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Theorem hbtlem1 42469
Description: Value of the leading coefficient sequence function. (Contributed by Stefan O'Rear, 31-Mar-2015.)
Hypotheses
Ref Expression
hbtlem.p 𝑃 = (Poly1β€˜π‘…)
hbtlem.u π‘ˆ = (LIdealβ€˜π‘ƒ)
hbtlem.s 𝑆 = (ldgIdlSeqβ€˜π‘…)
hbtlem.d 𝐷 = ( deg1 β€˜π‘…)
Assertion
Ref Expression
hbtlem1 ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ π‘ˆ ∧ 𝑋 ∈ β„•0) β†’ ((π‘†β€˜πΌ)β€˜π‘‹) = {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))})
Distinct variable groups:   𝑗,𝐼,π‘˜   𝑅,𝑗,π‘˜   𝑗,𝑋,π‘˜
Allowed substitution hints:   𝐷(𝑗,π‘˜)   𝑃(𝑗,π‘˜)   𝑆(𝑗,π‘˜)   π‘ˆ(𝑗,π‘˜)   𝑉(𝑗,π‘˜)

Proof of Theorem hbtlem1
Dummy variables 𝑖 π‘Ÿ π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hbtlem.s . . . . . 6 𝑆 = (ldgIdlSeqβ€˜π‘…)
2 elex 3488 . . . . . . 7 (𝑅 ∈ 𝑉 β†’ 𝑅 ∈ V)
3 fveq2 6891 . . . . . . . . . . . 12 (π‘Ÿ = 𝑅 β†’ (Poly1β€˜π‘Ÿ) = (Poly1β€˜π‘…))
4 hbtlem.p . . . . . . . . . . . 12 𝑃 = (Poly1β€˜π‘…)
53, 4eqtr4di 2785 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (Poly1β€˜π‘Ÿ) = 𝑃)
65fveq2d 6895 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜(Poly1β€˜π‘Ÿ)) = (LIdealβ€˜π‘ƒ))
7 hbtlem.u . . . . . . . . . 10 π‘ˆ = (LIdealβ€˜π‘ƒ)
86, 7eqtr4di 2785 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜(Poly1β€˜π‘Ÿ)) = π‘ˆ)
9 fveq2 6891 . . . . . . . . . . . . . . . 16 (π‘Ÿ = 𝑅 β†’ ( deg1 β€˜π‘Ÿ) = ( deg1 β€˜π‘…))
10 hbtlem.d . . . . . . . . . . . . . . . 16 𝐷 = ( deg1 β€˜π‘…)
119, 10eqtr4di 2785 . . . . . . . . . . . . . . 15 (π‘Ÿ = 𝑅 β†’ ( deg1 β€˜π‘Ÿ) = 𝐷)
1211fveq1d 6893 . . . . . . . . . . . . . 14 (π‘Ÿ = 𝑅 β†’ (( deg1 β€˜π‘Ÿ)β€˜π‘˜) = (π·β€˜π‘˜))
1312breq1d 5152 . . . . . . . . . . . . 13 (π‘Ÿ = 𝑅 β†’ ((( deg1 β€˜π‘Ÿ)β€˜π‘˜) ≀ π‘₯ ↔ (π·β€˜π‘˜) ≀ π‘₯))
1413anbi1d 629 . . . . . . . . . . . 12 (π‘Ÿ = 𝑅 β†’ (((( deg1 β€˜π‘Ÿ)β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯)) ↔ ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))))
1514rexbidv 3173 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (βˆƒπ‘˜ ∈ 𝑖 ((( deg1 β€˜π‘Ÿ)β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯)) ↔ βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))))
1615abbidv 2796 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((( deg1 β€˜π‘Ÿ)β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))} = {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))})
1716mpteq2dv 5244 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((( deg1 β€˜π‘Ÿ)β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))}) = (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))}))
188, 17mpteq12dv 5233 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (𝑖 ∈ (LIdealβ€˜(Poly1β€˜π‘Ÿ)) ↦ (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((( deg1 β€˜π‘Ÿ)β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))})) = (𝑖 ∈ π‘ˆ ↦ (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))})))
19 df-ldgis 42468 . . . . . . . 8 ldgIdlSeq = (π‘Ÿ ∈ V ↦ (𝑖 ∈ (LIdealβ€˜(Poly1β€˜π‘Ÿ)) ↦ (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((( deg1 β€˜π‘Ÿ)β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))})))
2018, 19, 7mptfvmpt 7234 . . . . . . 7 (𝑅 ∈ V β†’ (ldgIdlSeqβ€˜π‘…) = (𝑖 ∈ π‘ˆ ↦ (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))})))
212, 20syl 17 . . . . . 6 (𝑅 ∈ 𝑉 β†’ (ldgIdlSeqβ€˜π‘…) = (𝑖 ∈ π‘ˆ ↦ (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))})))
221, 21eqtrid 2779 . . . . 5 (𝑅 ∈ 𝑉 β†’ 𝑆 = (𝑖 ∈ π‘ˆ ↦ (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))})))
2322fveq1d 6893 . . . 4 (𝑅 ∈ 𝑉 β†’ (π‘†β€˜πΌ) = ((𝑖 ∈ π‘ˆ ↦ (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))}))β€˜πΌ))
2423fveq1d 6893 . . 3 (𝑅 ∈ 𝑉 β†’ ((π‘†β€˜πΌ)β€˜π‘‹) = (((𝑖 ∈ π‘ˆ ↦ (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))}))β€˜πΌ)β€˜π‘‹))
25243ad2ant1 1131 . 2 ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ π‘ˆ ∧ 𝑋 ∈ β„•0) β†’ ((π‘†β€˜πΌ)β€˜π‘‹) = (((𝑖 ∈ π‘ˆ ↦ (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))}))β€˜πΌ)β€˜π‘‹))
26 rexeq 3316 . . . . . . 7 (𝑖 = 𝐼 β†’ (βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯)) ↔ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))))
2726abbidv 2796 . . . . . 6 (𝑖 = 𝐼 β†’ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))} = {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))})
