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Theorem hbtlem2 40652
Description: Leading coefficient ideals are ideals. (Contributed by Stefan O'Rear, 1-Apr-2015.)
Hypotheses
Ref Expression
hbtlem.p 𝑃 = (Poly1𝑅)
hbtlem.u 𝑈 = (LIdeal‘𝑃)
hbtlem.s 𝑆 = (ldgIdlSeq‘𝑅)
hbtlem2.t 𝑇 = (LIdeal‘𝑅)
Assertion
Ref Expression
hbtlem2 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) ∈ 𝑇)

Proof of Theorem hbtlem2
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hbtlem.p . . 3 𝑃 = (Poly1𝑅)
2 hbtlem.u . . 3 𝑈 = (LIdeal‘𝑃)
3 hbtlem.s . . 3 𝑆 = (ldgIdlSeq‘𝑅)
4 eqid 2737 . . 3 ( deg1𝑅) = ( deg1𝑅)
51, 2, 3, 4hbtlem1 40651 . 2 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
6 eqid 2737 . . . . . . . . . . . 12 (Base‘𝑃) = (Base‘𝑃)
76, 2lidlss 20248 . . . . . . . . . . 11 (𝐼𝑈𝐼 ⊆ (Base‘𝑃))
873ad2ant2 1136 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → 𝐼 ⊆ (Base‘𝑃))
98sselda 3901 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑏𝐼) → 𝑏 ∈ (Base‘𝑃))
10 eqid 2737 . . . . . . . . . 10 (coe1𝑏) = (coe1𝑏)
11 eqid 2737 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
1210, 6, 1, 11coe1f 21132 . . . . . . . . 9 (𝑏 ∈ (Base‘𝑃) → (coe1𝑏):ℕ0⟶(Base‘𝑅))
139, 12syl 17 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑏𝐼) → (coe1𝑏):ℕ0⟶(Base‘𝑅))
14 simpl3 1195 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑏𝐼) → 𝑋 ∈ ℕ0)
1513, 14ffvelrnd 6905 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑏𝐼) → ((coe1𝑏)‘𝑋) ∈ (Base‘𝑅))
16 eleq1a 2833 . . . . . . 7 (((coe1𝑏)‘𝑋) ∈ (Base‘𝑅) → (𝑎 = ((coe1𝑏)‘𝑋) → 𝑎 ∈ (Base‘𝑅)))
1715, 16syl 17 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑏𝐼) → (𝑎 = ((coe1𝑏)‘𝑋) → 𝑎 ∈ (Base‘𝑅)))
1817adantld 494 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑏𝐼) → (((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) → 𝑎 ∈ (Base‘𝑅)))
1918rexlimdva 3203 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) → 𝑎 ∈ (Base‘𝑅)))
2019abssdv 3982 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ⊆ (Base‘𝑅))
211ply1ring 21169 . . . . . . . 8 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
22213ad2ant1 1135 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → 𝑃 ∈ Ring)
23 simp2 1139 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → 𝐼𝑈)
24 eqid 2737 . . . . . . . 8 (0g𝑃) = (0g𝑃)
252, 24lidl0cl 20250 . . . . . . 7 ((𝑃 ∈ Ring ∧ 𝐼𝑈) → (0g𝑃) ∈ 𝐼)
2622, 23, 25syl2anc 587 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (0g𝑃) ∈ 𝐼)
274, 1, 24deg1z 24985 . . . . . . . 8 (𝑅 ∈ Ring → (( deg1𝑅)‘(0g𝑃)) = -∞)
28273ad2ant1 1135 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (( deg1𝑅)‘(0g𝑃)) = -∞)
29 nn0ssre 12094 . . . . . . . . . 10 0 ⊆ ℝ
30 ressxr 10877 . . . . . . . . . 10 ℝ ⊆ ℝ*
3129, 30sstri 3910 . . . . . . . . 9 0 ⊆ ℝ*
32 simp3 1140 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → 𝑋 ∈ ℕ0)
3331, 32sseldi 3899 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → 𝑋 ∈ ℝ*)
34 mnfle 12726 . . . . . . . 8 (𝑋 ∈ ℝ* → -∞ ≤ 𝑋)
3533, 34syl 17 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → -∞ ≤ 𝑋)
3628, 35eqbrtrd 5075 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (( deg1𝑅)‘(0g𝑃)) ≤ 𝑋)
37 eqid 2737 . . . . . . . . . 10 (0g𝑅) = (0g𝑅)
381, 24, 37coe1z 21184 . . . . . . . . 9 (𝑅 ∈ Ring → (coe1‘(0g𝑃)) = (ℕ0 × {(0g𝑅)}))
39383ad2ant1 1135 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (coe1‘(0g𝑃)) = (ℕ0 × {(0g𝑅)}))
4039fveq1d 6719 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((coe1‘(0g𝑃))‘𝑋) = ((ℕ0 × {(0g𝑅)})‘𝑋))
