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Theorem hbtlem1 42612
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 3482 . . . . . . 7 (𝑅 ∈ 𝑉 β†’ 𝑅 ∈ V)
3 fveq2 6894 . . . . . . . . . . . 12 (π‘Ÿ = 𝑅 β†’ (Poly1β€˜π‘Ÿ) = (Poly1β€˜π‘…))
4 hbtlem.p . . . . . . . . . . . 12 𝑃 = (Poly1β€˜π‘…)
53, 4eqtr4di 2783 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (Poly1β€˜π‘Ÿ) = 𝑃)
65fveq2d 6898 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜(Poly1β€˜π‘Ÿ)) = (LIdealβ€˜π‘ƒ))
7 hbtlem.u . . . . . . . . . 10 π‘ˆ = (LIdealβ€˜π‘ƒ)
86, 7eqtr4di 2783 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜(Poly1β€˜π‘Ÿ)) = π‘ˆ)
9 fveq2 6894 . . . . . . . . . . . . . . . 16 (π‘Ÿ = 𝑅 β†’ ( deg1 β€˜π‘Ÿ) = ( deg1 β€˜π‘…))
10 hbtlem.d . . . . . . . . . . . . . . . 16 𝐷 = ( deg1 β€˜π‘…)
119, 10eqtr4di 2783 . . . . . . . . . . . . . . 15 (π‘Ÿ = 𝑅 β†’ ( deg1 β€˜π‘Ÿ) = 𝐷)
1211fveq1d 6896 . . . . . . . . . . . . . 14 (π‘Ÿ = 𝑅 β†’ (( deg1 β€˜π‘Ÿ)β€˜π‘˜) = (π·β€˜π‘˜))
1312breq1d 5158 . . . . . . . . . . . . 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 3169 . . . . . . . . . . 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 2794 . . . . . . . . . 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 5250 . . . . . . . . 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 5239 . . . . . . . 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 42611 . . . . . . . 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 7238 . . . . . . 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 2777 . . . . 5 (𝑅 ∈ 𝑉 β†’ 𝑆 = (𝑖 ∈ π‘ˆ ↦ (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))})))
2322fveq1d 6896 . . . 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 6896 . . 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 1130 . 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 3311 . . . . . . 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 2794 . . . . . 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 5250 . . . . 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 2725 . . . . 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 12508 . . . . . 6 β„•0 ∈ V
3130mptex 7233 . . . . 5 (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))}) ∈ V
3228, 29, 31fvmpt 7002 . . . 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 6896 . . 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 1131 . 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 2725 . . 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 5152 . . . . . 6 (π‘₯ = 𝑋 β†’ ((π·β€˜π‘˜) ≀ π‘₯ ↔ (π·β€˜π‘˜) ≀ 𝑋))
37 fveq2 6894 . . . . . . 7 (π‘₯ = 𝑋 β†’ ((coe1β€˜π‘˜)β€˜π‘₯) = ((coe1β€˜π‘˜)β€˜π‘‹))
3837eqeq2d 2736 . . . . . 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 3169 . . . 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 2794 . . 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 1135 . . 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 3074 . . . . 5 (βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)) β†’ βˆƒπ‘˜ ∈ 𝐼 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))
4544ss2abi 4060 . . . 4 {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))} βŠ† {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)}
46 abrexexg 7963 . . . . 5 (𝐼 ∈ π‘ˆ β†’ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)} ∈ V)
47463ad2ant2 1131 . . . 4 ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ π‘ˆ ∧ 𝑋 ∈ β„•0) β†’ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹)} ∈ V)
48 ssexg 5323 . . . 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 7025 . 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 2769 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 1084   = wceq 1533   ∈ wcel 2098  {cab 2702  βˆƒwrex 3060  Vcvv 3463   βŠ† wss 3945   class class class wbr 5148   ↦ cmpt 5231  β€˜cfv 6547   ≀ cle 11279  β„•0cn0 12502  LIdealclidl 21106  Poly1cpl1 22104  coe1cco1 22105   deg1 cdg1 26017  ldgIdlSeqcldgis 42610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5428  ax-un 7739  ax-cnex 11194  ax-1cn 11196  ax-addcl 11198
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3965  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  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 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7420  df-om 7870  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-nn 12243  df-n0 12503  df-ldgis 42611
This theorem is referenced by:  hbtlem2  42613  hbtlem4  42615  hbtlem3  42616  hbtlem5  42617  hbtlem6  42618
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