2827mpteq2dv 5244 . . . . 5 (𝑖 = 𝐼 β†’ (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))}) = (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))}))
29 eqid 2727 . . . . 5 (𝑖 ∈ π‘ˆ ↦ (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))})) = (𝑖 ∈ π‘ˆ ↦ (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))}))
30 nn0ex 12500 . . . . . 6 β„•0 ∈ V
3130mptex 7229 . . . . 5 (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))}) ∈ V
3228, 29, 31fvmpt 6999 . . . 4 (𝐼 ∈ π‘ˆ β†’ ((𝑖 ∈ π‘ˆ ↦ (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))}))β€˜πΌ) = (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))}))
3332fveq1d 6893 . . 3 (𝐼 ∈ π‘ˆ β†’ (((𝑖 ∈ π‘ˆ ↦ (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))}))β€˜πΌ)β€˜π‘‹) = ((π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))})β€˜π‘‹))
34333ad2ant2 1132 . 2 ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ π‘ˆ ∧ 𝑋 ∈ β„•0) β†’ (((𝑖 ∈ π‘ˆ ↦ (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))}))β€˜πΌ)β€˜π‘‹) = ((π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))})β€˜π‘‹))
35 eqid 2727 . . 3 (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))}) = (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))})
36 breq2 5146 . . . . . 6 (π‘₯ = 𝑋 β†’ ((π·β€˜π‘˜) ≀ π‘₯ ↔ (π·β€˜π‘˜) ≀ 𝑋))
37 fveq2 6891 . . . . . . 7 (π‘₯ = 𝑋 β†’ ((coe1β€˜π‘˜)β€˜π‘₯) = ((coe1β€˜π‘˜)β€˜π‘‹))
3837eqeq2d 2738 . . . . . 6 (π‘₯ = 𝑋 β†’ (𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯) ↔ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)))
3936, 38anbi12d 630 . . . . 5 (π‘₯ = 𝑋 β†’ (((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯)) ↔ ((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))))
4039rexbidv 3173 . . . 4 (π‘₯ = 𝑋 β†’ (βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯)) ↔ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))))
4140abbidv 2796 . . 3 (π‘₯ = 𝑋 β†’ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))} = {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))})
42 simp3 1136 . . 3 ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ π‘ˆ ∧ 𝑋 ∈ β„•0) β†’ 𝑋 ∈ β„•0)
43 simpr 484 . . . . . 6 (((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)) β†’ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))
4443reximi 3079 . . . . 5 (βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)) β†’ βˆƒπ‘˜ ∈ 𝐼 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))
4544ss2abi 4059 . . . 4 {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))} βŠ† {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)}
46 abrexexg 7958 . . . . 5 (𝐼 ∈ π‘ˆ β†’ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)} ∈ V)
47463ad2ant2 1132 . . . 4 ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ π‘ˆ ∧ 𝑋 ∈ β„•0) β†’ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)} ∈ V)
48 ssexg 5317 . . . 4 (({𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))} βŠ† {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)} ∧ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)} ∈ V) β†’ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))} ∈ V)
4945, 47, 48sylancr 586 . . 3 ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ π‘ˆ ∧ 𝑋 ∈ β„•0) β†’ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))} ∈ V)
5035, 41, 42, 49fvmptd3 7022 . 2 ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ π‘ˆ ∧ 𝑋 ∈ β„•0) β†’ ((π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))})β€˜π‘‹) = {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))})
5125, 34, 503eqtrd 2771 1 ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ π‘ˆ ∧ 𝑋 ∈ β„•0) β†’ ((π‘†β€˜πΌ)β€˜π‘‹) = {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  {cab 2704  βˆƒwrex 3065  Vcvv 3469   βŠ† wss 3944   class class class wbr 5142   ↦ cmpt 5225  β€˜cfv 6542   ≀ cle 11271  β„•0cn0 12494  LIdealclidl 21091  Poly1cpl1 22083  coe1cco1 22084   deg1 cdg1 25974  ldgIdlSeqcldgis 42467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-1cn 11188  ax-addcl 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-nn 12235  df-n0 12495  df-ldgis 42468
This theorem is referenced by:  hbtlem2  42470  hbtlem4  42472  hbtlem3  42473  hbtlem5  42474  hbtlem6  42475
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