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Theorem mpfind 20320
 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 2926 . . . 4 (𝜑𝐴 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅))
42mpfrcl 20298 . . . . . . . 8 (𝐴𝑄 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))
51, 4syl 17 . . . . . . 7 (𝜑 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))
6 eqid 2824 . . . . . . . 8 ((𝐼 evalSub 𝑆)‘𝑅) = ((𝐼 evalSub 𝑆)‘𝑅)
7 eqid 2824 . . . . . . . 8 (𝐼 mPoly (𝑆s 𝑅)) = (𝐼 mPoly (𝑆s 𝑅))
8 eqid 2824 . . . . . . . 8 (𝑆s 𝑅) = (𝑆s 𝑅)
9 eqid 2824 . . . . . . . 8 (𝑆s (𝐵m 𝐼)) = (𝑆s (𝐵m 𝐼))
10 mpfind.cb . . . . . . . 8 𝐵 = (Base‘𝑆)
116, 7, 8, 9, 10evlsrhm 20301 . . . . . . 7 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆s 𝑅)) RingHom (𝑆s (𝐵m 𝐼))))
12 eqid 2824 . . . . . . . 8 (Base‘(𝐼 mPoly (𝑆s 𝑅))) = (Base‘(𝐼 mPoly (𝑆s 𝑅)))
13 eqid 2824 . . . . . . . 8 (Base‘(𝑆s (𝐵m 𝐼))) = (Base‘(𝑆s (𝐵m 𝐼)))
1412, 13rhmf 19481 . . . . . . 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 6504 . . . . 5 (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆s 𝑅))))
17 fvelrnb 6717 . . . . 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 235 . . 3 (𝜑 → ∃𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))(((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴)
2015ffund 6507 . . . . . 6 (𝜑 → Fun ((𝐼 evalSub 𝑆)‘𝑅))
21 eqid 2824 . . . . . . 7 (Base‘(𝑆s 𝑅)) = (Base‘(𝑆s 𝑅))
22 eqid 2824 . . . . . . 7 (𝐼 mVar (𝑆s 𝑅)) = (𝐼 mVar (𝑆s 𝑅))
23 eqid 2824 . . . . . . 7 (+g‘(𝐼 mPoly (𝑆s 𝑅))) = (+g‘(𝐼 mPoly (𝑆s 𝑅)))
24 eqid 2824 . . . . . . 7 (.r‘(𝐼 mPoly (𝑆s 𝑅))) = (.r‘(𝐼 mPoly (𝑆s 𝑅)))
25 eqid 2824 . . . . . . 7 (algSc‘(𝐼 mPoly (𝑆s 𝑅))) = (algSc‘(𝐼 mPoly (𝑆s 𝑅)))
265simp1d 1139 . . . . . . . . . . . 12 (𝜑𝐼 ∈ V)
275simp2d 1140 . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ CRing)
285simp3d 1141 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ (SubRing‘𝑆))
298subrgcrng 19539 . . . . . . . . . . . . . 14 ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑆s 𝑅) ∈ CRing)
3027, 28, 29syl2anc 587 . . . . . . . . . . . . 13 (𝜑 → (𝑆s 𝑅) ∈ CRing)
31 crngring 19309 . . . . . . . . . . . . 13 ((𝑆s 𝑅) ∈ CRing → (𝑆s 𝑅) ∈ Ring)
3230, 31syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑆s 𝑅) ∈ Ring)
337mplring 20232 . . . . . . . . . . . 12 ((𝐼 ∈ V ∧ (𝑆s 𝑅) ∈ Ring) → (𝐼 mPoly (𝑆s 𝑅)) ∈ Ring)
3426, 32, 33syl2anc 587 . . . . . . . . . . 11 (𝜑 → (𝐼 mPoly (𝑆s 𝑅)) ∈ Ring)
3534adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (𝐼 mPoly (𝑆s 𝑅)) ∈ Ring)
36 simprl 770 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
37 elpreima 6819 . . . . . . . . . . . . . 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 484 . . . . . . . . . . . 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 235 . . . . . . . . . . 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 498 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝑖 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
42 simprr 772 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
43 elpreima 6819 . . . . . . . . . . . . . 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 484 . . . . . . . . . . . 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 235 . . . . . . . . . . 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 498 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝑗 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
