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Theorem hbtlem2 43136
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 43135 . 2 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = {𝑎 ∣ ∃𝑏𝐼 (((deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
6 eqid 2737 . . . . . . . . . . . 12 (Base‘𝑃) = (Base‘𝑃)
76, 2lidlss 21222 . . . . . . . . . . 11 (𝐼𝑈𝐼 ⊆ (Base‘𝑃))
873ad2ant2 1135 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → 𝐼 ⊆ (Base‘𝑃))
98sselda 3983 . . . . . . . . 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 22213 . . . . . . . . 9 (𝑏 ∈ (Base‘𝑃) → (coe1𝑏):ℕ0⟶(Base‘𝑅))
139, 12syl 17 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑏𝐼) → (coe1𝑏):ℕ0⟶(Base‘𝑅))
14 simpl3 1194 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑏𝐼) → 𝑋 ∈ ℕ0)
1513, 14ffvelcdmd 7105 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑏𝐼) → ((coe1𝑏)‘𝑋) ∈ (Base‘𝑅))
16 eleq1a 2836 . . . . . . 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 490 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑏𝐼) → ((((deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) → 𝑎 ∈ (Base‘𝑅)))
1918rexlimdva 3155 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (∃𝑏𝐼 (((deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) → 𝑎 ∈ (Base‘𝑅)))
2019abssdv 4068 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏𝐼 (((deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ⊆ (Base‘𝑅))
211ply1ring 22249 . . . . . . . 8 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
22213ad2ant1 1134 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → 𝑃 ∈ Ring)
23 simp2 1138 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → 𝐼𝑈)
24 eqid 2737 . . . . . . . 8 (0g𝑃) = (0g𝑃)
252, 24lidl0cl 21230 . . . . . . 7 ((𝑃 ∈ Ring ∧ 𝐼𝑈) → (0g𝑃) ∈ 𝐼)
2622, 23, 25syl2anc 584 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (0g𝑃) ∈ 𝐼)
274, 1, 24deg1z 26126 . . . . . . . 8 (𝑅 ∈ Ring → ((deg1𝑅)‘(0g𝑃)) = -∞)
28273ad2ant1 1134 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((deg1𝑅)‘(0g𝑃)) = -∞)
29 nn0ssre 12530 . . . . . . . . . 10 0 ⊆ ℝ
30 ressxr 11305 . . . . . . . . . 10 ℝ ⊆ ℝ*
3129, 30sstri 3993 . . . . . . . . 9 0 ⊆ ℝ*
32 simp3 1139 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → 𝑋 ∈ ℕ0)
3331, 32sselid 3981 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → 𝑋 ∈ ℝ*)
34 mnfle 13177 . . . . . . . 8 (𝑋 ∈ ℝ* → -∞ ≤ 𝑋)
3533, 34syl 17 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → -∞ ≤ 𝑋)
3628, 35eqbrtrd 5165 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((deg1𝑅)‘(0g𝑃)) ≤ 𝑋)
37 eqid 2737 . . . . . . . . . 10 (0g𝑅) = (0g𝑅)
381, 24, 37coe1z 22266 . . . . . . . . 9 (𝑅 ∈ Ring → (coe1‘(0g𝑃)) = (ℕ0 × {(0g𝑅)}))
39383ad2ant1 1134 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (coe1‘(0g𝑃)) = (ℕ0 × {(0g𝑅)}))
4039fveq1d 6908 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((coe1‘(0g𝑃))‘𝑋) = ((ℕ0 × {(0g𝑅)})‘𝑋))
41 fvex 6919 . . . . . . . . 9 (0g𝑅) ∈ V
4241fvconst2 7224 . . . . . . . 8 (𝑋 ∈ ℕ0 → ((ℕ0 × {(0g𝑅)})‘𝑋) = (0g𝑅))
43423ad2ant3 1136 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((ℕ0 × {(0g𝑅)})‘𝑋) = (0g𝑅))
4440, 43eqtr2d 2778 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (0g𝑅) = ((coe1‘(0g𝑃))‘𝑋))
45 fveq2 6906 . . . . . . . . 9 (𝑏 = (0g𝑃) → ((deg1𝑅)‘𝑏) = ((deg1𝑅)‘(0g𝑃)))
4645breq1d 5153 . . . . . . . 8 (𝑏 = (0g𝑃) → (((deg1𝑅)‘𝑏) ≤ 𝑋 ↔ ((deg1𝑅)‘(0g𝑃)) ≤ 𝑋))
47 fveq2 6906 . . . . . . . . . 10 (𝑏 = (0g𝑃) → (coe1𝑏) = (coe1‘(0g𝑃)))
4847fveq1d 6908 . . . . . . . . 9 (𝑏 = (0g𝑃) → ((coe1𝑏)‘𝑋) = ((coe1‘(0g𝑃))‘𝑋))
4948eqeq2d 2748 . . . . . . . 8 (𝑏 = (0g𝑃) → ((0g𝑅) = ((coe1𝑏)‘𝑋) ↔ (0g𝑅) = ((coe1‘(0g𝑃))‘𝑋)))
5046, 49anbi12d 632 . . . . . . 7 (𝑏 = (0g𝑃) → ((((deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋)) ↔ (((deg1𝑅)‘(0g𝑃)) ≤ 𝑋 ∧ (0g𝑅) = ((coe1‘(0g𝑃))‘𝑋))))
5150rspcev 3622 . . . . . 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 630 . . . . . . 7 (𝑎 = (0g𝑅) → ((((deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ (((deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋))))
5554rexbidv 3179 . . . . . 6 (𝑎 = (0g𝑅) → (∃𝑏𝐼 (((deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑏𝐼 (((deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋))))
5641, 55elab 3679 . . . . 5 ((0g𝑅) ∈ {𝑎 ∣ ∃𝑏𝐼 (((deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ↔ ∃𝑏𝐼 (((deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋)))
5752, 56sylibr 234 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (0g𝑅) ∈ {𝑎 ∣ ∃𝑏𝐼 (((deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
5857ne0d 4342 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏𝐼 (((deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ≠ ∅)
5922adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ ((deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ ((deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑃 ∈ Ring)
