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 37210
 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 2621 . . 3 ( deg1𝑅) = ( deg1𝑅)
51, 2, 3, 4hbtlem1 37209 . 2 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
6 eqid 2621 . . . . . . . . . . . 12 (Base‘𝑃) = (Base‘𝑃)
76, 2lidlss 19142 . . . . . . . . . . 11 (𝐼𝑈𝐼 ⊆ (Base‘𝑃))
873ad2ant2 1081 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → 𝐼 ⊆ (Base‘𝑃))
98sselda 3587 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑏𝐼) → 𝑏 ∈ (Base‘𝑃))
10 eqid 2621 . . . . . . . . . 10 (coe1𝑏) = (coe1𝑏)
11 eqid 2621 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
1210, 6, 1, 11coe1f 19513 . . . . . . . . 9 (𝑏 ∈ (Base‘𝑃) → (coe1𝑏):ℕ0⟶(Base‘𝑅))
139, 12syl 17 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑏𝐼) → (coe1𝑏):ℕ0⟶(Base‘𝑅))
14 simpl3 1064 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑏𝐼) → 𝑋 ∈ ℕ0)
1513, 14ffvelrnd 6321 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑏𝐼) → ((coe1𝑏)‘𝑋) ∈ (Base‘𝑅))
16 eleq1a 2693 . . . . . . 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 483 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ 𝑏𝐼) → (((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) → 𝑎 ∈ (Base‘𝑅)))
1918rexlimdva 3025 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) → 𝑎 ∈ (Base‘𝑅)))
2019abssdv 3660 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ⊆ (Base‘𝑅))
211ply1ring 19550 . . . . . . . 8 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
22213ad2ant1 1080 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → 𝑃 ∈ Ring)
23 simp2 1060 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → 𝐼𝑈)
24 eqid 2621 . . . . . . . 8 (0g𝑃) = (0g𝑃)
252, 24lidl0cl 19144 . . . . . . 7 ((𝑃 ∈ Ring ∧ 𝐼𝑈) → (0g𝑃) ∈ 𝐼)
2622, 23, 25syl2anc 692 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (0g𝑃) ∈ 𝐼)
274, 1, 24deg1z 23768 . . . . . . . 8 (𝑅 ∈ Ring → (( deg1𝑅)‘(0g𝑃)) = -∞)
28273ad2ant1 1080 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (( deg1𝑅)‘(0g𝑃)) = -∞)
29 nn0ssre 11248 . . . . . . . . . 10 0 ⊆ ℝ
30 ressxr 10035 . . . . . . . . . 10 ℝ ⊆ ℝ*
3129, 30sstri 3596 . . . . . . . . 9 0 ⊆ ℝ*
32 simp3 1061 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → 𝑋 ∈ ℕ0)
3331, 32sseldi 3585 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → 𝑋 ∈ ℝ*)
34 mnfle 11921 . . . . . . . 8 (𝑋 ∈ ℝ* → -∞ ≤ 𝑋)
3533, 34syl 17 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → -∞ ≤ 𝑋)
3628, 35eqbrtrd 4640 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (( deg1𝑅)‘(0g𝑃)) ≤ 𝑋)
37 eqid 2621 . . . . . . . . . 10 (0g𝑅) = (0g𝑅)
381, 24, 37coe1z 19565 . . . . . . . . 9 (𝑅 ∈ Ring → (coe1‘(0g𝑃)) = (ℕ0 × {(0g𝑅)}))
39383ad2ant1 1080 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (coe1‘(0g𝑃)) = (ℕ0 × {(0g𝑅)}))
4039fveq1d 6155 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((coe1‘(0g𝑃))‘𝑋) = ((ℕ0 × {(0g𝑅)})‘𝑋))
41 fvex 6163 . . . . . . . . 9 (0g𝑅) ∈ V
4241fvconst2 6429 . . . . . . . 8 (𝑋 ∈ ℕ0 → ((ℕ0 × {(0g𝑅)})‘𝑋) = (0g𝑅))
43423ad2ant3 1082 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((ℕ0 × {(0g𝑅)})‘𝑋) = (0g𝑅))
4440, 43eqtr2d 2656 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (0g𝑅) = ((coe1‘(0g𝑃))‘𝑋))
45 fveq2 6153 . . . . . . . . 9 (𝑏 = (0g𝑃) → (( deg1𝑅)‘𝑏) = (( deg1𝑅)‘(0g𝑃)))
4645breq1d 4628 . . . . . . . 8 (𝑏 = (0g𝑃) → ((( deg1𝑅)‘𝑏) ≤ 𝑋 ↔ (( deg1𝑅)‘(0g𝑃)) ≤ 𝑋))
47 fveq2 6153 . . . . . . . . . 10 (𝑏 = (0g𝑃) → (coe1𝑏) = (coe1‘(0g𝑃)))
4847fveq1d 6155 . . . . . . . . 9 (𝑏 = (0g𝑃) → ((coe1𝑏)‘𝑋) = ((coe1‘(0g𝑃))‘𝑋))
4948eqeq2d 2631 . . . . . . . 8 (𝑏 = (0g𝑃) → ((0g𝑅) = ((coe1𝑏)‘𝑋) ↔ (0g𝑅) = ((coe1‘(0g𝑃))‘𝑋)))
5046, 49anbi12d 746 . . . . . . 7 (𝑏 = (0g𝑃) → (((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋)) ↔ ((( deg1𝑅)‘(0g𝑃)) ≤ 𝑋 ∧ (0g𝑅) = ((coe1‘(0g𝑃))‘𝑋))))