4812, 23ringacl 19331 . . . . . . . . . 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 1368 . . . . . . . . 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 19480 . . . . . . . . . . . . . 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 484 . . . . . . . . . . . 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 2824 . . . . . . . . . . . . 13 (+g‘(𝑆s (𝐵m 𝐼))) = (+g‘(𝑆s (𝐵m 𝐼)))
5412, 23, 53ghmlin 18363 . . . . . . . . . . . 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 1368 . . . . . . . . . . 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 484 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝑆 ∈ CRing)
57 ovexd 7184 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (𝐵m 𝐼) ∈ V)
5815adantr 484 . . . . . . . . . . . . 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, 41ffvelrnd 6843 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ (Base‘(𝑆s (𝐵m 𝐼))))
6058, 47ffvelrnd 6843 . . . . . . . . . . . 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 16763 . . . . . . . . . . 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 2859 . . . . . . . . . 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 486 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → 𝜑)
65 fnfvelrn 6839 . . . . . . . . . . . . . 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 684 . . . . . . . . . . . . 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 2927 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄)
68 fvimacnvi 6813 . . . . . . . . . . . . 13 ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓})
6920, 36, 68syl2an2r 684 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓})
7067, 69jca 515 . . . . . . . . . . 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 6839 . . . . . . . . . . . . . 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 684 . . . . . . . . . . . . 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 2927 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄)
74 fvimacnvi 6813 . . . . . . . . . . . . 13 ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓})
7520, 42, 74syl2an2r 684 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}) ∧ 𝑗 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓})
7673, 75jca 515 . . . . . . . . . . 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 6674 . . . . . . . . . . . 12 (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ V
78 fvex 6674 . . . . . . . . . . . 12 (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ V
79 eleq1 2903 . . . . . . . . . . . . . . . 16 (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝑓𝑄 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄))
80 vex 3483 . . . . . . . . . . . . . . . . . 18 𝑓 ∈ V
81 mpfind.wc . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑓 → (𝜓𝜏))
8280, 81elab 3653 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ {𝑥𝜓} ↔ 𝜏)
83 eleq1 2903 . . . . . . . . . . . . . . . . 17 (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝑓 ∈ {𝑥𝜓} ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓}))
8482, 83bitr3id 288 . . . . . . . . . . . . . . . 16 (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → (𝜏 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓}))
8579, 84anbi12d 633 . . . . . . . . . . . . . . 15 (𝑓 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) → ((𝑓𝑄𝜏) ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑖) ∈ {𝑥𝜓})))
86 eleq1 2903 . . . . . . . . . . . . . . . 16 (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝑔𝑄 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄))