41 fvex 6730 . . . . . . . . 9 (0g𝑅) ∈ V
4241fvconst2 7019 . . . . . . . 8 (𝑋 ∈ ℕ0 → ((ℕ0 × {(0g𝑅)})‘𝑋) = (0g𝑅))
43423ad2ant3 1137 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((ℕ0 × {(0g𝑅)})‘𝑋) = (0g𝑅))
4440, 43eqtr2d 2778 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (0g𝑅) = ((coe1‘(0g𝑃))‘𝑋))
45 fveq2 6717 . . . . . . . . 9 (𝑏 = (0g𝑃) → (( deg1𝑅)‘𝑏) = (( deg1𝑅)‘(0g𝑃)))
4645breq1d 5063 . . . . . . . 8 (𝑏 = (0g𝑃) → ((( deg1𝑅)‘𝑏) ≤ 𝑋 ↔ (( deg1𝑅)‘(0g𝑃)) ≤ 𝑋))
47 fveq2 6717 . . . . . . . . . 10 (𝑏 = (0g𝑃) → (coe1𝑏) = (coe1‘(0g𝑃)))
4847fveq1d 6719 . . . . . . . . 9 (𝑏 = (0g𝑃) → ((coe1𝑏)‘𝑋) = ((coe1‘(0g𝑃))‘𝑋))
4948eqeq2d 2748 . . . . . . . 8 (𝑏 = (0g𝑃) → ((0g𝑅) = ((coe1𝑏)‘𝑋) ↔ (0g𝑅) = ((coe1‘(0g𝑃))‘𝑋)))
5046, 49anbi12d 634 . . . . . . 7 (𝑏 = (0g𝑃) → (((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋)) ↔ ((( deg1𝑅)‘(0g𝑃)) ≤ 𝑋 ∧ (0g𝑅) = ((coe1‘(0g𝑃))‘𝑋))))
5150rspcev 3537 . . . . . 6 (((0g𝑃) ∈ 𝐼 ∧ ((( deg1𝑅)‘(0g𝑃)) ≤ 𝑋 ∧ (0g𝑅) = ((coe1‘(0g𝑃))‘𝑋))) → ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋)))
5226, 36, 44, 51syl12anc 837 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋)))
53 eqeq1 2741 . . . . . . . 8 (𝑎 = (0g𝑅) → (𝑎 = ((coe1𝑏)‘𝑋) ↔ (0g𝑅) = ((coe1𝑏)‘𝑋)))
5453anbi2d 632 . . . . . . 7 (𝑎 = (0g𝑅) → (((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋))))
5554rexbidv 3216 . . . . . 6 (𝑎 = (0g𝑅) → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋))))
5641, 55elab 3587 . . . . 5 ((0g𝑅) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ↔ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋)))
5752, 56sylibr 237 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (0g𝑅) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
5857ne0d 4250 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ≠ ∅)
5922adantr 484 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑃 ∈ Ring)
60 simpl2 1194 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝐼𝑈)
61 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (algSc‘𝑃) = (algSc‘𝑃)
621, 61, 11, 6ply1sclf 21206 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑅 ∈ Ring → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
63623ad2ant1 1135 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
6463adantr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
65 simprl 771 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑐 ∈ (Base‘𝑅))
6664, 65ffvelrnd 6905 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → ((algSc‘𝑃)‘𝑐) ∈ (Base‘𝑃))
67 simprll 779 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋))) → 𝑓𝐼)
6867adantl 485 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑓𝐼)
69 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . . 24 (.r𝑃) = (.r𝑃)
702, 6, 69lidlmcl 20255 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑃 ∈ Ring ∧ 𝐼𝑈) ∧ (((algSc‘𝑃)‘𝑐) ∈ (Base‘𝑃) ∧ 𝑓𝐼)) → (((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓) ∈ 𝐼)
7159, 60, 66, 68, 70syl22anc 839 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓) ∈ 𝐼)
72 simprrl 781 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋))) → 𝑔𝐼)
7372adantl 485 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑔𝐼)
74 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . 23 (+g𝑃) = (+g𝑃)
752, 74lidlacl 20251 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑃 ∈ Ring ∧ 𝐼𝑈) ∧ ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓) ∈ 𝐼𝑔𝐼)) → ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) ∈ 𝐼)
