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Theorem hbtlem1 42167
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 3491 . . . . . . 7 (𝑅 ∈ 𝑉 β†’ 𝑅 ∈ V)
3 fveq2 6890 . . . . . . . . . . . 12 (π‘Ÿ = 𝑅 β†’ (Poly1β€˜π‘Ÿ) = (Poly1β€˜π‘…))
4 hbtlem.p . . . . . . . . . . . 12 𝑃 = (Poly1β€˜π‘…)
53, 4eqtr4di 2788 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (Poly1β€˜π‘Ÿ) = 𝑃)
65fveq2d 6894 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜(Poly1β€˜π‘Ÿ)) = (LIdealβ€˜π‘ƒ))
7 hbtlem.u . . . . . . . . . 10 π‘ˆ = (LIdealβ€˜π‘ƒ)
86, 7eqtr4di 2788 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜(Poly1β€˜π‘Ÿ)) = π‘ˆ)
9 fveq2 6890 . . . . . . . . . . . . . . . 16 (π‘Ÿ = 𝑅 β†’ ( deg1 β€˜π‘Ÿ) = ( deg1 β€˜π‘…))
10 hbtlem.d . . . . . . . . . . . . . . . 16 𝐷 = ( deg1 β€˜π‘…)
119, 10eqtr4di 2788 . . . . . . . . . . . . . . 15 (π‘Ÿ = 𝑅 β†’ ( deg1 β€˜π‘Ÿ) = 𝐷)
1211fveq1d 6892 . . . . . . . . . . . . . 14 (π‘Ÿ = 𝑅 β†’ (( deg1 β€˜π‘Ÿ)β€˜π‘˜) = (π·β€˜π‘˜))
1312breq1d 5157 . . . . . . . . . . . . 13 (π‘Ÿ = 𝑅 β†’ ((( deg1 β€˜π‘Ÿ)β€˜π‘˜) ≀ π‘₯ ↔ (π·β€˜π‘˜) ≀ π‘₯))
1413anbi1d 628 . . . . . . . . . . . 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 3176 . . . . . . . . . . 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 2799 . . . . . . . . . 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 5249 . . . . . . . . 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 5238 . . . . . . . 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 42166 . . . . . . . 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 7231 . . . . . . 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 2782 . . . . 5 (𝑅 ∈ 𝑉 β†’ 𝑆 = (𝑖 ∈ π‘ˆ ↦ (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))})))
2322fveq1d 6892 . . . 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 6892 . . 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 3319 . . . . . . 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 2799 . . . . . 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 5249 . . . . 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 2730 . . . . 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 12482 . . . . . 6 β„•0 ∈ V
3130mptex 7226 . . . . 5 (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))}) ∈ V
3228, 29, 31fvmpt 6997 . . . 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 6892 . . 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 2730 . . 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 5151 . . . . . 6 (π‘₯ = 𝑋 β†’ ((π·β€˜π‘˜) ≀ π‘₯ ↔ (π·β€˜π‘˜) ≀ 𝑋))
37 fveq2 6890 . . . . . . 7 (π‘₯ = 𝑋 β†’ ((coe1β€˜π‘˜)β€˜π‘₯) = ((coe1β€˜π‘˜)β€˜π‘‹))
3837eqeq2d 2741 . . . . . 6 (π‘₯ = 𝑋 β†’ (𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯) ↔ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)))
3936, 38anbi12d 629 . . . . 5 (π‘₯ = 𝑋 β†’ (((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯)) ↔ ((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))))
4039rexbidv 3176 . . . 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 2799 . . 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 483 . . . . . 6 (((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)) β†’ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))
4443reximi 3082 . . . . 5 (βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)) β†’ βˆƒπ‘˜ ∈ 𝐼 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))
4544ss2abi 4062 . . . 4 {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))} βŠ† {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)}
46 abrexexg 7949 . . . . 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 5322 . . . 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 585 . . 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 7020 . 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 2774 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  {cab 2707  βˆƒwrex 3068  Vcvv 3472   βŠ† wss 3947   class class class wbr 5147   ↦ cmpt 5230  β€˜cfv 6542   ≀ cle 11253  β„•0cn0 12476  LIdealclidl 20928  Poly1cpl1 21920  coe1cco1 21921   deg1 cdg1 25804  ldgIdlSeqcldgis 42165
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-1cn 11170  ax-addcl 11172
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  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 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-nn 12217  df-n0 12477  df-ldgis 42166
This theorem is referenced by:  hbtlem2  42168  hbtlem4  42170  hbtlem3  42171  hbtlem5  42172  hbtlem6  42173
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