Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hbtlem2 Structured version   Visualization version   GIF version

Theorem hbtlem2 38195
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 2806 . . 3 ( deg1𝑅) = ( deg1𝑅)
51, 2, 3, 4hbtlem1 38194 . 2 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
6 eqid 2806 . . . . . . . . . . . 12 (Base‘𝑃) = (Base‘𝑃)
76, 2lidlss 19419 . . . . . . . . . . 11 (𝐼𝑈𝐼 ⊆ (Base‘𝑃))
873ad2ant2 1157 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → 𝐼 ⊆ (Base‘𝑃))
98sselda 3798 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑏𝐼) → 𝑏 ∈ (Base‘𝑃))
10 eqid 2806 . . . . . . . . . 10 (coe1𝑏) = (coe1𝑏)
11 eqid 2806 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
1210, 6, 1, 11coe1f 19789 . . . . . . . . 9 (𝑏 ∈ (Base‘𝑃) → (coe1𝑏):ℕ0⟶(Base‘𝑅))
139, 12syl 17 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑏𝐼) → (coe1𝑏):ℕ0⟶(Base‘𝑅))
14 simpl3 1239 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑏𝐼) → 𝑋 ∈ ℕ0)
1513, 14ffvelrnd 6582 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑏𝐼) → ((coe1𝑏)‘𝑋) ∈ (Base‘𝑅))
16 eleq1a 2880 . . . . . . 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 480 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑏𝐼) → (((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) → 𝑎 ∈ (Base‘𝑅)))
1918rexlimdva 3219 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) → 𝑎 ∈ (Base‘𝑅)))
2019abssdv 3873 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ⊆ (Base‘𝑅))
211ply1ring 19826 . . . . . . . 8 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
22213ad2ant1 1156 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → 𝑃 ∈ Ring)
23 simp2 1160 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → 𝐼𝑈)
24 eqid 2806 . . . . . . . 8 (0g𝑃) = (0g𝑃)
252, 24lidl0cl 19421 . . . . . . 7 ((𝑃 ∈ Ring ∧ 𝐼𝑈) → (0g𝑃) ∈ 𝐼)
2622, 23, 25syl2anc 575 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (0g𝑃) ∈ 𝐼)
274, 1, 24deg1z 24061 . . . . . . . 8 (𝑅 ∈ Ring → (( deg1𝑅)‘(0g𝑃)) = -∞)
28273ad2ant1 1156 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (( deg1𝑅)‘(0g𝑃)) = -∞)
29 nn0ssre 11563 . . . . . . . . . 10 0 ⊆ ℝ
30 ressxr 10368 . . . . . . . . . 10 ℝ ⊆ ℝ*
3129, 30sstri 3807 . . . . . . . . 9 0 ⊆ ℝ*
32 simp3 1161 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → 𝑋 ∈ ℕ0)
3331, 32sseldi 3796 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → 𝑋 ∈ ℝ*)
34 mnfle 12185 . . . . . . . 8 (𝑋 ∈ ℝ* → -∞ ≤ 𝑋)
3533, 34syl 17 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → -∞ ≤ 𝑋)
3628, 35eqbrtrd 4866 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (( deg1𝑅)‘(0g𝑃)) ≤ 𝑋)
37 eqid 2806 . . . . . . . . . 10 (0g𝑅) = (0g𝑅)
381, 24, 37coe1z 19841 . . . . . . . . 9 (𝑅 ∈ Ring → (coe1‘(0g𝑃)) = (ℕ0 × {(0g𝑅)}))
39383ad2ant1 1156 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (coe1‘(0g𝑃)) = (ℕ0 × {(0g𝑅)}))
4039fveq1d 6410 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((coe1‘(0g𝑃))‘𝑋) = ((ℕ0 × {(0g𝑅)})‘𝑋))
41 fvex 6421 . . . . . . . . 9 (0g𝑅) ∈ V
4241fvconst2 6694 . . . . . . . 8 (𝑋 ∈ ℕ0 → ((ℕ0 × {(0g𝑅)})‘𝑋) = (0g𝑅))
43423ad2ant3 1158 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((ℕ0 × {(0g𝑅)})‘𝑋) = (0g𝑅))
4440, 43eqtr2d 2841 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (0g𝑅) = ((coe1‘(0g𝑃))‘𝑋))
45 fveq2 6408 . . . . . . . . 9 (𝑏 = (0g𝑃) → (( deg1𝑅)‘𝑏) = (( deg1𝑅)‘(0g𝑃)))
4645breq1d 4854 . . . . . . . 8 (𝑏 = (0g𝑃) → ((( deg1𝑅)‘𝑏) ≤ 𝑋 ↔ (( deg1𝑅)‘(0g𝑃)) ≤ 𝑋))
47 fveq2 6408 . . . . . . . . . 10 (𝑏 = (0g𝑃) → (coe1𝑏) = (coe1‘(0g𝑃)))
4847fveq1d 6410 . . . . . . . . 9 (𝑏 = (0g𝑃) → ((coe1𝑏)‘𝑋) = ((coe1‘(0g𝑃))‘𝑋))