7659, 60, 71, 73, 75syl22anc 839 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) ∈ 𝐼)
77 simpl1 1193 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑅 ∈ Ring)
788adantr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝐼 ⊆ (Base‘𝑃))
7978, 68sseldd 3902 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑓 ∈ (Base‘𝑃))
806, 69ringcl 19579 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃 ∈ Ring ∧ ((algSc‘𝑃)‘𝑐) ∈ (Base‘𝑃) ∧ 𝑓 ∈ (Base‘𝑃)) → (((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓) ∈ (Base‘𝑃))
8159, 66, 79, 80syl3anc 1373 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓) ∈ (Base‘𝑃))
8278, 73sseldd 3902 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑔 ∈ (Base‘𝑃))
83 simpl3 1195 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑋 ∈ ℕ0)
8431, 83sseldi 3899 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑋 ∈ ℝ*)
854, 1, 6deg1xrcl 24980 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓) ∈ (Base‘𝑃) → (( deg1𝑅)‘(((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)) ∈ ℝ*)
8681, 85syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1𝑅)‘(((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)) ∈ ℝ*)
874, 1, 6deg1xrcl 24980 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 ∈ (Base‘𝑃) → (( deg1𝑅)‘𝑓) ∈ ℝ*)
8879, 87syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1𝑅)‘𝑓) ∈ ℝ*)
894, 1, 11, 6, 69, 61deg1mul3le 25014 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 ∈ Ring ∧ 𝑐 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑃)) → (( deg1𝑅)‘(((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)) ≤ (( deg1𝑅)‘𝑓))
9077, 65, 79, 89syl3anc 1373 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1𝑅)‘(((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)) ≤ (( deg1𝑅)‘𝑓))
91 simprlr 780 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋))) → (( deg1𝑅)‘𝑓) ≤ 𝑋)
9291adantl 485 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1𝑅)‘𝑓) ≤ 𝑋)
9386, 88, 84, 90, 92xrletrd 12752 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1𝑅)‘(((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)) ≤ 𝑋)
94 simprrr 782 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋))) → (( deg1𝑅)‘𝑔) ≤ 𝑋)
9594adantl 485 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1𝑅)‘𝑔) ≤ 𝑋)
961, 4, 77, 6, 74, 81, 82, 84, 93, 95deg1addle2 25000 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1𝑅)‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔)) ≤ 𝑋)
97 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . . 24 (+g𝑅) = (+g𝑅)
981, 6, 74, 97coe1addfv 21186 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ (((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓) ∈ (Base‘𝑃) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ 𝑋 ∈ ℕ0) → ((coe1‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔))‘𝑋) = (((coe1‘(((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓))‘𝑋)(+g𝑅)((coe1𝑔)‘𝑋)))
9977, 81, 82, 83, 98syl31anc 1375 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → ((coe1‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔))‘𝑋) = (((coe1‘(((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓))‘𝑋)(+g𝑅)((coe1𝑔)‘𝑋)))
100 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . . . 25 (.r𝑅) = (.r𝑅)
1011, 6, 11, 61, 69, 100coe1sclmulfv 21204 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 ∈ Ring ∧ (𝑐 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑋 ∈ ℕ0) → ((coe1‘(((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓))‘𝑋) = (𝑐(.r𝑅)((coe1𝑓)‘𝑋)))