87 vex 3483 . . . . . . . . . . . . . . . . . 18 𝑔 ∈ V
88 mpfind.wd . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑔 → (𝜓𝜂))
8987, 88elab 3653 . . . . . . . . . . . . . . . . 17 (𝑔 ∈ {𝑥𝜓} ↔ 𝜂)
90 eleq1 2903 . . . . . . . . . . . . . . . . 17 (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝑔 ∈ {𝑥𝜓} ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓}))
9189, 90bitr3id 288 . . . . . . . . . . . . . . . 16 (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → (𝜂 ↔ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓}))
9286, 91anbi12d 633 . . . . . . . . . . . . . . 15 (𝑔 = (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) → ((𝑔𝑄𝜂) ↔ ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ 𝑄 ∧ (((𝐼 evalSub 𝑆)‘𝑅)‘𝑗) ∈ {𝑥𝜓})))
9385, 92bi2anan9 638 . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . 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 7182 . . . . . . . . . . . . . . 15 (𝑓f + 𝑔) ∈ V
96 mpfind.we . . . . . . . . . . . . . . 15 (𝑥 = (𝑓f + 𝑔) → (𝜓𝜁))
9795, 96elab 3653 . . . . . . . . . . . . . 14 ((𝑓f + 𝑔) ∈ {𝑥𝜓} ↔ 𝜁)
98 oveq12 7158 . . . . . . . . . . . . . . 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 2900 . . . . . . . . . . . . . 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 288 . . . . . . . . . . . . 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 348 . . . . . . . . . . . 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 3547 . . . . . . . . . . 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 2916 . . . . . . . . 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 6819 . . . . . . . . . . 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 484 . . . . . . . . 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 712 . . . . . . . 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 714 . . . . . . 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 19314 . . . . . . . . . 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 1368 . . . . . . . . 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 2824 . . . . . . . . . . . . . . 15 (mulGrp‘(𝐼 mPoly (𝑆s 𝑅))) = (mulGrp‘(𝐼 mPoly (𝑆s 𝑅)))
114 eqid 2824 . . . . . . . . . . . . . . 15 (mulGrp‘(𝑆s (𝐵m 𝐼))) = (mulGrp‘(𝑆s (𝐵m 𝐼)))
115113, 114rhmmhm 19477 . . . . . . . . . . . . . 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 484 . . . . . . . . . . . 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 19245 . . . . . . . . . . . . 13 (Base‘(𝐼 mPoly (𝑆s 𝑅))) = (Base‘(mulGrp‘(𝐼 mPoly (𝑆s 𝑅))))
119113, 24mgpplusg 19243 . . . . . . . . . . . . 13 (.r‘(𝐼 mPoly (𝑆s 𝑅))) = (+g‘(mulGrp‘(𝐼 mPoly (𝑆s 𝑅))))
120 eqid 2824 . . . . . . . . . . . . . 14 (.r‘(𝑆s (𝐵m 𝐼))) = (.r‘(𝑆s (𝐵m 𝐼)))
121114, 120mgpplusg 19243 . . . . . . . . . . . . 13 (.r‘(𝑆s (𝐵m 𝐼))) = (+g‘(mulGrp‘(𝑆s (𝐵m 𝐼))))
122118, 119, 121mhmlin 17963 . . . . . . . . . . . 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 1368 . . . . . . . . . . 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 16764 . . . . . . . . . . 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 2859 . . . . . . . . . 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 7182 . . . . . . . . . . . . . . 15 (𝑓f · 𝑔) ∈ V
128 mpfind.wf . . . . . . . . . . . . . . 15 (𝑥 = (𝑓f · 𝑔) → (𝜓𝜎))
129127, 128elab 3653 . . . . . . . . . . . . . 14 ((𝑓f · 𝑔) ∈ {𝑥𝜓} ↔ 𝜎)