4948eqeq2d 2816 . . . . . . . 8 (𝑏 = (0g𝑃) → ((0g𝑅) = ((coe1𝑏)‘𝑋) ↔ (0g𝑅) = ((coe1‘(0g𝑃))‘𝑋)))
5046, 49anbi12d 618 . . . . . . 7 (𝑏 = (0g𝑃) → (((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋)) ↔ ((( deg1𝑅)‘(0g𝑃)) ≤ 𝑋 ∧ (0g𝑅) = ((coe1‘(0g𝑃))‘𝑋))))
5150rspcev 3502 . . . . . 6 (((0g𝑃) ∈ 𝐼 ∧ ((( deg1𝑅)‘(0g𝑃)) ≤ 𝑋 ∧ (0g𝑅) = ((coe1‘(0g𝑃))‘𝑋))) → ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋)))
5226, 36, 44, 51syl12anc 856 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋)))
53 eqeq1 2810 . . . . . . . 8 (𝑎 = (0g𝑅) → (𝑎 = ((coe1𝑏)‘𝑋) ↔ (0g𝑅) = ((coe1𝑏)‘𝑋)))
5453anbi2d 616 . . . . . . 7 (𝑎 = (0g𝑅) → (((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋))))
5554rexbidv 3240 . . . . . 6 (𝑎 = (0g𝑅) → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋))))
5641, 55elab 3545 . . . . 5 ((0g𝑅) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ↔ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋)))
5752, 56sylibr 225 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (0g𝑅) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
5857ne0d 4123 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ≠ ∅)
5922adantr 468 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑃 ∈ Ring)
60 simpl2 1237 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝐼𝑈)
61 eqid 2806 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (algSc‘𝑃) = (algSc‘𝑃)
621, 61, 11, 6ply1sclf 19863 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑅 ∈ Ring → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
63623ad2ant1 1156 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
6463adantr 468 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
65 simprl 778 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑐 ∈ (Base‘𝑅))
6664, 65ffvelrnd 6582 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → ((algSc‘𝑃)‘𝑐) ∈ (Base‘𝑃))
67 simprll 788 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋))) → 𝑓𝐼)
6867adantl 469 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑓𝐼)
69 eqid 2806 . . . . . . . . . . . . . . . . . . . . . . . 24 (.r𝑃) = (.r𝑃)
702, 6, 69lidlmcl 19426 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑃 ∈ Ring ∧ 𝐼𝑈) ∧ (((algSc‘𝑃)‘𝑐) ∈ (Base‘𝑃) ∧ 𝑓𝐼)) → (((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓) ∈ 𝐼)
7159, 60, 66, 68, 70syl22anc 858 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓) ∈ 𝐼)
72 simprrl 790 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋))) → 𝑔𝐼)
7372adantl 469 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑔𝐼)
74 eqid 2806 . . . . . . . . . . . . . . . . . . . . . . 23 (+g𝑃) = (+g𝑃)
752, 74lidlacl 19422 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑃 ∈ Ring ∧ 𝐼𝑈) ∧ ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓) ∈ 𝐼𝑔𝐼)) → ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) ∈ 𝐼)
7659, 60, 71, 73, 75syl22anc 858 . . . . . . . . . . . . . . . . . . . . 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 1235 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑅 ∈ Ring)
788adantr 468 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝐼 ⊆ (Base‘𝑃))
7978, 68sseldd 3799 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑓 ∈ (Base‘𝑃))
806, 69ringcl 18763 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃 ∈ Ring ∧ ((algSc‘𝑃)‘𝑐) ∈ (Base‘𝑃) ∧ 𝑓 ∈ (Base‘𝑃)) → (((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓) ∈ (Base‘𝑃))
8159, 66, 79, 80syl3anc 1483 . . . . . . . . . . . . . . . . . . . . . 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 3799 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑔 ∈ (Base‘𝑃))
83 simpl3 1239 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑋 ∈ ℕ0)