10277, 65, 79, 83, 101syl121anc 1377 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → ((coe1‘(((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓))‘𝑋) = (𝑐(.r𝑅)((coe1𝑓)‘𝑋)))
103102oveq1d 7228 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (((coe1‘(((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓))‘𝑋)(+g𝑅)((coe1𝑔)‘𝑋)) = ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)))
10499, 103eqtr2d 2778 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) = ((coe1‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔))‘𝑋))
105 fveq2 6717 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) → (( deg1𝑅)‘𝑏) = (( deg1𝑅)‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔)))
106105breq1d 5063 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) → ((( deg1𝑅)‘𝑏) ≤ 𝑋 ↔ (( deg1𝑅)‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔)) ≤ 𝑋))
107 fveq2 6717 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) → (coe1𝑏) = (coe1‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔)))
108107fveq1d 6719 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) → ((coe1𝑏)‘𝑋) = ((coe1‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔))‘𝑋))
109108eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) → (((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) = ((coe1𝑏)‘𝑋) ↔ ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) = ((coe1‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔))‘𝑋)))
110106, 109anbi12d 634 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) → (((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) = ((coe1𝑏)‘𝑋)) ↔ ((( deg1𝑅)‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔)) ≤ 𝑋 ∧ ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) = ((coe1‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔))‘𝑋))))
111110rspcev 3537 . . . . . . . . . . . . . . . . . . . . 21 ((((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) ∈ 𝐼 ∧ ((( deg1𝑅)‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔)) ≤ 𝑋 ∧ ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) = ((coe1‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔))‘𝑋))) → ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) = ((coe1𝑏)‘𝑋)))
11276, 96, 104, 111syl12anc 837 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) = ((coe1𝑏)‘𝑋)))
113 ovex 7246 . . . . . . . . . . . . . . . . . . . . 21 ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) ∈ V
114 eqeq1 2741 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) → (𝑎 = ((coe1𝑏)‘𝑋) ↔ ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) = ((coe1𝑏)‘𝑋)))
115114anbi2d 632 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) → (((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) = ((coe1𝑏)‘𝑋))))
116115rexbidv 3216 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) = ((coe1𝑏)‘𝑋))))
117113, 116elab 3587 . . . . . . . . . . . . . . . . . . . 20 (((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ↔ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) = ((coe1𝑏)‘𝑋)))
118112, 117sylibr 237 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
119118exp45 442 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (𝑐 ∈ (Base‘𝑅) → ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) → ((𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋) → ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))}))))
