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Theorem iblcnlem 25176
Description: Expand out the universal quantifier in isibl2 25154. (Contributed by Mario Carneiro, 6-Aug-2014.)
Hypotheses
Ref Expression
itgcnlem.r 𝑅 = (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜π΅)), (β„œβ€˜π΅), 0)))
itgcnlem.s 𝑆 = (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜π΅)), -(β„œβ€˜π΅), 0)))
itgcnlem.t 𝑇 = (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜π΅)), (β„‘β€˜π΅), 0)))
itgcnlem.u π‘ˆ = (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜π΅)), -(β„‘β€˜π΅), 0)))
itgcnlem.v ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ 𝑉)
Assertion
Ref Expression
iblcnlem (πœ‘ β†’ ((π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ 𝐿1 ↔ ((π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ π‘ˆ ∈ ℝ))))
Distinct variable groups:   π‘₯,𝐴   πœ‘,π‘₯   π‘₯,𝑉
Allowed substitution hints:   𝐡(π‘₯)   𝑅(π‘₯)   𝑆(π‘₯)   𝑇(π‘₯)   π‘ˆ(π‘₯)

Proof of Theorem iblcnlem
StepHypRef Expression
1 iblmbf 25155 . . 3 ((π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ 𝐿1 β†’ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn)
21a1i 11 . 2 (πœ‘ β†’ ((π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ 𝐿1 β†’ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn))
3 simp1 1137 . . 3 (((π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ π‘ˆ ∈ ℝ)) β†’ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn)
43a1i 11 . 2 (πœ‘ β†’ (((π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ π‘ˆ ∈ ℝ)) β†’ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn))
5 eqid 2733 . . . . . 6 (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) = (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0)))
6 eqid 2733 . . . . . 6 (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) = (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0)))
7 eqid 2733 . . . . . 6 (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) = (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0)))
8 eqid 2733 . . . . . 6 (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) = (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0)))
9 0cn 11155 . . . . . . . 8 0 ∈ β„‚
109elimel 4559 . . . . . . 7 if(𝐡 ∈ β„‚, 𝐡, 0) ∈ β„‚
1110a1i 11 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ if(𝐡 ∈ β„‚, 𝐡, 0) ∈ β„‚)
125, 6, 7, 8, 11iblcnlem1 25175 . . . . 5 (πœ‘ β†’ ((π‘₯ ∈ 𝐴 ↦ if(𝐡 ∈ β„‚, 𝐡, 0)) ∈ 𝐿1 ↔ ((π‘₯ ∈ 𝐴 ↦ if(𝐡 ∈ β„‚, 𝐡, 0)) ∈ MblFn ∧ ((∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) ∈ ℝ ∧ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) ∈ ℝ) ∧ ((∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) ∈ ℝ ∧ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) ∈ ℝ))))
1312adantr 482 . . . 4 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ ((π‘₯ ∈ 𝐴 ↦ if(𝐡 ∈ β„‚, 𝐡, 0)) ∈ 𝐿1 ↔ ((π‘₯ ∈ 𝐴 ↦ if(𝐡 ∈ β„‚, 𝐡, 0)) ∈ MblFn ∧ ((∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) ∈ ℝ ∧ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) ∈ ℝ) ∧ ((∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) ∈ ℝ ∧ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) ∈ ℝ))))
14 eqid 2733 . . . . . 6 𝐴 = 𝐴
15 mbff 25012 . . . . . . . . 9 ((π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn β†’ (π‘₯ ∈ 𝐴 ↦ 𝐡):dom (π‘₯ ∈ 𝐴 ↦ 𝐡)βŸΆβ„‚)
16 eqid 2733 . . . . . . . . . . . 12 (π‘₯ ∈ 𝐴 ↦ 𝐡) = (π‘₯ ∈ 𝐴 ↦ 𝐡)
17 itgcnlem.v . . . . . . . . . . . 12 ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ 𝑉)
1816, 17dmmptd 6650 . . . . . . . . . . 11 (πœ‘ β†’ dom (π‘₯ ∈ 𝐴 ↦ 𝐡) = 𝐴)