8431, 83sseldi 3796 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑋 ∈ ℝ*)
854, 1, 6deg1xrcl 24056 . . . . . . . . . . . . . . . . . . . . . . . 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 24056 . . . . . . . . . . . . . . . . . . . . . . . 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 24090 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 ∈ Ring ∧ 𝑐 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑃)) → (( deg1𝑅)‘(((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)) ≤ (( deg1𝑅)‘𝑓))
9077, 65, 79, 89syl3anc 1483 . . . . . . . . . . . . . . . . . . . . . . 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 789 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋))) → (( deg1𝑅)‘𝑓) ≤ 𝑋)
9291adantl 469 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1𝑅)‘𝑓) ≤ 𝑋)
9386, 88, 84, 90, 92xrletrd 12211 . . . . . . . . . . . . . . . . . . . . . 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 791 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋))) → (( deg1𝑅)‘𝑔) ≤ 𝑋)
9594adantl 469 . . . . . . . . . . . . . . . . . . . . . 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 24076 . . . . . . . . . . . . . . . . . . . . 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 2806 . . . . . . . . . . . . . . . . . . . . . . . 24 (+g𝑅) = (+g𝑅)
981, 6, 74, 97coe1addfv 19843 . . . . . . . . . . . . . . . . . . . . . . 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 1485 . . . . . . . . . . . . . . . . . . . . . 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 2806 . . . . . . . . . . . . . . . . . . . . . . . . 25 (.r𝑅) = (.r𝑅)
1011, 6, 11, 61, 69, 100coe1sclmulfv 19861 . . . . . . . . . . . . . . . . . . . . . . . 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 1487 . . . . . . . . . . . . . . . . . . . . . . 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 6889 . . . . . . . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . . . . . . . . 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 6408 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) → (( deg1𝑅)‘𝑏) = (( deg1𝑅)‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔)))
106105breq1d 4854 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) → ((( deg1𝑅)‘𝑏) ≤ 𝑋 ↔ (( deg1𝑅)‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔)) ≤ 𝑋))
107 fveq2 6408 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) → (coe1𝑏) = (coe1‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔)))
108107fveq1d 6410 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) → ((coe1𝑏)‘𝑋) = ((coe1‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔))‘𝑋))
109108eqeq2d 2816 . . . . . . . . . . . . . . . . . . . . . . 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 618 . . . . . . . . . . . . . . . . . . . . . 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 3502 . . . . . . . . . . . . . . . . . . . . 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 856 . . . . . . . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . . . . . . 21 ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) ∈ V
114 eqeq1 2810 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) → (𝑎 = ((coe1𝑏)‘𝑋) ↔ ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) = ((coe1𝑏)‘𝑋)))
