𝜓})
180164, 179eqeltrd 2838 . . . . . . . . 9 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖)) ∈ {𝑥𝜓})
181 elpreima 7008 . . . . . . . . . . 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})))
18216, 181syl 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})))
183182adantr 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})))
184155, 180, 183mpbir2and 711 . . . . . . . 8 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
185184adantlr 713 . . . . . . 7 (((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) ∧ 𝑖 ∈ (Base‘(𝑆s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
18626adantr 481 . . . . . . . . . 10 ((𝜑𝑖𝐼) → 𝐼 ∈ V)
18732adantr 481 . . . . . . . . . 10 ((𝜑𝑖𝐼) → (𝑆s 𝑅) ∈ Ring)
188 simpr 485 . . . . . . . . . 10 ((𝜑𝑖𝐼) → 𝑖𝐼)
1897, 22, 12, 186, 187, 188mvrcl 21421 . . . . . . . . 9 ((𝜑𝑖𝐼) → ((𝐼 mVar (𝑆s 𝑅))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
19027adantr 481 . . . . . . . . . . 11 ((𝜑𝑖𝐼) → 𝑆 ∈ CRing)
19128adantr 481 . . . . . . . . . . 11 ((𝜑𝑖𝐼) → 𝑅 ∈ (SubRing‘𝑆))
1926, 22, 8, 10, 186, 190, 191, 188evlsvar 21500 . . . . . . . . . 10 ((𝜑𝑖𝐼) → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆s 𝑅))‘𝑖)) = (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑖)))
193 mpfind.pr . . . . . . . . . . . . . 14 ((𝜑𝑓𝐼) → 𝜃)
194167mptex 7173 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑓)) ∈ V
195 mpfind.wb . . . . . . . . . . . . . . 15 (𝑥 = (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑓)) → (𝜓𝜃))
196194, 195elab 3630 . . . . . . . . . . . . . 14 ((𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑓)) ∈ {𝑥𝜓} ↔ 𝜃)
197193, 196sylibr 233 . . . . . . . . . . . . 13 ((𝜑𝑓𝐼) → (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑓)) ∈ {𝑥𝜓})
198197ralrimiva 3143 . . . . . . . . . . . 12 (𝜑 → ∀𝑓𝐼 (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑓)) ∈ {𝑥𝜓})
199 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑓 = 𝑖 → (𝑔𝑓) = (𝑔𝑖))
200199mpteq2dv 5207 . . . . . . . . . . . . . 14 (𝑓 = 𝑖 → (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑓)) = (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑖)))
201200eleq1d 2822 . . . . . . . . . . . . 13 (𝑓 = 𝑖 → ((𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑓)) ∈ {𝑥𝜓} ↔ (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑖)) ∈ {𝑥𝜓}))
202201cbvralvw 3225 . . . . . . . . . . . 12 (∀𝑓𝐼 (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑓)) ∈ {𝑥𝜓} ↔ ∀𝑖𝐼 (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑖)) ∈ {𝑥𝜓})
203198, 202sylib 217 . . . . . . . . . . 11 (𝜑 → ∀𝑖𝐼 (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑖)) ∈ {𝑥𝜓})
204203r19.21bi 3234 . . . . . . . . . 10 ((𝜑𝑖𝐼) → (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑖)) ∈ {𝑥𝜓})
205192, 204eqeltrd 2838 . . . . . . . . 9 ((𝜑𝑖𝐼) → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆s 𝑅))‘𝑖)) ∈ {𝑥𝜓})
206 elpreima 7008 . . . . . . . . . . 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})))
20716, 206syl 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})))
208207adantr 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})))
209189, 205, 208mpbir2and 711 . . . . . . . 8 ((𝜑𝑖𝐼) → ((𝐼 mVar (𝑆s 𝑅))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
210209adantlr 713 . . . . . . 7 (((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) ∧ 𝑖𝐼) → ((𝐼 mVar (𝑆s 𝑅))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
211 simpr 485 . . . . . . 7 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
21226adantr 481 . . . . . . 7 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → 𝐼 ∈ V)
21330adantr 481 . . . . . . 7 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → (𝑆s 𝑅) ∈ CRing)
21421, 22, 7, 23, 24, 25, 12, 110, 142, 185, 210, 211, 212, 213mplind 21478 . . . . . 6 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → 𝑦 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
215 fvimacnvi 7002 . . . . . 6 ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑦 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥𝜓})
21620, 214, 215syl2an2r 683 . . . . 5 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥𝜓})
217 eleq1 2825 . . . . 5 ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴 → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥𝜓} ↔ 𝐴 ∈ {𝑥𝜓}))
218216, 217syl5ibcom 244 . . . 4 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴𝐴 ∈ {𝑥𝜓}))
219218rexlimdva 3152 . . 3 (𝜑 → (∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴𝐴 ∈ {𝑥𝜓}))
22019, 219mpd 15 . 2 (𝜑𝐴 ∈ {𝑥𝜓})
221 mpfind.wg . . . 4 (𝑥 = 𝐴 → (𝜓𝜌))
222221elabg 3628 . . 3 (𝐴𝑄 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜌))
2231, 222syl 17 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜌))
224220, 223mpbid 231 1 (𝜑𝜌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  {cab 2713  wral 3064  wrex 3073  Vcvv 3445  wss 3910  {csn 4586  cmpt 5188   × cxp 5631  ccnv 5632  ran crn 5634  cima 5636  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  f cof 7615  m cmap 8765  Basecbs 17083  s cress 17112  +gcplusg 17133  .rcmulr 17134  Scalarcsca 17136  s cpws 17328   MndHom cmhm 18599   GrpHom cghm 19005  mulGrpcmgp 19896  Ringcrg 19964  CRingccrg 19965   RingHom crh 20143  SubRingcsubrg 20218  AssAlgcasa 21256  algSccascl 21258   mVar cmvr 21307   mPoly cmpl 21308   evalSub ces 21480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-srg 19918  df-ring 19966  df-cring 19967  df-rnghom 20146  df-subrg 20220  df-lmod 20324  df-lss 20393  df-lsp 20433  df-assa 21259  df-asp 21260  df-ascl 21261  df-psr 21311  df-mvr 21312  df-mpl 21313  df-evls 21482
This theorem is referenced by:  pf1ind  21721  mzpmfp  41056
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