130 oveq12 7158 . . . . . . . . . . . . . . 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 2900 . . . . . . . . . . . . . 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 288 . . . . . . . . . . . . 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 348 . . . . . . . . . . . 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 3547 . . . . . . . . . . 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 2916 . . . . . . . . 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 6819 . . . . . . . . . . 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 484 . . . . . . . . 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 712 . . . . . . . 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 714 . . . . . . 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 20235 . . . . . . . . . . . . 13 ((𝐼 ∈ V ∧ (𝑆s 𝑅) ∈ CRing) → (𝐼 mPoly (𝑆s 𝑅)) ∈ AssAlg)
14426, 30, 143syl2anc 587 . . . . . . . . . . . 12 (𝜑 → (𝐼 mPoly (𝑆s 𝑅)) ∈ AssAlg)
145 eqid 2824 . . . . . . . . . . . . 13 (Scalar‘(𝐼 mPoly (𝑆s 𝑅))) = (Scalar‘(𝐼 mPoly (𝑆s 𝑅)))
14625, 145asclrhm 20119 . . . . . . . . . . . 12 ((𝐼 mPoly (𝑆s 𝑅)) ∈ AssAlg → (algSc‘(𝐼 mPoly (𝑆s 𝑅))) ∈ ((Scalar‘(𝐼 mPoly (𝑆s 𝑅))) RingHom (𝐼 mPoly (𝑆s 𝑅))))
147 eqid 2824 . . . . . . . . . . . . 13 (Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅)))) = (Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅))))
148147, 12rhmf 19481 . . . . . . . . . . . 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 484 . . . . . . . . . 10 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → (algSc‘(𝐼 mPoly (𝑆s 𝑅))):(Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅))))⟶(Base‘(𝐼 mPoly (𝑆s 𝑅))))
1517, 26, 30mplsca 20225 . . . . . . . . . . . . 13 (𝜑 → (𝑆s 𝑅) = (Scalar‘(𝐼 mPoly (𝑆s 𝑅))))
152151fveq2d 6665 . . . . . . . . . . . 12 (𝜑 → (Base‘(𝑆s 𝑅)) = (Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅)))))
153152eleq2d 2901 . . . . . . . . . . 11 (𝜑 → (𝑖 ∈ (Base‘(𝑆s 𝑅)) ↔ 𝑖 ∈ (Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅))))))
154153biimpa 480 . . . . . . . . . 10 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → 𝑖 ∈ (Base‘(Scalar‘(𝐼 mPoly (𝑆s 𝑅)))))
155150, 154ffvelrnd 6843 . . . . . . . . 9 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
15626adantr 484 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → 𝐼 ∈ V)
15727adantr 484 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → 𝑆 ∈ CRing)
15828adantr 484 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → 𝑅 ∈ (SubRing‘𝑆))
15910subrgss 19536 . . . . . . . . . . . . . 14 (𝑅 ∈ (SubRing‘𝑆) → 𝑅𝐵)
1608, 10ressbas2 16555 . . . . . . . . . . . . . 14 (𝑅𝐵𝑅 = (Base‘(𝑆s 𝑅)))
16128, 159, 1603syl 18 . . . . . . . . . . . . 13 (𝜑𝑅 = (Base‘(𝑆s 𝑅)))
162161eleq2d 2901 . . . . . . . . . . . 12 (𝜑 → (𝑖𝑅𝑖 ∈ (Base‘(𝑆s 𝑅))))
163162biimpar 481 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → 𝑖𝑅)
1646, 7, 8, 10, 25, 156, 157, 158, 163evlssca 20302 . . . . . . . . . 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 3177 . . . . . . . . . . . . 13 (𝜑 → ∀𝑓𝑅 𝜒)
167 ovex 7182 . . . . . . . . . . . . . . . . 17 (𝐵m 𝐼) ∈ V
168 snex 5319 . . . . . . . . . . . . . . . . 17 {𝑓} ∈ V
169167, 168xpex 7470 . . . . . . . . . . . . . . . 16 ((𝐵m 𝐼) × {𝑓}) ∈ V
170 mpfind.wa . . . . . . . . . . . . . . . 16 (𝑥 = ((𝐵m 𝐼) × {𝑓}) → (𝜓𝜒))
171169, 170elab 3653 . . . . . . . . . . . . . . 15 (((𝐵m 𝐼) × {𝑓}) ∈ {𝑥𝜓} ↔ 𝜒)
172 sneq 4560 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝑖 → {𝑓} = {𝑖})
173172xpeq2d 5572 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑖 → ((𝐵m 𝐼) × {𝑓}) = ((𝐵m 𝐼) × {𝑖}))
174173eleq1d 2900 . . . . . . . . . . . . . . 15 (𝑓 = 𝑖 → (((𝐵m 𝐼) × {𝑓}) ∈ {𝑥𝜓} ↔ ((𝐵m 𝐼) × {𝑖}) ∈ {𝑥𝜓}))