115114anbi2d 616 . . . . . . . . . . . . . . . . . . . . . 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 3240 . . . . . . . . . . . . . . . . . . . . 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 3545 . . . . . . . . . . . . . . . . . . . 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 225 . . . . . . . . . . . . . . . . . . 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 427 . . . . . . . . . . . . . . . . . 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 395 . . . . . . . . . . . . . . . . 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 433 . . . . . . . . . . . . . . . 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 395 . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . 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 6882 . . . . . . . . . . . . . . 15 (𝑒 = ((coe1𝑔)‘𝑋) → ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)𝑒) = ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)))
125124eleq1d 2870 . . . . . . . . . . . . . 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 238 . . . . . . . . . . . . 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 443 . . . . . . . . . . . 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 3219 . . . . . . . . . . 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 2018 . . . . . . . . . 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 2810 . . . . . . . . . . . . . 14 (𝑎 = 𝑒 → (𝑎 = ((coe1𝑏)‘𝑋) ↔ 𝑒 = ((coe1𝑏)‘𝑋)))
131130anbi2d 616 . . . . . . . . . . . . 13 (𝑎 = 𝑒 → (((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))))
132131rexbidv 3240 . . . . . . . . . . . 12 (𝑎 = 𝑒 → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))))
133 fveq2 6408 . . . . . . . . . . . . . . 15 (𝑏 = 𝑔 → (( deg1𝑅)‘𝑏) = (( deg1𝑅)‘𝑔))
134133breq1d 4854 . . . . . . . . . . . . . 14 (𝑏 = 𝑔 → ((( deg1𝑅)‘𝑏) ≤ 𝑋 ↔ (( deg1𝑅)‘𝑔) ≤ 𝑋))
135 fveq2 6408 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑔 → (coe1𝑏) = (coe1𝑔))
136135fveq1d 6410 . . . . . . . . . . . . . . 15 (𝑏 = 𝑔 → ((coe1𝑏)‘𝑋) = ((coe1𝑔)‘𝑋))
137136eqeq2d 2816 . . . . . . . . . . . . . 14 (𝑏 = 𝑔 → (𝑒 = ((coe1𝑏)‘𝑋) ↔ 𝑒 = ((coe1𝑔)‘𝑋)))
138134, 137anbi12d 618 . . . . . . . . . . . . 13 (𝑏 = 𝑔 → (((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋)) ↔ ((( deg1𝑅)‘𝑔) ≤ 𝑋𝑒 = ((coe1𝑔)‘𝑋))))
139138cbvrexv 3361 . . . . . . . . . . . 12 (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑔𝐼 ((( deg1𝑅)‘𝑔) ≤ 𝑋𝑒 = ((coe1𝑔)‘𝑋)))
140132, 139syl6bb 278 . . . . . . . . . . 11 (𝑎 = 𝑒 → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑔𝐼 ((( deg1𝑅)‘𝑔) ≤ 𝑋𝑒 = ((coe1𝑔)‘𝑋))))
141140ralab 3563 . . . . . . . . . 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 225 . . . . . . . . 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 6882 . . . . . . . . . . . 12 (𝑑 = ((coe1𝑓)‘𝑋) → (𝑐(.r𝑅)𝑑) = (𝑐(.r𝑅)((coe1𝑓)‘𝑋)))
144143oveq1d 6889 . . . . . . . . . . 11 (𝑑 = ((coe1𝑓)‘𝑋) → ((𝑐(.r𝑅)𝑑)(+g𝑅)𝑒) = ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)𝑒))
145144eleq1d 2870 . . . . . . . . . 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 3174 . . . . . . . . 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 238 . . . . . . . 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 443 . . . . . . 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 3219 . . . . . 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 2018 . . . . 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 2810 . . . . . . . . 9 (𝑎 = 𝑑 → (𝑎 = ((coe1𝑏)‘𝑋) ↔ 𝑑 = ((coe1𝑏)‘𝑋)))
152151anbi2d 616 . . . . . . . 8 (𝑎 = 𝑑 → (((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))))
153152rexbidv 3240 . . . . . . 7 (𝑎 = 𝑑 → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))))
154 fveq2 6408 . . . . . . . . . 10 (𝑏 = 𝑓 → (( deg1𝑅)‘𝑏) = (( deg1𝑅)‘𝑓))
155154breq1d 4854 . . . . . . . . 9 (𝑏 = 𝑓 → ((( deg1𝑅)‘𝑏) ≤ 𝑋 ↔ (( deg1𝑅)‘𝑓) ≤ 𝑋))
156 fveq2 6408 . . . . . . . . . . 11 (𝑏 = 𝑓 → (coe1𝑏) = (coe1𝑓))
157156fveq1d 6410 . . . . . . . . . 10 (𝑏 = 𝑓 → ((coe1𝑏)‘𝑋) = ((coe1𝑓)‘𝑋))
