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Theorem ftc1anclem2 33139
 Description: Lemma for ftc1anc 33146- restriction of an integrable function to the absolute value of its real or imaginary part. (Contributed by Brendan Leahy, 19-Jun-2018.) (Revised by Brendan Leahy, 8-Aug-2018.)
Assertion
Ref Expression
ftc1anclem2 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1𝐺 ∈ {ℜ, ℑ}) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) ∈ ℝ)
Distinct variable groups:   𝑡,𝐹   𝑡,𝐴   𝑡,𝐺

Proof of Theorem ftc1anclem2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elpri 4170 . . 3 (𝐺 ∈ {ℜ, ℑ} → (𝐺 = ℜ ∨ 𝐺 = ℑ))
2 fveq1 6149 . . . . . . . . . 10 (𝐺 = ℜ → (𝐺‘(𝐹𝑡)) = (ℜ‘(𝐹𝑡)))
32fveq2d 6154 . . . . . . . . 9 (𝐺 = ℜ → (abs‘(𝐺‘(𝐹𝑡))) = (abs‘(ℜ‘(𝐹𝑡))))
43ifeq1d 4078 . . . . . . . 8 (𝐺 = ℜ → if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0) = if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0))
54mpteq2dv 4707 . . . . . . 7 (𝐺 = ℜ → (𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0)))
65fveq2d 6154 . . . . . 6 (𝐺 = ℜ → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) = (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0))))
76adantl 482 . . . . 5 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝐺 = ℜ) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) = (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0))))
8 ffvelrn 6315 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (𝐹𝑡) ∈ ℂ)
98recld 13871 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (ℜ‘(𝐹𝑡)) ∈ ℝ)
109adantlr 750 . . . . . . . . 9 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝑡𝐴) → (ℜ‘(𝐹𝑡)) ∈ ℝ)
11 simpl 473 . . . . . . . . . . . . 13 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → 𝐹:𝐴⟶ℂ)
1211feqmptd 6208 . . . . . . . . . . . 12 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → 𝐹 = (𝑡𝐴 ↦ (𝐹𝑡)))
13 simpr 477 . . . . . . . . . . . 12 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → 𝐹 ∈ 𝐿1)
1412, 13eqeltrrd 2699 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (𝐹𝑡)) ∈ 𝐿1)
158iblcn 23478 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → ((𝑡𝐴 ↦ (𝐹𝑡)) ∈ 𝐿1 ↔ ((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ 𝐿1 ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ 𝐿1)))
1615biimpa 501 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ (𝑡𝐴 ↦ (𝐹𝑡)) ∈ 𝐿1) → ((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ 𝐿1 ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ 𝐿1))
1714, 16syldan 487 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → ((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ 𝐿1 ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ 𝐿1))
1817simpld 475 . . . . . . . . 9 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ 𝐿1)
199recnd 10015 . . . . . . . . . . . 12 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (ℜ‘(𝐹𝑡)) ∈ ℂ)
20 eqidd 2622 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) = (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))))
21 absf 14014 . . . . . . . . . . . . . 14 abs:ℂ⟶ℝ
2221a1i 11 . . . . . . . . . . . . 13 (𝐹:𝐴⟶ℂ → abs:ℂ⟶ℝ)
2322feqmptd 6208 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → abs = (𝑥 ∈ ℂ ↦ (abs‘𝑥)))
24 fveq2 6150 . . . . . . . . . . . 12 (𝑥 = (ℜ‘(𝐹𝑡)) → (abs‘𝑥) = (abs‘(ℜ‘(𝐹𝑡))))
2519, 20, 23, 24fmptco 6354 . . . . . . . . . . 11 (𝐹:𝐴⟶ℂ → (abs ∘ (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡)))) = (𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))))
2625adantr 481 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (abs ∘ (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡)))) = (𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))))
27 eqid 2621 . . . . . . . . . . . . 13 (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) = (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡)))
289, 27fmptd 6343 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))):𝐴⟶ℝ)
2928adantr 481 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))):𝐴⟶ℝ)
30 iblmbf 23447 . . . . . . . . . . . . . . 15 (𝐹 ∈ 𝐿1𝐹 ∈ MblFn)
3130adantl 482 . . . . . . . . . . . . . 14 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → 𝐹 ∈ MblFn)
3212, 31eqeltrrd 2699 . . . . . . . . . . . . 13 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (𝐹𝑡)) ∈ MblFn)