120119imp 410 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) → ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) → ((𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋) → ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})))
121120exp5c 448 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) → (𝑓𝐼 → ((( deg1𝑅)‘𝑓) ≤ 𝑋 → (𝑔𝐼 → ((( deg1𝑅)‘𝑔) ≤ 𝑋 → ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})))))
122121imp 410 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓𝐼) → ((( deg1𝑅)‘𝑓) ≤ 𝑋 → (𝑔𝐼 → ((( deg1𝑅)‘𝑔) ≤ 𝑋 → ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))}))))
123122imp41 429 . . . . . . . . . . . . . 14 (((((((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓𝐼) ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ 𝑔𝐼) ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋) → ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
124 oveq2 7221 . . . . . . . . . . . . . . 15 (𝑒 = ((coe1𝑔)‘𝑋) → ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)𝑒) = ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)))
125124eleq1d 2822 . . . . . . . . . . . . . 14 (𝑒 = ((coe1𝑔)‘𝑋) → (((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ↔ ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))}))
126123, 125syl5ibrcom 250 . . . . . . . . . . . . 13 (((((((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓𝐼) ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ 𝑔𝐼) ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋) → (𝑒 = ((coe1𝑔)‘𝑋) → ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))}))
127126expimpd 457 . . . . . . . . . . . 12 ((((((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓𝐼) ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ 𝑔𝐼) → (((( deg1𝑅)‘𝑔) ≤ 𝑋𝑒 = ((coe1𝑔)‘𝑋)) → ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))}))
128127rexlimdva 3203 . . . . . . . . . . 11 (((((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓𝐼) ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) → (∃𝑔𝐼 ((( deg1𝑅)‘𝑔) ≤ 𝑋𝑒 = ((coe1𝑔)‘𝑋)) → ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))}))
129128alrimiv 1935 . . . . . . . . . 10 (((((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓𝐼) ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) → ∀𝑒(∃𝑔𝐼 ((( deg1𝑅)‘𝑔) ≤ 𝑋𝑒 = ((coe1𝑔)‘𝑋)) → ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))}))
130 eqeq1 2741 . . . . . . . . . . . . . 14 (𝑎 = 𝑒 → (𝑎 = ((coe1𝑏)‘𝑋) ↔ 𝑒 = ((coe1𝑏)‘𝑋)))
131130anbi2d 632 . . . . . . . . . . . . 13 (𝑎 = 𝑒 → (((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))))
132131rexbidv 3216 . . . . . . . . . . . 12 (𝑎 = 𝑒 → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))))
133 fveq2 6717 . . . . . . . . . . . . . . 15 (𝑏 = 𝑔 → (( deg1𝑅)‘𝑏) = (( deg1𝑅)‘𝑔))
134133breq1d 5063 . . . . . . . . . . . . . 14 (𝑏 = 𝑔 → ((( deg1𝑅)‘𝑏) ≤ 𝑋 ↔ (( deg1𝑅)‘𝑔) ≤ 𝑋))
135 fveq2 6717 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑔 → (coe1𝑏) = (coe1𝑔))
136135fveq1d 6719 . . . . . . . . . . . . . . 15 (𝑏 = 𝑔 → ((coe1𝑏)‘𝑋) = ((coe1𝑔)‘𝑋))
137136eqeq2d 2748 . . . . . . . . . . . . . 14 (𝑏 = 𝑔 → (𝑒 = ((coe1𝑏)‘𝑋) ↔ 𝑒 = ((coe1𝑔)‘𝑋)))
138134, 137anbi12d 634 . . . . . . . . . . . . 13 (𝑏 = 𝑔 → (((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋)) ↔ ((( deg1𝑅)‘𝑔) ≤ 𝑋𝑒 = ((coe1𝑔)‘𝑋))))
139138cbvrexvw 3359 . . . . . . . . . . . 12 (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑔𝐼 ((( deg1𝑅)‘𝑔) ≤ 𝑋𝑒 = ((coe1𝑔)‘𝑋)))