5150rspcev 3298 . . . . . 6 (((0g𝑃) ∈ 𝐼 ∧ ((( deg1𝑅)‘(0g𝑃)) ≤ 𝑋 ∧ (0g𝑅) = ((coe1‘(0g𝑃))‘𝑋))) → ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋)))
5226, 36, 44, 51syl12anc 1321 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋)))
53 eqeq1 2625 . . . . . . . 8 (𝑎 = (0g𝑅) → (𝑎 = ((coe1𝑏)‘𝑋) ↔ (0g𝑅) = ((coe1𝑏)‘𝑋)))
5453anbi2d 739 . . . . . . 7 (𝑎 = (0g𝑅) → (((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋))))
5554rexbidv 3046 . . . . . 6 (𝑎 = (0g𝑅) → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋))))
5641, 55elab 3337 . . . . 5 ((0g𝑅) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ↔ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋 ∧ (0g𝑅) = ((coe1𝑏)‘𝑋)))
5752, 56sylibr 224 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (0g𝑅) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
58 ne0i 3902 . . . 4 ((0g𝑅) ∈ {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} → {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ≠ ∅)
5957, 58syl 17 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ≠ ∅)
6022adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑃 ∈ Ring)
61 simpl2 1063 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝐼𝑈)
62 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (algSc‘𝑃) = (algSc‘𝑃)
631, 62, 11, 6ply1sclf 19587 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑅 ∈ Ring → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
64633ad2ant1 1080 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
6564adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
66 simprl 793 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑐 ∈ (Base‘𝑅))
6765, 66ffvelrnd 6321 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → ((algSc‘𝑃)‘𝑐) ∈ (Base‘𝑃))
68 simprll 801 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋))) → 𝑓𝐼)
6968adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑓𝐼)
70 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . 24 (.r𝑃) = (.r𝑃)
712, 6, 70lidlmcl 19149 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑃 ∈ Ring ∧ 𝐼𝑈) ∧ (((algSc‘𝑃)‘𝑐) ∈ (Base‘𝑃) ∧ 𝑓𝐼)) → (((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓) ∈ 𝐼)
7260, 61, 67, 69, 71syl22anc 1324 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓) ∈ 𝐼)
73 simprrl 803 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋))) → 𝑔𝐼)
7473adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑔𝐼)
75 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . 23 (+g𝑃) = (+g𝑃)
762, 75lidlacl 19145 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑃 ∈ Ring ∧ 𝐼𝑈) ∧ ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓) ∈ 𝐼𝑔𝐼)) → ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) ∈ 𝐼)
7760, 61, 72, 74, 76syl22anc 1324 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) ∈ 𝐼)
78 simpl1 1062 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑅 ∈ Ring)
798adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝐼 ⊆ (Base‘𝑃))
8079, 69sseldd 3588 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑓 ∈ (Base‘𝑃))
816, 70ringcl 18493 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃 ∈ Ring ∧ ((algSc‘𝑃)‘𝑐) ∈ (Base‘𝑃) ∧ 𝑓 ∈ (Base‘𝑃)) → (((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓) ∈ (Base‘𝑃))
8260, 67, 80, 81syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓) ∈ (Base‘𝑃))
8379, 74sseldd 3588 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑔 ∈ (Base‘𝑃))
84 simpl3 1064 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑋 ∈ ℕ0)
8531, 84sseldi 3585 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → 𝑋 ∈ ℝ*)
864, 1, 6deg1xrcl 23763 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓) ∈ (Base‘𝑃) → (( deg1𝑅)‘(((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)) ∈ ℝ*)