338ismbfcn2 23319 . . . . . . . . . . . . . 14 (𝐹:𝐴⟶ℂ → ((𝑡𝐴 ↦ (𝐹𝑡)) ∈ MblFn ↔ ((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn)))
3433biimpa 501 . . . . . . . . . . . . 13 ((𝐹:𝐴⟶ℂ ∧ (𝑡𝐴 ↦ (𝐹𝑡)) ∈ MblFn) → ((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn))
3532, 34syldan 487 . . . . . . . . . . . 12 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → ((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn))
3635simpld 475 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn)
37 ftc1anclem1 33138 . . . . . . . . . . 11 (((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))):𝐴⟶ℝ ∧ (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn) → (abs ∘ (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡)))) ∈ MblFn)
3829, 36, 37syl2anc 692 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (abs ∘ (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡)))) ∈ MblFn)
3926, 38eqeltrrd 2699 . . . . . . . . 9 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))) ∈ MblFn)
4010, 18, 39iblabsnc 33127 . . . . . . . 8 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))) ∈ 𝐿1)
4119abscld 14112 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (abs‘(ℜ‘(𝐹𝑡))) ∈ ℝ)
4219absge0d 14120 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → 0 ≤ (abs‘(ℜ‘(𝐹𝑡))))
4341, 42iblpos 23472 . . . . . . . . 9 (𝐹:𝐴⟶ℂ → ((𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))) ∈ 𝐿1 ↔ ((𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))) ∈ MblFn ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0))) ∈ ℝ)))
4443adantr 481 . . . . . . . 8 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → ((𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))) ∈ 𝐿1 ↔ ((𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))) ∈ MblFn ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0))) ∈ ℝ)))
4540, 44mpbid 222 . . . . . . 7 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → ((𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))) ∈ MblFn ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0))) ∈ ℝ))
4645simprd 479 . . . . . 6 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0))) ∈ ℝ)
4746adantr 481 . . . . 5 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝐺 = ℜ) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0))) ∈ ℝ)
487, 47eqeltrd 2698 . . . 4 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝐺 = ℜ) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) ∈ ℝ)
49 fveq1 6149 . . . . . . . . . 10 (𝐺 = ℑ → (𝐺‘(𝐹𝑡)) = (ℑ‘(𝐹𝑡)))
5049fveq2d 6154 . . . . . . . . 9 (𝐺 = ℑ → (abs‘(𝐺‘(𝐹𝑡))) = (abs‘(ℑ‘(𝐹𝑡))))
5150ifeq1d 4078 . . . . . . . 8 (𝐺 = ℑ → if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0) = if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0))
5251mpteq2dv 4707 . . . . . . 7 (𝐺 = ℑ → (𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0)))
5352fveq2d 6154 . . . . . 6 (𝐺 = ℑ → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) = (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0))))
5453adantl 482 . . . . 5 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝐺 = ℑ) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) = (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0))))
558imcld 13872 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (ℑ‘(𝐹𝑡)) ∈ ℝ)
5655recnd 10015 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (ℑ‘(𝐹𝑡)) ∈ ℂ)
5756adantlr 750 . . . . . . . . 9 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝑡𝐴) → (ℑ‘(𝐹𝑡)) ∈ ℂ)
5817simprd 479 . . . . . . . . 9 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ 𝐿1)
59 eqidd 2622 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) = (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))))
60 fveq2 6150 . . . . . . . . . . . 12 (𝑥 = (ℑ‘(𝐹𝑡)) → (abs‘𝑥) = (abs‘(ℑ‘(𝐹𝑡))))
6156, 59, 23, 60fmptco 6354 . . . . . . . . . . 11 (𝐹:𝐴⟶ℂ → (abs ∘ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡)))) = (𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))))
6261adantr 481 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (abs ∘ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡)))) = (𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))))
63 eqid 2621 . . . . . . . . . . . . 13 (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) = (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡)))