140132, 139bitrdi 290 . . . . . . . . . . 11 (𝑎 = 𝑒 → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑔𝐼 ((( deg1𝑅)‘𝑔) ≤ 𝑋𝑒 = ((coe1𝑔)‘𝑋))))
141140ralab 3606 . . . . . . . . . 10 (∀𝑒 ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ↔ ∀𝑒(∃𝑔𝐼 ((( deg1𝑅)‘𝑔) ≤ 𝑋𝑒 = ((coe1𝑔)‘𝑋)) → ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))}))
142129, 141sylibr 237 . . . . . . . . 9 (((((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓𝐼) ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) → ∀𝑒 ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
143 oveq2 7221 . . . . . . . . . . . 12 (𝑑 = ((coe1𝑓)‘𝑋) → (𝑐(.r𝑅)𝑑) = (𝑐(.r𝑅)((coe1𝑓)‘𝑋)))
144143oveq1d 7228 . . . . . . . . . . 11 (𝑑 = ((coe1𝑓)‘𝑋) → ((𝑐(.r𝑅)𝑑)(+g𝑅)𝑒) = ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)𝑒))
145144eleq1d 2822 . . . . . . . . . 10 (𝑑 = ((coe1𝑓)‘𝑋) → (((𝑐(.r𝑅)𝑑)(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ↔ ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))}))
146145ralbidv 3118 . . . . . . . . 9 (𝑑 = ((coe1𝑓)‘𝑋) → (∀𝑒 ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ((𝑐(.r𝑅)𝑑)(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ↔ ∀𝑒 ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))}))
147142, 146syl5ibrcom 250 . . . . . . . 8 (((((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓𝐼) ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) → (𝑑 = ((coe1𝑓)‘𝑋) → ∀𝑒 ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ((𝑐(.r𝑅)𝑑)(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))}))
148147expimpd 457 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓𝐼) → (((( deg1𝑅)‘𝑓) ≤ 𝑋𝑑 = ((coe1𝑓)‘𝑋)) → ∀𝑒 ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ((𝑐(.r𝑅)𝑑)(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))}))
149148rexlimdva 3203 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) → (∃𝑓𝐼 ((( deg1𝑅)‘𝑓) ≤ 𝑋𝑑 = ((coe1𝑓)‘𝑋)) → ∀𝑒 ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ((𝑐(.r𝑅)𝑑)(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))}))
150149alrimiv 1935 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) → ∀𝑑(∃𝑓𝐼 ((( deg1𝑅)‘𝑓) ≤ 𝑋𝑑 = ((coe1𝑓)‘𝑋)) → ∀𝑒 ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ((𝑐(.r𝑅)𝑑)(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))}))
151 eqeq1 2741 . . . . . . . . 9 (𝑎 = 𝑑 → (𝑎 = ((coe1𝑏)‘𝑋) ↔ 𝑑 = ((coe1𝑏)‘𝑋)))
152151anbi2d 632 . . . . . . . 8 (𝑎 = 𝑑 → (((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))))
153152rexbidv 3216 . . . . . . 7 (𝑎 = 𝑑 → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))))
154 fveq2 6717 . . . . . . . . . 10 (𝑏 = 𝑓 → (( deg1𝑅)‘𝑏) = (( deg1𝑅)‘𝑓))
155154breq1d 5063 . . . . . . . . 9 (𝑏 = 𝑓 → ((( deg1𝑅)‘𝑏) ≤ 𝑋 ↔ (( deg1𝑅)‘𝑓) ≤ 𝑋))
156 fveq2 6717 . . . . . . . . . . 11 (𝑏 = 𝑓 → (coe1𝑏) = (coe1𝑓))
157156fveq1d 6719 . . . . . . . . . 10 (𝑏 = 𝑓 → ((coe1𝑏)‘𝑋) = ((coe1𝑓)‘𝑋))
158157eqeq2d 2748 . . . . . . . . 9 (𝑏 = 𝑓 → (𝑑 = ((coe1𝑏)‘𝑋) ↔ 𝑑 = ((coe1𝑓)‘𝑋)))
159155, 158anbi12d 634 . . . . . . . 8 (𝑏 = 𝑓 → (((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋)) ↔ ((( deg1𝑅)‘𝑓) ≤ 𝑋𝑑 = ((coe1𝑓)‘𝑋))))
160159cbvrexvw 3359 . . . . . . 7 (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑓𝐼 ((( deg1𝑅)‘𝑓) ≤ 𝑋𝑑 = ((coe1𝑓)‘𝑋)))
161153, 160bitrdi 290 . . . . . 6 (𝑎 = 𝑑 → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑓𝐼 ((( deg1𝑅)‘𝑓) ≤ 𝑋𝑑 = ((coe1𝑓)‘𝑋))))
