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Theorem iblcnlem 25305
Description: Expand out the universal quantifier in isibl2 25283. (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 25284 . . 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 1136 . . 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 2732 . . . . . 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 2732 . . . . . 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 2732 . . . . . 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 2732 . . . . . 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 11205 . . . . . . . 8 0 ∈ β„‚
109elimel 4597 . . . . . . 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 25304 . . . . 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 481 . . . 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 2732 . . . . . 6 𝐴 = 𝐴
15 mbff 25141 . . . . . . . . 9 ((π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn β†’ (π‘₯ ∈ 𝐴 ↦ 𝐡):dom (π‘₯ ∈ 𝐴 ↦ 𝐡)βŸΆβ„‚)
16 eqid 2732 . . . . . . . . . . . 12 (π‘₯ ∈ 𝐴 ↦ 𝐡) = (π‘₯ ∈ 𝐴 ↦ 𝐡)
17 itgcnlem.v . . . . . . . . . . . 12 ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ 𝑉)
1816, 17dmmptd 6695 . . . . . . . . . . 11 (πœ‘ β†’ dom (π‘₯ ∈ 𝐴 ↦ 𝐡) = 𝐴)
1918feq2d 6703 . . . . . . . . . 10 (πœ‘ β†’ ((π‘₯ ∈ 𝐴 ↦ 𝐡):dom (π‘₯ ∈ 𝐴 ↦ 𝐡)βŸΆβ„‚ ↔ (π‘₯ ∈ 𝐴 ↦ 𝐡):π΄βŸΆβ„‚))
2019biimpa 477 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡):dom (π‘₯ ∈ 𝐴 ↦ 𝐡)βŸΆβ„‚) β†’ (π‘₯ ∈ 𝐴 ↦ 𝐡):π΄βŸΆβ„‚)
2115, 20sylan2 593 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ (π‘₯ ∈ 𝐴 ↦ 𝐡):π΄βŸΆβ„‚)
2216fmpt 7109 . . . . . . . 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 4534 . . . . . . . 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 5240 . . . . . 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 587 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ (π‘₯ ∈ 𝐴 ↦ if(𝐡 ∈ β„‚, 𝐡, 0)) = (π‘₯ ∈ 𝐴 ↦ 𝐡))
2928eleq1d 2818 . . . 4 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ ((π‘₯ ∈ 𝐴 ↦ if(𝐡 ∈ β„‚, 𝐡, 0)) ∈ 𝐿1 ↔ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ 𝐿1))
3028eleq1d 2818 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn) β†’ ((π‘₯ ∈ 𝐴 ↦ if(𝐡 ∈ β„‚, 𝐡, 0)) ∈ MblFn ↔ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn))
31 eqid 2732 . . . . . . . . . 10 ℝ = ℝ
3224imim2i 16 . . . . . . . . . . . . . . . 16 ((π‘₯ ∈ 𝐴 β†’ 𝐡 ∈ β„‚) β†’ (π‘₯ ∈ 𝐴 β†’ if(𝐡 ∈ β„‚, 𝐡, 0) = 𝐡))
3332imp 407 . . . . . . . . . . . . . . 15 (((π‘₯ ∈ 𝐴 β†’ 𝐡 ∈ β„‚) ∧ π‘₯ ∈ 𝐴) β†’ if(𝐡 ∈ β„‚, 𝐡, 0) = 𝐡)
3433fveq2d 6895 . . . . . . . . . . . . . 14 (((π‘₯ ∈ 𝐴 β†’ 𝐡 ∈ β„‚) ∧ π‘₯ ∈ 𝐴) β†’ (β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)) = (β„œβ€˜π΅))
3534ibllem 25281 . . . . . . . . . . . . 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 5240 . . . . . . . . . 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 587 . . . . . . . . 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 6895 . . . . . . . 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 2790 . . . . . . 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 2818 . . . . . 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 11453 . . . . . . . . . . . . . 14 (((π‘₯ ∈ 𝐴 β†’ 𝐡 ∈ β„‚) ∧ π‘₯ ∈ 𝐴) β†’ -(β„œβ€˜if(𝐡 ∈ β„‚, 𝐡, 0)) = -(β„œβ€˜π΅))
4645ibllem 25281 . . . . . . . . . . . . 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 5240 . . . . . . . . . 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 587 . . . . . . . . 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 6895 . . . . . . . 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 2790 . . . . . . 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 2818 . . . . . 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 631 . . . . 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 6895 . . . . . . . . . . . . . 14 (((π‘₯ ∈ 𝐴 β†’ 𝐡 ∈ β„‚) ∧ π‘₯ ∈ 𝐴) β†’ (β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)) = (β„‘β€˜π΅))
5857ibllem 25281 . . . . . . . . . . . . 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 5240 . . . . . . . . . 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 587 . . . . . . . . 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 6895 . . . . . . . 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 2790 . . . . . . 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 2818 . . . . . 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 11453 . . . . . . . . . . . . . 14 (((π‘₯ ∈ 𝐴 β†’ 𝐡 ∈ β„‚) ∧ π‘₯ ∈ 𝐴) β†’ -(β„‘β€˜if(𝐡 ∈ β„‚, 𝐡, 0)) = -(β„‘β€˜π΅))
6968ibllem 25281 . . . . . . . . . . . . 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 5240 . . . . . . . . . 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 587 . . . . . . . . 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 6895 . . . . . . . 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 2790 . . . . . . 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 2818 . . . . . 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 631 . . . . 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 1436 . . . 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 308 . . 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 413 . 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 380 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  ifcif 4528   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5676  βŸΆwf 6539  β€˜cfv 6543  β„‚cc 11107  β„cr 11108  0cc0 11109   ≀ cle 11248  -cneg 11444  β„œcre 15043  β„‘cim 15044  MblFncmbf 25130  βˆ«2citg2 25132  πΏ1cibl 25133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-pm 8822  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-2 12274  df-3 12275  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13484  df-seq 13966  df-exp 14027  df-cj 15045  df-re 15046  df-im 15047  df-mbf 25135  df-ibl 25138
This theorem is referenced by:  itgcnlem  25306  iblrelem  25307  ibladd  25337  ibladdnc  36540
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