60 simpl2 1193 . . . . . . . . . . . . . . . . . . . . . 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 22288 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑅 ∈ Ring → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
63623ad2ant1 1134 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
6463adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 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, 65ffvelcdmd 7105 . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . 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 21235 . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . 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 21231 . . . . . . . . . . . . . . . . . . . . . 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 1192 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ ((deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ ((deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑅 ∈ Ring)
788adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ ((deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ ((deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝐼 ⊆ (Base‘𝑃))
7978, 68sseldd 3984 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ ((deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ ((deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑓 ∈ (Base‘𝑃))
806, 69ringcl 20247 . . . . . . . . . . . . . . . . . . . . . . 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 3984 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ ((deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ ((deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑔 ∈ (Base‘𝑃))
83 simpl3 1194 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ ((deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ ((deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑋 ∈ ℕ0)
8431, 83sselid 3981 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ ((deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ ((deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑋 ∈ ℝ*)
854, 1, 6deg1xrcl 26121 . . . . . . . . . . . . . . . . . . . . . . . 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 26121 . . . . . . . . . . . . . . . . . . . . . . . 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 26156 . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ ((deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ ((deg1𝑅)‘𝑔) ≤ 𝑋)))) → ((deg1𝑅)‘𝑓) ≤ 𝑋)
9386, 88, 84, 90, 92xrletrd 13204 . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . 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 26141 . . . . . . . . . . . . . . . . . . . . 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 22268 . . . . . . . . . . . . . . . . . . . . . . 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 22286 . . . . . . . . . . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) → ((deg1𝑅)‘𝑏) = ((deg1𝑅)‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔)))
106105breq1d 5153 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) → (((deg1𝑅)‘𝑏) ≤ 𝑋 ↔ ((deg1𝑅)‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔)) ≤ 𝑋))
107 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) → (coe1𝑏) = (coe1‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔)))
108107fveq1d 6908 . . . . . . . . . . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . . . . . . . . . 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 3622 . . . . . . . . . . . . . . . . . . . . 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 7464 . . . . . . . . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . . . . . . . . . . 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 3179 . . . . . . . . . . . . . . . . . . . . 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 3679 . . . . . . . . . . . . . . . . . . . 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 234 . . . . . . . . . . . . . . . . . . 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 438 . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . 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 425 . . . . . . . . . . . . . 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 7439 . . . . . . . . . . . . . . 15 (𝑒 = ((coe1𝑔)‘𝑋) → ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)𝑒) = ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)))
125124eleq1d 2826 . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . 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 453 . . . . . . . . . . . 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 3155 . . . . . . . . . . 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 1927 . . . . . . . . . 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 630 . . . . . . . . . . . . 13 (𝑎 = 𝑒 → ((((deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ (((deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))))