175171, 174bitr3id 288 . . . . . . . . . . . . . 14 (𝑓 = 𝑖 → (𝜒 ↔ ((𝐵m 𝐼) × {𝑖}) ∈ {𝑥𝜓}))
176175cbvralvw 3434 . . . . . . . . . . . . 13 (∀𝑓𝑅 𝜒 ↔ ∀𝑖𝑅 ((𝐵m 𝐼) × {𝑖}) ∈ {𝑥𝜓})
177166, 176sylib 221 . . . . . . . . . . . 12 (𝜑 → ∀𝑖𝑅 ((𝐵m 𝐼) × {𝑖}) ∈ {𝑥𝜓})
178177r19.21bi 3203 . . . . . . . . . . 11 ((𝜑𝑖𝑅) → ((𝐵m 𝐼) × {𝑖}) ∈ {𝑥𝜓})
179163, 178syldan 594 . . . . . . . . . 10 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → ((𝐵m 𝐼) × {𝑖}) ∈ {𝑥𝜓})
180164, 179eqeltrd 2916 . . . . . . . . 9 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → (((𝐼 evalSub 𝑆)‘𝑅)‘((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖)) ∈ {𝑥𝜓})
181 elpreima 6819 . . . . . . . . . . 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 484 . . . . . . . . 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 712 . . . . . . . 8 ((𝜑𝑖 ∈ (Base‘(𝑆s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
185184adantlr 714 . . . . . . 7 (((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) ∧ 𝑖 ∈ (Base‘(𝑆s 𝑅))) → ((algSc‘(𝐼 mPoly (𝑆s 𝑅)))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
18626adantr 484 . . . . . . . . . 10 ((𝜑𝑖𝐼) → 𝐼 ∈ V)
18732adantr 484 . . . . . . . . . 10 ((𝜑𝑖𝐼) → (𝑆s 𝑅) ∈ Ring)
188 simpr 488 . . . . . . . . . 10 ((𝜑𝑖𝐼) → 𝑖𝐼)
1897, 22, 12, 186, 187, 188mvrcl 20229 . . . . . . . . 9 ((𝜑𝑖𝐼) → ((𝐼 mVar (𝑆s 𝑅))‘𝑖) ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
19027adantr 484 . . . . . . . . . . 11 ((𝜑𝑖𝐼) → 𝑆 ∈ CRing)
19128adantr 484 . . . . . . . . . . 11 ((𝜑𝑖𝐼) → 𝑅 ∈ (SubRing‘𝑆))
1926, 22, 8, 10, 186, 190, 191, 188evlsvar 20303 . . . . . . . . . 10 ((𝜑𝑖𝐼) → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆s 𝑅))‘𝑖)) = (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑖)))
193 mpfind.pr . . . . . . . . . . . . . 14 ((𝜑𝑓𝐼) → 𝜃)
194167mptex 6977 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑓)) ∈ V
195 mpfind.wb . . . . . . . . . . . . . . 15 (𝑥 = (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑓)) → (𝜓𝜃))
196194, 195elab 3653 . . . . . . . . . . . . . 14 ((𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑓)) ∈ {𝑥𝜓} ↔ 𝜃)
197193, 196sylibr 237 . . . . . . . . . . . . 13 ((𝜑𝑓𝐼) → (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑓)) ∈ {𝑥𝜓})
198197ralrimiva 3177 . . . . . . . . . . . 12 (𝜑 → ∀𝑓𝐼 (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑓)) ∈ {𝑥𝜓})
199 fveq2 6661 . . . . . . . . . . . . . . 15 (𝑓 = 𝑖 → (𝑔𝑓) = (𝑔𝑖))
200199mpteq2dv 5148 . . . . . . . . . . . . . 14 (𝑓 = 𝑖 → (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑓)) = (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑖)))
201200eleq1d 2900 . . . . . . . . . . . . 13 (𝑓 = 𝑖 → ((𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑓)) ∈ {𝑥𝜓} ↔ (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑖)) ∈ {𝑥𝜓}))
202201cbvralvw 3434 . . . . . . . . . . . 12 (∀𝑓𝐼 (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑓)) ∈ {𝑥𝜓} ↔ ∀𝑖𝐼 (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑖)) ∈ {𝑥𝜓})
203198, 202sylib 221 . . . . . . . . . . 11 (𝜑 → ∀𝑖𝐼 (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑖)) ∈ {𝑥𝜓})
204203r19.21bi 3203 . . . . . . . . . 10 ((𝜑𝑖𝐼) → (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑖)) ∈ {𝑥𝜓})
205192, 204eqeltrd 2916 . . . . . . . . 9 ((𝜑𝑖𝐼) → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆s 𝑅))‘𝑖)) ∈ {𝑥𝜓})
206 elpreima 6819 . . . . . . . . . . 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 484 . . . . . . . . 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 712 . . . . . . . 8 ((𝜑𝑖𝐼) → ((𝐼 mVar (𝑆s 𝑅))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