1918feq2d 6658 . . . . . . . . . 10 (πœ‘ β†’ ((π‘₯ ∈ 𝐴 ↦ 𝐡):dom (π‘₯ ∈ 𝐴 ↦ 𝐡)βŸΆβ„‚ ↔ (π‘₯ ∈ 𝐴 ↦ 𝐡):π΄βŸΆβ„‚))
2019biimpa 478 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡):dom (π‘₯ ∈ 𝐴 ↦ 𝐡)βŸΆβ„‚) β†’ (π‘₯ ∈ 𝐴 ↦ 𝐡):π΄βŸΆβ„‚)
2115, 20sylan2 594 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ (π‘₯ ∈ 𝐴 ↦ 𝐡):π΄βŸΆβ„‚)
2216fmpt 7062 . . . . . . . 8 (βˆ€π‘₯ ∈ 𝐴 𝐡 ∈ β„‚ ↔ (π‘₯ ∈ 𝐴 ↦ 𝐡):π΄βŸΆβ„‚)
2321, 22sylibr 233 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ βˆ€π‘₯ ∈ 𝐴 𝐡 ∈ β„‚)
24 iftrue 4496 . . . . . . . 8 (𝐡 ∈ β„‚ β†’ if(𝐡 ∈ β„‚, 𝐡, 0) = 𝐡)
2524ralimi 3083 . . . . . . 7 (βˆ€π‘₯ ∈ 𝐴 𝐡 ∈ β„‚ β†’ βˆ€π‘₯ ∈ 𝐴 if(𝐡 ∈ β„‚, 𝐡, 0) = 𝐡)
2623, 25syl 17 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ βˆ€π‘₯ ∈ 𝐴 if(𝐡 ∈ β„‚, 𝐡, 0) = 𝐡)
27 mpteq12 5201 . . . . . 6 ((𝐴 = 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 if(𝐡 ∈ β„‚, 𝐡, 0) = 𝐡) β†’ (π‘₯ ∈ 𝐴 ↦ if(𝐡 ∈ β„‚, 𝐡, 0)) = (π‘₯ ∈ 𝐴 ↦ 𝐡))
2814, 26, 27sylancr 588 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ (π‘₯ ∈ 𝐴 ↦ if(𝐡 ∈ β„‚, 𝐡, 0)) = (π‘₯ ∈ 𝐴 ↦ 𝐡))
2928eleq1d 2819 . . . 4 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ ((π‘₯ ∈ 𝐴 ↦ if(𝐡 ∈ β„‚, 𝐡, 0)) ∈ 𝐿1 ↔ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ 𝐿1))
3028eleq1d 2819 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ ((π‘₯ ∈ 𝐴 ↦ if(𝐡 ∈ β„‚, 𝐡, 0)) ∈ MblFn ↔ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn))
31 eqid 2733 . . . . . . . . . 10 ℝ = ℝ
3224imim2i 16 . . . . . . . . . . . . . . . 16 ((π‘₯ ∈ 𝐴 β†’ 𝐡 ∈ β„‚) β†’ (π‘₯ ∈ 𝐴 β†’ if(𝐡 ∈ β„‚, 𝐡, 0) = 𝐡))
3332imp 408 . . . . . . . . . . . . . . 15 (((π‘₯ ∈ 𝐴 β†’ 𝐡 ∈ β„‚) ∧ π‘₯ ∈ 𝐴) β†’ if(𝐡 ∈ β„‚, 𝐡, 0) = 𝐡)
3433fveq2d 6850 . . . . . . . . . . . . . 14 (((π‘₯ ∈ 𝐴 β†’ 𝐡 ∈ β„‚) ∧ π‘₯ ∈ 𝐴) β†’ (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)) = (β„œβ€˜π΅))
3534ibllem 25152 . . . . . . . . . . . . 13 ((π‘₯ ∈ 𝐴 β†’ 𝐡 ∈ β„‚) β†’ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜π΅)), (β„œβ€˜π΅), 0))
3635a1d 25 . . . . . . . . . . . 12 ((π‘₯ ∈ 𝐴 β†’ 𝐡 ∈ β„‚) β†’ (π‘₯ ∈ ℝ β†’ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜π΅)), (β„œβ€˜π΅), 0)))
3736ralimi2 3078 . . . . . . . . . . 11 (βˆ€π‘₯ ∈ 𝐴 𝐡 ∈ β„‚ β†’ βˆ€π‘₯ ∈ ℝ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜π΅)), (β„œβ€˜π΅), 0))
3823, 37syl 17 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ βˆ€π‘₯ ∈ ℝ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜π΅)), (β„œβ€˜π΅), 0))
39 mpteq12 5201 . . . . . . . . . 10 ((ℝ = ℝ ∧ βˆ€π‘₯ ∈ ℝ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜π΅)), (β„œβ€˜π΅), 0)) β†’ (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0)) = (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜π΅)), (β„œβ€˜π΅), 0)))
4031, 38, 39sylancr 588 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0)) = (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜π΅)), (β„œβ€˜π΅), 0)))
4140fveq2d 6850 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) = (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜π΅)), (β„œβ€˜π΅), 0))))
42 itgcnlem.r . . . . . . . 8 𝑅 = (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜π΅)), (β„œβ€˜π΅), 0)))
4341, 42eqtr4di 2791 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) = 𝑅)
4443eleq1d 2819 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ ((∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) ∈ ℝ ↔ 𝑅 ∈ ℝ))
4534negeqd 11403 . . . . . . . . . . . . . 14 (((π‘₯ ∈ 𝐴 β†’ 𝐡 ∈ β„‚) ∧ π‘₯ ∈ 𝐴) β†’ -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)) = -(β„œβ€˜π΅))
4645ibllem 25152 . . . . . . . . . . . . 13 ((π‘₯ ∈ 𝐴 β†’ 𝐡 ∈ β„‚) β†’ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜π΅)), -(β„œβ€˜π΅), 0))