158157eqeq2d 2816 . . . . . . . . 9 (𝑏 = 𝑓 → (𝑑 = ((coe1𝑏)‘𝑋) ↔ 𝑑 = ((coe1𝑓)‘𝑋)))
159155, 158anbi12d 618 . . . . . . . 8 (𝑏 = 𝑓 → (((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋)) ↔ ((( deg1𝑅)‘𝑓) ≤ 𝑋𝑑 = ((coe1𝑓)‘𝑋))))
160159cbvrexv 3361 . . . . . . 7 (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑓𝐼 ((( deg1𝑅)‘𝑓) ≤ 𝑋𝑑 = ((coe1𝑓)‘𝑋)))
161153, 160syl6bb 278 . . . . . 6 (𝑎 = 𝑑 → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑓𝐼 ((( deg1𝑅)‘𝑓) ≤ 𝑋𝑑 = ((coe1𝑓)‘𝑋))))
162161ralab 3563 . . . . 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 225 . . . 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 3154 . . 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 19420 . . 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 1436 . 2 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ∈ 𝑇)
1685, 167eqeltrd 2885 1 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) ∈ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100  wal 1635   = wceq 1637  wcel 2156  {cab 2792  wne 2978  wral 3096  wrex 3097  wss 3769  c0 4116  {csn 4370   class class class wbr 4844   × cxp 5309  wf 6097  cfv 6101  (class class class)co 6874  cr 10220  -∞cmnf 10357  *cxr 10358  cle 10360  0cn0 11559  Basecbs 16068  +gcplusg 16153  .rcmulr 16154  0gc0g 16305  Ringcrg 18749  LIdealclidl 19379  algSccascl 19520  Poly1cpl1 19755  coe1cco1 19756   deg1 cdg1 24028  ldgIdlSeqcldgis 38192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-inf2 8785  ax-cnex 10277  ax-resscn 10278  ax-1cn 10279  ax-icn 10280  ax-addcl 10281  ax-addrcl 10282  ax-mulcl 10283  ax-mulrcl 10284  ax-mulcom 10285  ax-addass 10286  ax-mulass 10287  ax-distr 10288  ax-i2m1 10289  ax-1ne0 10290  ax-1rid 10291  ax-rnegex 10292  ax-rrecex 10293  ax-cnre 10294  ax-pre-lttri 10295  ax-pre-lttrn 10296  ax-pre-ltadd 10297  ax-pre-mulgt0 10298  ax-pre-sup 10299  ax-addf 10300  ax-mulf 10301
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-iin 4715  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-of 7127  df-ofr 7128  df-om 7296  df-1st 7398  df-2nd 7399  df-supp 7530  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-2o 7797  df-oadd 7800  df-er 7979  df-map 8094  df-pm 8095  df-ixp 8146  df-en 8193  df-dom 8194  df-sdom 8195  df-fin 8196  df-fsupp 8515  df-sup 8587  df-oi 8654  df-card 9048  df-pnf 10361  df-mnf 10362  df-xr 10363  df-ltxr 10364  df-le 10365  df-sub 10553  df-neg 10554  df-nn 11306  df-2 11364  df-3 11365  df-4 11366  df-5 11367  df-6 11368  df-7 11369  df-8 11370  df-9 11371  df-n0 11560  df-z 11644  df-dec 11760  df-uz 11905  df-fz 12550  df-fzo 12690  df-seq 13025  df-hash 13338  df-struct 16070  df-ndx 16071  df-slot 16072  df-base 16074  df-sets 16075  df-ress 16076  df-plusg 16166  df-mulr 16167  df-starv 16168  df-sca 16169  df-vsca 16170  df-ip 16171  df-tset 16172  df-ple 16173  df-ds 16175  df-unif 16176  df-0g 16307  df-gsum 16308  df-mre 16451  df-mrc 16452  df-acs 16454  df-mgm 17447  df-sgrp 17489  df-mnd 17500  df-mhm 17540  df-submnd 17541  df-grp 17630  df-minusg 17631  df-sbg 17632  df-mulg 17746  df-subg 17793  df-ghm 17860  df-cntz 17951  df-cmn 18396  df-abl 18397  df-mgp 18692  df-ur 18704  df-ring 18751  df-cring 18752  df-subrg 18982  df-lmod 19069  df-lss 19137  df-sra 19381  df-rgmod 19382  df-lidl 19383  df-ascl 19523  df-psr 19565  df-mvr 19566  df-mpl 19567  df-opsr 19569  df-psr1 19758  df-vr1 19759  df-ply1 19760  df-coe1 19761  df-cnfld 19955  df-mdeg 24029  df-deg1 24030  df-ldgis 38193
This theorem is referenced by:  hbtlem7  38196  hbtlem6  38200
  Copyright terms: Public domain W3C validator