162161ralab 3606 . . . . 5 (∀𝑑 ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))}∀𝑒 ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ((𝑐(.r𝑅)𝑑)(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ↔ ∀𝑑(∃𝑓𝐼 ((( deg1𝑅)‘𝑓) ≤ 𝑋𝑑 = ((coe1𝑓)‘𝑋)) → ∀𝑒 ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ((𝑐(.r𝑅)𝑑)(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))}))
163150, 162sylibr 237 . . . 4 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) → ∀𝑑 ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))}∀𝑒 ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ((𝑐(.r𝑅)𝑑)(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
164163ralrimiva 3105 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ∀𝑐 ∈ (Base‘𝑅)∀𝑑 ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))}∀𝑒 ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ((𝑐(.r𝑅)𝑑)(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
165 hbtlem2.t . . . 4 𝑇 = (LIdeal‘𝑅)
166165, 11, 97, 100islidl 20249 . . 3 ({𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ∈ 𝑇 ↔ ({𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ⊆ (Base‘𝑅) ∧ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ≠ ∅ ∧ ∀𝑐 ∈ (Base‘𝑅)∀𝑑 ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))}∀𝑒 ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ((𝑐(.r𝑅)𝑑)(+g𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))}))
16720, 58, 164, 166syl3anbrc 1345 . 2 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ∈ 𝑇)
1685, 167eqeltrd 2838 1 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) ∈ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089  wal 1541   = wceq 1543  wcel 2110  {cab 2714  wne 2940  wral 3061  wrex 3062  wss 3866  c0 4237  {csn 4541   class class class wbr 5053   × cxp 5549  wf 6376  cfv 6380  (class class class)co 7213  cr 10728  -∞cmnf 10865  *cxr 10866  cle 10868  0cn0 12090  Basecbs 16760  +gcplusg 16802  .rcmulr 16803  0gc0g 16944  Ringcrg 19562  LIdealclidl 20207  algSccascl 20814  Poly1cpl1 21098  coe1cco1 21099   deg1 cdg1 24949  ldgIdlSeqcldgis 40649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807  ax-addf 10808  ax-mulf 10809
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-ofr 7470  df-om 7645  df-1st 7761  df-2nd 7762  df-supp 7904  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-map 8510  df-pm 8511  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fsupp 8986  df-sup 9058  df-oi 9126  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-z 12177  df-dec 12294  df-uz 12439  df-fz 13096  df-fzo 13239  df-seq 13575  df-hash 13897  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-starv 16817  df-sca 16818  df-vsca 16819  df-ip 16820  df-tset 16821  df-ple 16822  df-ds 16824  df-unif 16825  df-0g 16946  df-gsum 16947  df-mre 17089  df-mrc 17090  df-acs 17092  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-mhm 18218  df-submnd 18219  df-grp 18368  df-minusg 18369  df-sbg 18370  df-mulg 18489  df-subg 18540  df-ghm 18620  df-cntz 18711  df-cmn 19172  df-abl 19173  df-mgp 19505  df-ur 19517  df-ring 19564  df-cring 19565  df-subrg 19798  df-lmod 19901  df-lss 19969  df-sra 20209  df-rgmod 20210  df-lidl 20211  df-cnfld 20364  df-ascl 20817  df-psr 20868  df-mvr 20869  df-mpl 20870  df-opsr 20872  df-psr1 21101  df-vr1 21102  df-ply1 21103  df-coe1 21104  df-mdeg 24950  df-deg1 24951  df-ldgis 40650
This theorem is referenced by:  hbtlem7  40653  hbtlem6  40657
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