8782, 86syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1𝑅)‘(((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)) ∈ ℝ*)
884, 1, 6deg1xrcl 23763 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 ∈ (Base‘𝑃) → (( deg1𝑅)‘𝑓) ∈ ℝ*)
8980, 88syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1𝑅)‘𝑓) ∈ ℝ*)
904, 1, 11, 6, 70, 62deg1mul3le 23797 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 ∈ Ring ∧ 𝑐 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑃)) → (( deg1𝑅)‘(((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)) ≤ (( deg1𝑅)‘𝑓))
9178, 66, 80, 90syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1𝑅)‘(((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)) ≤ (( deg1𝑅)‘𝑓))
92 simprlr 802 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋))) → (( deg1𝑅)‘𝑓) ≤ 𝑋)
9392adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1𝑅)‘𝑓) ≤ 𝑋)
9487, 89, 85, 91, 93xrletrd 11945 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1𝑅)‘(((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)) ≤ 𝑋)
95 simprrr 804 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋))) → (( deg1𝑅)‘𝑔) ≤ 𝑋)
9695adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓𝐼 ∧ (( deg1𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔𝐼 ∧ (( deg1𝑅)‘𝑔) ≤ 𝑋)))) → (( deg1𝑅)‘𝑔) ≤ 𝑋)
971, 4, 78, 6, 75, 82, 83, 85, 94, 96deg1addle2 23783 . . . . . . . . . . . . . . . . . . . . 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)
98 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . 24 (+g𝑅) = (+g𝑅)
991, 6, 75, 98coe1addfv 19567 . . . . . . . . . . . . . . . . . . . . . . 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)))
10078, 82, 83, 84, 99syl31anc 1326 . . . . . . . . . . . . . . . . . . . . . 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)))
101 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . . 25 (.r𝑅) = (.r𝑅)
1021, 6, 11, 62, 70, 101coe1sclmulfv 19585 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 ∈ Ring ∧ (𝑐 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑋 ∈ ℕ0) → ((coe1‘(((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓))‘𝑋) = (𝑐(.r𝑅)((coe1𝑓)‘𝑋)))
10378, 66, 80, 84, 102syl121anc 1328 . . . . . . . . . . . . . . . . . . . . . . 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)))
104103oveq1d 6625 . . . . . . . . . . . . . . . . . . . . . 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)))
105100, 104eqtr2d 2656 . . . . . . . . . . . . . . . . . . . . 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))
106 fveq2 6153 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) → (( deg1𝑅)‘𝑏) = (( deg1𝑅)‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔)))
107106breq1d 4628 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) → ((( deg1𝑅)‘𝑏) ≤ 𝑋 ↔ (( deg1𝑅)‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔)) ≤ 𝑋))
108 fveq2 6153 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) → (coe1𝑏) = (coe1‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔)))
109108fveq1d 6155 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔) → ((coe1𝑏)‘𝑋) = ((coe1‘((((algSc‘𝑃)‘𝑐)(.r𝑃)𝑓)(+g𝑃)𝑔))‘𝑋))
110109eqeq2d 2631 . . . . . . . . . . . . . . . . . . . . . . 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)))
111107, 110anbi12d 746 . . . . . . . . . . . . . . . . . . . . . 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))))
112111rspcev 3298 . . . . . . . . . . . . . . . . . . . . 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)))
11377, 97, 105, 112syl12anc 1321 . . . . . . . . . . . . . . . . . . . 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)))
114 ovex 6638 . . . . . . . . . . . . . . . . . . . . 21 ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) ∈ V
115 eqeq1 2625 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) → (𝑎 = ((coe1𝑏)‘𝑋) ↔ ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)) = ((coe1𝑏)‘𝑋)))