210209adantlr 714 . . . . . . 7 (((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) ∧ 𝑖𝐼) → ((𝐼 mVar (𝑆s 𝑅))‘𝑖) ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
211 simpr 488 . . . . . . 7 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → 𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅))))
21226adantr 484 . . . . . . 7 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → 𝐼 ∈ V)
21330adantr 484 . . . . . . 7 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → (𝑆s 𝑅) ∈ CRing)
21421, 22, 7, 23, 24, 25, 12, 110, 142, 185, 210, 211, 212, 213mplind 20282 . . . . . 6 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → 𝑦 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓}))
215 fvimacnvi 6813 . . . . . 6 ((Fun ((𝐼 evalSub 𝑆)‘𝑅) ∧ 𝑦 ∈ (((𝐼 evalSub 𝑆)‘𝑅) “ {𝑥𝜓})) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥𝜓})
21620, 214, 215syl2an2r 684 . . . . 5 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥𝜓})
217 eleq1 2903 . . . . 5 ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴 → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) ∈ {𝑥𝜓} ↔ 𝐴 ∈ {𝑥𝜓}))
218216, 217syl5ibcom 248 . . . 4 ((𝜑𝑦 ∈ (Base‘(𝐼 mPoly (𝑆s 𝑅)))) → ((((𝐼 evalSub 𝑆)‘𝑅)‘𝑦) = 𝐴𝐴 ∈ {𝑥𝜓}))
219218rexlimdva 3276 . . 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 3652 . . 3 (𝐴𝑄 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜌))
2231, 222syl 17 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜌))
224220, 223mpbid 235 1 (𝜑𝜌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  {cab 2802  ∀wral 3133  ∃wrex 3134  Vcvv 3480   ⊆ wss 3919  {csn 4550   ↦ cmpt 5132   × cxp 5540  ◡ccnv 5541  ran crn 5543   “ cima 5545  Fun wfun 6337   Fn wfn 6338  ⟶wf 6339  ‘cfv 6343  (class class class)co 7149   ∘f cof 7401   ↑m cmap 8402  Basecbs 16483   ↾s cress 16484  +gcplusg 16565  .rcmulr 16566  Scalarcsca 16568   ↑s cpws 16720   MndHom cmhm 17954   GrpHom cghm 18355  mulGrpcmgp 19239  Ringcrg 19297  CRingccrg 19298   RingHom crh 19467  SubRingcsubrg 19531  AssAlgcasa 20082  algSccascl 20084   mVar cmvr 20132   mPoly cmpl 20133   evalSub ces 20284 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-ofr 7404  df-om 7575  df-1st 7684  df-2nd 7685  df-supp 7827  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-ixp 8458  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-fsupp 8831  df-sup 8903  df-oi 8971  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-5 11700  df-6 11701  df-7 11702  df-8 11703  df-9 11704  df-n0 11895  df-z 11979  df-dec 12096  df-uz 12241  df-fz 12895  df-fzo 13038  df-seq 13374  df-hash 13696  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-sca 16581  df-vsca 16582  df-ip 16583  df-tset 16584  df-ple 16585  df-ds 16587  df-hom 16589  df-cco 16590  df-0g 16715  df-gsum 16716  df-prds 16721  df-pws 16723  df-mre 16857  df-mrc 16858  df-acs 16860  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-mhm 17956  df-submnd 17957  df-grp 18106  df-minusg 18107  df-sbg 18108  df-mulg 18225  df-subg 18276  df-ghm 18356  df-cntz 18447  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-srg 19256  df-ring 19299  df-cring 19300  df-rnghom 19470  df-subrg 19533  df-lmod 19636  df-lss 19704  df-lsp 19744  df-assa 20085  df-asp 20086  df-ascl 20087  df-psr 20136  df-mvr 20137  df-mpl 20138  df-evls 20286 This theorem is referenced by:  pf1ind  20518  mzpmfp  39604
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