6455, 63fmptd 6343 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))):𝐴⟶ℝ)
6564adantr 481 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))):𝐴⟶ℝ)
6635simprd 479 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn)
67 ftc1anclem1 33138 . . . . . . . . . . 11 (((𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))):𝐴⟶ℝ ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn) → (abs ∘ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡)))) ∈ MblFn)
6865, 66, 67syl2anc 692 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (abs ∘ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡)))) ∈ MblFn)
6962, 68eqeltrrd 2699 . . . . . . . . 9 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))) ∈ MblFn)
7057, 58, 69iblabsnc 33127 . . . . . . . 8 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))) ∈ 𝐿1)
7156abscld 14112 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (abs‘(ℑ‘(𝐹𝑡))) ∈ ℝ)
7256absge0d 14120 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → 0 ≤ (abs‘(ℑ‘(𝐹𝑡))))
7371, 72iblpos 23472 . . . . . . . . 9 (𝐹:𝐴⟶ℂ → ((𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))) ∈ 𝐿1 ↔ ((𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))) ∈ MblFn ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0))) ∈ ℝ)))
7473adantr 481 . . . . . . . 8 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → ((𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))) ∈ 𝐿1 ↔ ((𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))) ∈ MblFn ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0))) ∈ ℝ)))
7570, 74mpbid 222 . . . . . . 7 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → ((𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))) ∈ MblFn ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0))) ∈ ℝ))
7675simprd 479 . . . . . 6 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0))) ∈ ℝ)
7776adantr 481 . . . . 5 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝐺 = ℑ) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0))) ∈ ℝ)
7854, 77eqeltrd 2698 . . . 4 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝐺 = ℑ) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) ∈ ℝ)
7948, 78jaodan 825 . . 3 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ (𝐺 = ℜ ∨ 𝐺 = ℑ)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) ∈ ℝ)
801, 79sylan2 491 . 2 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝐺 ∈ {ℜ, ℑ}) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) ∈ ℝ)
81803impa 1256 1 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1𝐺 ∈ {ℜ, ℑ}) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) ∈ ℝ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ifcif 4060  {cpr 4152   ↦ cmpt 4675   ∘ ccom 5080  ⟶wf 5845  ‘cfv 5849  ℂcc 9881  ℝcr 9882  0cc0 9883  ℜcre 13774  ℑcim 13775  abscabs 13911  MblFncmbf 23296  ∫2citg2 23298  𝐿1cibl 23299 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-inf2 8485  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960  ax-pre-sup 9961  ax-addf 9962 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-disj 4586  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-of 6853  df-ofr 6854  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-2o 7509  df-oadd 7512  df-er 7690  df-map 7807  df-pm 7808  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-fi 8264  df-sup 8295  df-inf 8296  df-oi 8362  df-card 8712  df-cda 8937  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-div 10632  df-nn 10968  df-2 11026  df-3 11027  df-n0 11240  df-z 11325  df-uz 11635  df-q 11736  df-rp 11780  df-xneg 11893  df-xadd 11894  df-xmul 11895  df-ioo 12124  df-ico 12126  df-icc 12127  df-fz 12272  df-fzo 12410  df-fl 12536  df-seq 12745  df-exp 12804  df-hash 13061  df-cj 13776  df-re 13777  df-im 13778  df-sqrt 13912  df-abs 13913  df-clim 14156  df-sum 14354  df-rest 16007  df-topgen 16028  df-psmet 19660  df-xmet 19661  df-met 19662  df-bl 19663  df-mopn 19664  df-top 20621  df-topon 20638  df-bases 20664  df-cmp 21103  df-ovol 23146  df-vol 23147  df-mbf 23301  df-itg1 23302  df-itg2 23303  df-ibl 23304  df-0p 23350 This theorem is referenced by:  ftc1anclem8  33145
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