116115anbi2d 739 . . . . . . . . . . . . . . . . . . . . . 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))))
117116rexbidv 3046 . . . . . . . . . . . . . . . . . . . . 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))))
118114, 117elab 3337 . . . . . . . . . . . . . . . . . . . 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)))
119113, 118sylibr 224 . . . . . . . . . . . . . . . . . . 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))})
120119exp45 641 . . . . . . . . . . . . . . . . . 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))}))))
121120imp 445 . . . . . . . . . . . . . . . . 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))})))
122121exp5c 643 . . . . . . . . . . . . . . . 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))})))))
123122imp 445 . . . . . . . . . . . . . . 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))}))))
124123imp41 618 . . . . . . . . . . . . . 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))})
125 oveq2 6618 . . . . . . . . . . . . . . 15 (𝑒 = ((coe1𝑔)‘𝑋) → ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)𝑒) = ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)((coe1𝑔)‘𝑋)))
126125eleq1d 2683 . . . . . . . . . . . . . 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))}))
127124, 126syl5ibrcom 237 . . . . . . . . . . . . 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))}))
128127expimpd 628 . . . . . . . . . . . 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))}))
129128rexlimdva 3025 . . . . . . . . . . 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))}))
130129alrimiv 1852 . . . . . . . . . 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))}))
131 eqeq1 2625 . . . . . . . . . . . . . 14 (𝑎 = 𝑒 → (𝑎 = ((coe1𝑏)‘𝑋) ↔ 𝑒 = ((coe1𝑏)‘𝑋)))
132131anbi2d 739 . . . . . . . . . . . . 13 (𝑎 = 𝑒 → (((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))))
133132rexbidv 3046 . . . . . . . . . . . 12 (𝑎 = 𝑒 → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋))))
134 fveq2 6153 . . . . . . . . . . . . . . 15 (𝑏 = 𝑔 → (( deg1𝑅)‘𝑏) = (( deg1𝑅)‘𝑔))
135134breq1d 4628 . . . . . . . . . . . . . 14 (𝑏 = 𝑔 → ((( deg1𝑅)‘𝑏) ≤ 𝑋 ↔ (( deg1𝑅)‘𝑔) ≤ 𝑋))
136 fveq2 6153 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑔 → (coe1𝑏) = (coe1𝑔))
137136fveq1d 6155 . . . . . . . . . . . . . . 15 (𝑏 = 𝑔 → ((coe1𝑏)‘𝑋) = ((coe1𝑔)‘𝑋))
138137eqeq2d 2631 . . . . . . . . . . . . . 14 (𝑏 = 𝑔 → (𝑒 = ((coe1𝑏)‘𝑋) ↔ 𝑒 = ((coe1𝑔)‘𝑋)))
139135, 138anbi12d 746 . . . . . . . . . . . . 13 (𝑏 = 𝑔 → (((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋)) ↔ ((( deg1𝑅)‘𝑔) ≤ 𝑋𝑒 = ((coe1𝑔)‘𝑋))))
140139cbvrexv 3163 . . . . . . . . . . . 12 (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑒 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑔𝐼 ((( deg1𝑅)‘𝑔) ≤ 𝑋𝑒 = ((coe1𝑔)‘𝑋)))
141133, 140syl6bb 276 . . . . . . . . . . 11 (𝑎 = 𝑒 → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑔𝐼 ((( deg1𝑅)‘𝑔) ≤ 𝑋𝑒 = ((coe1𝑔)‘𝑋))))
142141ralab 3353 . . . . . . . . . 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))}))
143130, 142sylibr 224 . . . . . . . . 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))})
144 oveq2 6618 . . . . . . . . . . . 12 (𝑑 = ((coe1𝑓)‘𝑋) → (𝑐(.r𝑅)𝑑) = (𝑐(.r𝑅)((coe1𝑓)‘𝑋)))
145144oveq1d 6625 . . . . . . . . . . 11 (𝑑 = ((coe1𝑓)‘𝑋) → ((𝑐(.r𝑅)𝑑)(+g𝑅)𝑒) = ((𝑐(.r𝑅)((coe1𝑓)‘𝑋))(+g𝑅)𝑒))
146145eleq1d 2683 . . . . . . . . . 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))}))
147146ralbidv 2981 . . . . . . . . 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))}))
148143, 147syl5ibrcom 237 . . . . . . . 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))}))
149148expimpd 628 . . . . . . 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))}))