4746a1d 25 . . . . . . . . . . . 12 ((π‘₯ ∈ 𝐴 β†’ 𝐡 ∈ β„‚) β†’ (π‘₯ ∈ ℝ β†’ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜π΅)), -(β„œβ€˜π΅), 0)))
4847ralimi2 3078 . . . . . . . . . . 11 (βˆ€π‘₯ ∈ 𝐴 𝐡 ∈ β„‚ β†’ βˆ€π‘₯ ∈ ℝ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜π΅)), -(β„œβ€˜π΅), 0))
4923, 48syl 17 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ βˆ€π‘₯ ∈ ℝ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜π΅)), -(β„œβ€˜π΅), 0))
50 mpteq12 5201 . . . . . . . . . 10 ((ℝ = ℝ ∧ βˆ€π‘₯ ∈ ℝ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜π΅)), -(β„œβ€˜π΅), 0)) β†’ (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0)) = (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜π΅)), -(β„œβ€˜π΅), 0)))
5131, 49, 50sylancr 588 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0)) = (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜π΅)), -(β„œβ€˜π΅), 0)))
5251fveq2d 6850 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) = (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜π΅)), -(β„œβ€˜π΅), 0))))
53 itgcnlem.s . . . . . . . 8 𝑆 = (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜π΅)), -(β„œβ€˜π΅), 0)))
5452, 53eqtr4di 2791 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) = 𝑆)
5554eleq1d 2819 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ ((∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) ∈ ℝ ↔ 𝑆 ∈ ℝ))
5644, 55anbi12d 632 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ (((∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) ∈ ℝ ∧ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) ∈ ℝ) ↔ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ)))
5733fveq2d 6850 . . . . . . . . . . . . . 14 (((π‘₯ ∈ 𝐴 β†’ 𝐡 ∈ β„‚) ∧ π‘₯ ∈ 𝐴) β†’ (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)) = (β„‘β€˜π΅))
5857ibllem 25152 . . . . . . . . . . . . 13 ((π‘₯ ∈ 𝐴 β†’ 𝐡 ∈ β„‚) β†’ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜π΅)), (β„‘β€˜π΅), 0))
5958a1d 25 . . . . . . . . . . . 12 ((π‘₯ ∈ 𝐴 β†’ 𝐡 ∈ β„‚) β†’ (π‘₯ ∈ ℝ β†’ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜π΅)), (β„‘β€˜π΅), 0)))
6059ralimi2 3078 . . . . . . . . . . 11 (βˆ€π‘₯ ∈ 𝐴 𝐡 ∈ β„‚ β†’ βˆ€π‘₯ ∈ ℝ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜π΅)), (β„‘β€˜π΅), 0))
6123, 60syl 17 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ βˆ€π‘₯ ∈ ℝ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜π΅)), (β„‘β€˜π΅), 0))
62 mpteq12 5201 . . . . . . . . . 10 ((ℝ = ℝ ∧ βˆ€π‘₯ ∈ ℝ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜π΅)), (β„‘β€˜π΅), 0)) β†’ (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0)) = (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜π΅)), (β„‘β€˜π΅), 0)))
6331, 61, 62sylancr 588 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0)) = (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜π΅)), (β„‘β€˜π΅), 0)))
6463fveq2d 6850 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) = (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜π΅)), (β„‘β€˜π΅), 0))))
65 itgcnlem.t . . . . . . . 8 𝑇 = (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜π΅)), (β„‘β€˜π΅), 0)))
6664, 65eqtr4di 2791 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) = 𝑇)
6766eleq1d 2819 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ ((∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) ∈ ℝ ↔ 𝑇 ∈ ℝ))
6857negeqd 11403 . . . . . . . . . . . . . 14 (((π‘₯ ∈ 𝐴 β†’ 𝐡 ∈ β„‚) ∧ π‘₯ ∈ 𝐴) β†’ -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)) = -(β„‘β€˜π΅))