132131rexbidv 3179 . . . . . . . . . . . 12 (𝑎 = 𝑒 → (∃𝑏𝐼 (((deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑏𝐼 (((deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))))
133 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑏 = 𝑔 → ((deg1𝑅)‘𝑏) = ((deg1𝑅)‘𝑔))
134133breq1d 5153 . . . . . . . . . . . . . 14 (𝑏 = 𝑔 → (((deg1𝑅)‘𝑏) ≤ 𝑋 ↔ ((deg1𝑅)‘𝑔) ≤ 𝑋))
135 fveq2 6906 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑔 → (coe1𝑏) = (coe1𝑔))
136135fveq1d 6908 . . . . . . . . . . . . . . 15 (𝑏 = 𝑔 → ((coe1𝑏)‘𝑋) = ((coe1𝑔)‘𝑋))
137136eqeq2d 2748 . . . . . . . . . . . . . 14 (𝑏 = 𝑔 → (𝑒 = ((coe1𝑏)‘𝑋) ↔ 𝑒 = ((coe1𝑔)‘𝑋)))
138134, 137anbi12d 632 . . . . . . . . . . . . 13 (𝑏 = 𝑔 → ((((deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋)) ↔ (((deg1𝑅)‘𝑔) ≤ 𝑋𝑒 = ((coe1𝑔)‘𝑋))))
139138cbvrexvw 3238 . . . . . . . . . . . 12 (∃𝑏𝐼 (((deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑔𝐼 (((deg1𝑅)‘𝑔) ≤ 𝑋𝑒 = ((coe1𝑔)‘𝑋)))
140132, 139bitrdi 287 . . . . . . . . . . 11 (𝑎 = 𝑒 → (∃𝑏𝐼 (((deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑔𝐼 (((deg1𝑅)‘𝑔) ≤ 𝑋𝑒 = ((coe1𝑔)‘𝑋))))
141140ralab 3697 . . . . . . . . . 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 234 . . . . . . . . 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 7439 . . . . . . . . . . . 12 (𝑑 = ((coe1𝑓)‘𝑋) → (𝑐(.r𝑅)𝑑) = (𝑐(.r𝑅)((coe1𝑓)‘𝑋)))
144143oveq1d 7446 . . . . . . . . . . 11 (𝑑 = ((coe1𝑓)‘𝑋) → ((𝑐(.r𝑅)𝑑)(+g𝑅)𝑒) = ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)𝑒))
145144eleq1d 2826 . . . . . . . . . 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 3178 . . . . . . . . 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 247 . . . . . . . 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 453 . . . . . . 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 3155 . . . . . 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 1927 . . . . 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 630 . . . . . . . 8 (𝑎 = 𝑑 → ((((deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ (((deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))))
153152rexbidv 3179 . . . . . . 7 (𝑎 = 𝑑 → (∃𝑏𝐼 (((deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑏𝐼 (((deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))))
154 fveq2 6906 . . . . . . . . . 10 (𝑏 = 𝑓 → ((deg1𝑅)‘𝑏) = ((deg1𝑅)‘𝑓))
155154breq1d 5153 . . . . . . . . 9 (𝑏 = 𝑓 → (((deg1𝑅)‘𝑏) ≤ 𝑋 ↔ ((deg1𝑅)‘𝑓) ≤ 𝑋))
156 fveq2 6906 . . . . . . . . . . 11 (𝑏 = 𝑓 → (coe1𝑏) = (coe1𝑓))
157156fveq1d 6908 . . . . . . . . . 10 (𝑏 = 𝑓 → ((coe1𝑏)‘𝑋) = ((coe1𝑓)‘𝑋))
158157eqeq2d 2748 . . . . . . . . 9 (𝑏 = 𝑓 → (𝑑 = ((coe1𝑏)‘𝑋) ↔ 𝑑 = ((coe1𝑓)‘𝑋)))
159155, 158anbi12d 632 . . . . . . . 8 (𝑏 = 𝑓 → ((((deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋)) ↔ (((deg1𝑅)‘𝑓) ≤ 𝑋𝑑 = ((coe1𝑓)‘𝑋))))
160159cbvrexvw 3238 . . . . . . 7 (∃𝑏𝐼 (((deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑓𝐼 (((deg1𝑅)‘𝑓) ≤ 𝑋𝑑 = ((coe1𝑓)‘𝑋)))
161153, 160bitrdi 287 . . . . . 6 (𝑎 = 𝑑 → (∃𝑏𝐼 (((deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑓𝐼 (((deg1𝑅)‘𝑓) ≤ 𝑋𝑑 = ((coe1𝑓)‘𝑋))))
162161ralab 3697 . . . . 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 234 . . . 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 3146 . . 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 21225 . . 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 1344 . 2 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏𝐼 (((deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ∈ 𝑇)
1685, 167eqeltrd 2841 1 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) ∈ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087  wal 1538   = wceq 1540  wcel 2108  {cab 2714  wne 2940  wral 3061  wrex 3070  wss 3951  c0 4333  {csn 4626   class class class wbr 5143   × cxp 5683  wf 6557  cfv 6561  (class class class)co 7431  cr 11154  -∞cmnf 11293  *cxr 11294  cle 11296  0cn0 12526  Basecbs 17247  +gcplusg 17297  .rcmulr 17298  0gc0g 17484  Ringcrg 20230  LIdealclidl 21216  algSccascl 21872  Poly1cpl1 22178  coe1cco1 22179  deg1cdg1 26093  ldgIdlSeqcldgis 43133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-cring 20233  df-subrng 20546  df-subrg 20570  df-lmod 20860  df-lss 20930  df-sra 21172  df-rgmod 21173  df-lidl 21218  df-cnfld 21365  df-ascl 21875  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-psr1 22181  df-vr1 22182  df-ply1 22183  df-coe1 22184  df-mdeg 26094  df-deg1 26095  df-ldgis 43134
This theorem is referenced by:  hbtlem7  43137  hbtlem6  43141
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