150149rexlimdva 3025 . . . . . 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))}))
151150alrimiv 1852 . . . . 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))}))
152 eqeq1 2625 . . . . . . . . 9 (𝑎 = 𝑑 → (𝑎 = ((coe1𝑏)‘𝑋) ↔ 𝑑 = ((coe1𝑏)‘𝑋)))
153152anbi2d 739 . . . . . . . 8 (𝑎 = 𝑑 → (((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))))
154153rexbidv 3046 . . . . . . 7 (𝑎 = 𝑑 → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋))))
155 fveq2 6153 . . . . . . . . . 10 (𝑏 = 𝑓 → (( deg1𝑅)‘𝑏) = (( deg1𝑅)‘𝑓))
156155breq1d 4628 . . . . . . . . 9 (𝑏 = 𝑓 → ((( deg1𝑅)‘𝑏) ≤ 𝑋 ↔ (( deg1𝑅)‘𝑓) ≤ 𝑋))
157 fveq2 6153 . . . . . . . . . . 11 (𝑏 = 𝑓 → (coe1𝑏) = (coe1𝑓))
158157fveq1d 6155 . . . . . . . . . 10 (𝑏 = 𝑓 → ((coe1𝑏)‘𝑋) = ((coe1𝑓)‘𝑋))
159158eqeq2d 2631 . . . . . . . . 9 (𝑏 = 𝑓 → (𝑑 = ((coe1𝑏)‘𝑋) ↔ 𝑑 = ((coe1𝑓)‘𝑋)))
160156, 159anbi12d 746 . . . . . . . 8 (𝑏 = 𝑓 → (((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋)) ↔ ((( deg1𝑅)‘𝑓) ≤ 𝑋𝑑 = ((coe1𝑓)‘𝑋))))
161160cbvrexv 3163 . . . . . . 7 (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑑 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑓𝐼 ((( deg1𝑅)‘𝑓) ≤ 𝑋𝑑 = ((coe1𝑓)‘𝑋)))
162154, 161syl6bb 276 . . . . . 6 (𝑎 = 𝑑 → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) ↔ ∃𝑓𝐼 ((( deg1𝑅)‘𝑓) ≤ 𝑋𝑑 = ((coe1𝑓)‘𝑋))))
163162ralab 3353 . . . . 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))}))
164151, 163sylibr 224 . . . 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))})
165164ralrimiva 2961 . . 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))})
166 hbtlem2.t . . . 4 𝑇 = (LIdeal‘𝑅)
167166, 11, 98, 101islidl 19143 . . 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))}))
16820, 59, 165, 167syl3anbrc 1244 . 2 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ∈ 𝑇)
1695, 168eqeltrd 2698 1 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) ∈ 𝑇)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036  ∀wal 1478   = wceq 1480   ∈ wcel 1987  {cab 2607   ≠ wne 2790  ∀wral 2907  ∃wrex 2908   ⊆ wss 3559  ∅c0 3896  {csn 4153   class class class wbr 4618   × cxp 5077  ⟶wf 5848  ‘cfv 5852  (class class class)co 6610  ℝcr 9887  -∞cmnf 10024  ℝ*cxr 10025   ≤ cle 10027  ℕ0cn0 11244  Basecbs 15792  +gcplusg 15873  .rcmulr 15874  0gc0g 16032  Ringcrg 18479  LIdealclidl 19102  algSccascl 19243  Poly1cpl1 19479  coe1cco1 19480   deg1 cdg1 23735  ldgIdlSeqcldgis 37207 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966  ax-addf 9967  ax-mulf 9968 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-ofr 6858  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-ixp 7861  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-fsupp 8228  df-sup 8300  df-oi 8367  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-z 11330  df-dec 11446  df-uz 11640  df-fz 12277  df-fzo 12415  df-seq 12750  df-hash 13066  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-mulr 15887  df-starv 15888  df-sca 15889  df-vsca 15890  df-ip 15891  df-tset 15892  df-ple 15893  df-ds 15896  df-unif 15897  df-0g 16034  df-gsum 16035  df-mre 16178  df-mrc 16179  df-acs 16181  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-mhm 17267  df-submnd 17268  df-grp 17357  df-minusg 17358  df-sbg 17359  df-mulg 17473  df-subg 17523  df-ghm 17590  df-cntz 17682  df-cmn 18127  df-abl 18128  df-mgp 18422  df-ur 18434  df-ring 18481  df-cring 18482  df-subrg 18710  df-lmod 18797  df-lss 18865  df-sra 19104  df-rgmod 19105  df-lidl 19106  df-ascl 19246  df-psr 19288  df-mvr 19289  df-mpl 19290  df-opsr 19292  df-psr1 19482  df-vr1 19483  df-ply1 19484  df-coe1 19485  df-cnfld 19679  df-mdeg 23736  df-deg1 23737  df-ldgis 37208 This theorem is referenced by:  hbtlem7  37211  hbtlem6  37215
 Copyright terms: Public domain W3C validator