6968ibllem 25152 . . . . . . . . . . . . 13 ((π‘₯ ∈ 𝐴 β†’ 𝐡 ∈ β„‚) β†’ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜π΅)), -(β„‘β€˜π΅), 0))
7069a1d 25 . . . . . . . . . . . 12 ((π‘₯ ∈ 𝐴 β†’ 𝐡 ∈ β„‚) β†’ (π‘₯ ∈ ℝ β†’ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜π΅)), -(β„‘β€˜π΅), 0)))
7170ralimi2 3078 . . . . . . . . . . 11 (βˆ€π‘₯ ∈ 𝐴 𝐡 ∈ β„‚ β†’ βˆ€π‘₯ ∈ ℝ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜π΅)), -(β„‘β€˜π΅), 0))
7223, 71syl 17 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ βˆ€π‘₯ ∈ ℝ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜π΅)), -(β„‘β€˜π΅), 0))
73 mpteq12 5201 . . . . . . . . . 10 ((ℝ = ℝ ∧ βˆ€π‘₯ ∈ ℝ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜π΅)), -(β„‘β€˜π΅), 0)) β†’ (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0)) = (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜π΅)), -(β„‘β€˜π΅), 0)))
7431, 72, 73sylancr 588 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0)) = (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜π΅)), -(β„‘β€˜π΅), 0)))
7574fveq2d 6850 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) = (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜π΅)), -(β„‘β€˜π΅), 0))))
76 itgcnlem.u . . . . . . . 8 π‘ˆ = (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜π΅)), -(β„‘β€˜π΅), 0)))
7775, 76eqtr4di 2791 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) = π‘ˆ)
7877eleq1d 2819 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ ((∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) ∈ ℝ ↔ π‘ˆ ∈ ℝ))
7967, 78anbi12d 632 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ (((∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) ∈ ℝ ∧ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) ∈ ℝ) ↔ (𝑇 ∈ ℝ ∧ π‘ˆ ∈ ℝ)))
8030, 56, 793anbi123d 1437 . . . 4 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ (((π‘₯ ∈ 𝐴 ↦ if(𝐡 ∈ β„‚, 𝐡, 0)) ∈ MblFn ∧ ((∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) ∈ ℝ ∧ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) ∈ ℝ) ∧ ((∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) ∈ ℝ ∧ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0))), -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)), 0))) ∈ ℝ)) ↔ ((π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ π‘ˆ ∈ ℝ))))
8113, 29, 803bitr3d 309 . . 3 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ ((π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ 𝐿1 ↔ ((π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ π‘ˆ ∈ ℝ))))
8281ex 414 . 2 (πœ‘ β†’ ((π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn β†’ ((π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ 𝐿1 ↔ ((π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ π‘ˆ ∈ ℝ)))))
832, 4, 82pm5.21ndd 381 1 (πœ‘ β†’ ((π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ 𝐿1 ↔ ((π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ π‘ˆ ∈ ℝ))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  ifcif 4490   class class class wbr 5109   ↦ cmpt 5192  dom cdm 5637  βŸΆwf 6496  β€˜cfv 6500  β„‚cc 11057  β„cr 11058  0cc0 11059   ≀ cle 11198  -cneg 11394  β„œcre 14991  β„‘cim 14992  MblFncmbf 25001  βˆ«2citg2 25003  πΏ1cibl 25004
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-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-pm 8774  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-n0 12422  df-z 12508  df-uz 12772  df-fz 13434  df-seq 13916  df-exp 13977  df-cj 14993  df-re 14994  df-im 14995  df-mbf 25006  df-ibl 25009
This theorem is referenced by:  itgcnlem  25177  iblrelem  25178  ibladd  25208  ibladdnc  36185
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