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Theorem hbtlem1 41865
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 3493 . . . . . . 7 (𝑅 ∈ 𝑉 β†’ 𝑅 ∈ V)
3 fveq2 6892 . . . . . . . . . . . 12 (π‘Ÿ = 𝑅 β†’ (Poly1β€˜π‘Ÿ) = (Poly1β€˜π‘…))
4 hbtlem.p . . . . . . . . . . . 12 𝑃 = (Poly1β€˜π‘…)
53, 4eqtr4di 2791 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (Poly1β€˜π‘Ÿ) = 𝑃)
65fveq2d 6896 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜(Poly1β€˜π‘Ÿ)) = (LIdealβ€˜π‘ƒ))
7 hbtlem.u . . . . . . . . . 10 π‘ˆ = (LIdealβ€˜π‘ƒ)
86, 7eqtr4di 2791 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜(Poly1β€˜π‘Ÿ)) = π‘ˆ)
9 fveq2 6892 . . . . . . . . . . . . . . . 16 (π‘Ÿ = 𝑅 β†’ ( deg1 β€˜π‘Ÿ) = ( deg1 β€˜π‘…))
10 hbtlem.d . . . . . . . . . . . . . . . 16 𝐷 = ( deg1 β€˜π‘…)
119, 10eqtr4di 2791 . . . . . . . . . . . . . . 15 (π‘Ÿ = 𝑅 β†’ ( deg1 β€˜π‘Ÿ) = 𝐷)
1211fveq1d 6894 . . . . . . . . . . . . . 14 (π‘Ÿ = 𝑅 β†’ (( deg1 β€˜π‘Ÿ)β€˜π‘˜) = (π·β€˜π‘˜))
1312breq1d 5159 . . . . . . . . . . . . 13 (π‘Ÿ = 𝑅 β†’ ((( deg1 β€˜π‘Ÿ)β€˜π‘˜) ≀ π‘₯ ↔ (π·β€˜π‘˜) ≀ π‘₯))
1413anbi1d 631 . . . . . . . . . . . 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 3179 . . . . . . . . . . 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 2802 . . . . . . . . . 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 5251 . . . . . . . . 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 5240 . . . . . . . 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 41864 . . . . . . . 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 7230 . . . . . . 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 2785 . . . . 5 (𝑅 ∈ 𝑉 β†’ 𝑆 = (𝑖 ∈ π‘ˆ ↦ (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))})))
2322fveq1d 6894 . . . 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 6894 . . 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 1134 . 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 3322 . . . . . . 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 2802 . . . . . 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 5251 . . . . 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 2733 . . . . 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 12478 . . . . . 6 β„•0 ∈ V
3130mptex 7225 . . . . 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 6894 . . 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 1135 . 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 2733 . . 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 5153 . . . . . 6 (π‘₯ = 𝑋 β†’ ((π·β€˜π‘˜) ≀ π‘₯ ↔ (π·β€˜π‘˜) ≀ 𝑋))
37 fveq2 6892 . . . . . . 7 (π‘₯ = 𝑋 β†’ ((coe1β€˜π‘˜)β€˜π‘₯) = ((coe1β€˜π‘˜)β€˜π‘‹))
3837eqeq2d 2744 . . . . . 6 (π‘₯ = 𝑋 β†’ (𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯) ↔ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)))
3936, 38anbi12d 632 . . . . 5 (π‘₯ = 𝑋 β†’ (((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯)) ↔ ((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))))
4039rexbidv 3179 . . . 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 2802 . . 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 1139 . . 3 ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ π‘ˆ ∧ 𝑋 ∈ β„•0) β†’ 𝑋 ∈ β„•0)
43 simpr 486 . . . . . 6 (((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)) β†’ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))
4443reximi 3085 . . . . 5 (βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)) β†’ βˆƒπ‘˜ ∈ 𝐼 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))
4544ss2abi 4064 . . . 4 {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))} βŠ† {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)}
46 abrexexg 7947 . . . . 5 (𝐼 ∈ π‘ˆ β†’ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)} ∈ V)
47463ad2ant2 1135 . . . 4 ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ π‘ˆ ∧ 𝑋 ∈ β„•0) β†’ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)} ∈ V)
48 ssexg 5324 . . . 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 588 . . 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 2777 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆƒwrex 3071  Vcvv 3475   βŠ† wss 3949   class class class wbr 5149   ↦ cmpt 5232  β€˜cfv 6544   ≀ cle 11249  β„•0cn0 12472  LIdealclidl 20783  Poly1cpl1 21701  coe1cco1 21702   deg1 cdg1 25569  ldgIdlSeqcldgis 41863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-1cn 11168  ax-addcl 11170
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-nn 12213  df-n0 12473  df-ldgis 41864
This theorem is referenced by:  hbtlem2  41866  hbtlem4  41868  hbtlem3  41869  hbtlem5  41870  hbtlem6  41871
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