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Theorem ftc1anclem2 32656
 Description: Lemma for ftc1anc 32663- 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 4145 . . 3 (𝐺 ∈ {ℜ, ℑ} → (𝐺 = ℜ ∨ 𝐺 = ℑ))
2 fveq1 6102 . . . . . . . . . 10 (𝐺 = ℜ → (𝐺‘(𝐹𝑡)) = (ℜ‘(𝐹𝑡)))
32fveq2d 6107 . . . . . . . . 9 (𝐺 = ℜ → (abs‘(𝐺‘(𝐹𝑡))) = (abs‘(ℜ‘(𝐹𝑡))))
43ifeq1d 4054 . . . . . . . 8 (𝐺 = ℜ → if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0) = if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0))
54mpteq2dv 4673 . . . . . . 7 (𝐺 = ℜ → (𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0)))
65fveq2d 6107 . . . . . 6 (𝐺 = ℜ → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) = (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0))))
76adantl 481 . . . . 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 6265 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (𝐹𝑡) ∈ ℂ)
98recld 13782 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (ℜ‘(𝐹𝑡)) ∈ ℝ)
109adantlr 747 . . . . . . . . 9 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝑡𝐴) → (ℜ‘(𝐹𝑡)) ∈ ℝ)
11 simpl 472 . . . . . . . . . . . . 13 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → 𝐹:𝐴⟶ℂ)
1211feqmptd 6159 . . . . . . . . . . . 12 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → 𝐹 = (𝑡𝐴 ↦ (𝐹𝑡)))
13 simpr 476 . . . . . . . . . . . 12 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → 𝐹 ∈ 𝐿1)
1412, 13eqeltrrd 2689 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (𝐹𝑡)) ∈ 𝐿1)
158iblcn 23371 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → ((𝑡𝐴 ↦ (𝐹𝑡)) ∈ 𝐿1 ↔ ((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ 𝐿1 ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ 𝐿1)))
1615biimpa 500 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ (𝑡𝐴 ↦ (𝐹𝑡)) ∈ 𝐿1) → ((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ 𝐿1 ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ 𝐿1))
1714, 16syldan 486 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → ((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ 𝐿1 ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ 𝐿1))
1817simpld 474 . . . . . . . . 9 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ 𝐿1)
199recnd 9947 . . . . . . . . . . . 12 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (ℜ‘(𝐹𝑡)) ∈ ℂ)
20 eqidd 2611 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) = (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))))
21 absf 13925 . . . . . . . . . . . . . 14 abs:ℂ⟶ℝ
2221a1i 11 . . . . . . . . . . . . 13 (𝐹:𝐴⟶ℂ → abs:ℂ⟶ℝ)
2322feqmptd 6159 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → abs = (𝑥 ∈ ℂ ↦ (abs‘𝑥)))
24 fveq2 6103 . . . . . . . . . . . 12 (𝑥 = (ℜ‘(𝐹𝑡)) → (abs‘𝑥) = (abs‘(ℜ‘(𝐹𝑡))))
2519, 20, 23, 24fmptco 6303 . . . . . . . . . . 11 (𝐹:𝐴⟶ℂ → (abs ∘ (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡)))) = (𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))))
2625adantr 480 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (abs ∘ (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡)))) = (𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))))
27 eqid 2610 . . . . . . . . . . . . 13 (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) = (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡)))
289, 27fmptd 6292 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))):𝐴⟶ℝ)
2928adantr 480 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))):𝐴⟶ℝ)
30 iblmbf 23340 . . . . . . . . . . . . . . 15 (𝐹 ∈ 𝐿1𝐹 ∈ MblFn)
3130adantl 481 . . . . . . . . . . . . . 14 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → 𝐹 ∈ MblFn)
3212, 31eqeltrrd 2689 . . . . . . . . . . . . 13 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (𝐹𝑡)) ∈ MblFn)
338ismbfcn2 23212 . . . . . . . . . . . . . 14 (𝐹:𝐴⟶ℂ → ((𝑡𝐴 ↦ (𝐹𝑡)) ∈ MblFn ↔ ((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn)))
3433biimpa 500 . . . . . . . . . . . . 13 ((𝐹:𝐴⟶ℂ ∧ (𝑡𝐴 ↦ (𝐹𝑡)) ∈ MblFn) → ((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn))
3532, 34syldan 486 . . . . . . . . . . . 12 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → ((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn))
3635simpld 474 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn)
37 ftc1anclem1 32655 . . . . . . . . . . 11 (((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))):𝐴⟶ℝ ∧ (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn) → (abs ∘ (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡)))) ∈ MblFn)
3829, 36, 37syl2anc 691 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (abs ∘ (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡)))) ∈ MblFn)
3926, 38eqeltrrd 2689 . . . . . . . . 9 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))) ∈ MblFn)
4010, 18, 39iblabsnc 32644 . . . . . . . 8 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))) ∈ 𝐿1)
4119abscld 14023 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (abs‘(ℜ‘(𝐹𝑡))) ∈ ℝ)
4219absge0d 14031 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → 0 ≤ (abs‘(ℜ‘(𝐹𝑡))))
4341, 42iblpos 23365 . . . . . . . . 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 480 . . . . . . . 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 221 . . . . . . 7 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → ((𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))) ∈ MblFn ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0))) ∈ ℝ))
4645simprd 478 . . . . . 6 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0))) ∈ ℝ)
4746adantr 480 . . . . 5 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝐺 = ℜ) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0))) ∈ ℝ)
487, 47eqeltrd 2688 . . . 4 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝐺 = ℜ) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) ∈ ℝ)
49 fveq1 6102 . . . . . . . . . 10 (𝐺 = ℑ → (𝐺‘(𝐹𝑡)) = (ℑ‘(𝐹𝑡)))
5049fveq2d 6107 . . . . . . . . 9 (𝐺 = ℑ → (abs‘(𝐺‘(𝐹𝑡))) = (abs‘(ℑ‘(𝐹𝑡))))
5150ifeq1d 4054 . . . . . . . 8 (𝐺 = ℑ → if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0) = if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0))
5251mpteq2dv 4673 . . . . . . 7 (𝐺 = ℑ → (𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0)))
5352fveq2d 6107 . . . . . 6 (𝐺 = ℑ → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) = (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0))))
5453adantl 481 . . . . 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 13783 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (ℑ‘(𝐹𝑡)) ∈ ℝ)
5655recnd 9947 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (ℑ‘(𝐹𝑡)) ∈ ℂ)
5756adantlr 747 . . . . . . . . 9 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝑡𝐴) → (ℑ‘(𝐹𝑡)) ∈ ℂ)
5817simprd 478 . . . . . . . . 9 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ 𝐿1)
59 eqidd 2611 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) = (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))))
60 fveq2 6103 . . . . . . . . . . . 12 (𝑥 = (ℑ‘(𝐹𝑡)) → (abs‘𝑥) = (abs‘(ℑ‘(𝐹𝑡))))
6156, 59, 23, 60fmptco 6303 . . . . . . . . . . 11 (𝐹:𝐴⟶ℂ → (abs ∘ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡)))) = (𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))))
6261adantr 480 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (abs ∘ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡)))) = (𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))))
63 eqid 2610 . . . . . . . . . . . . 13 (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) = (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡)))
6455, 63fmptd 6292 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))):𝐴⟶ℝ)
6564adantr 480 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))):𝐴⟶ℝ)
6635simprd 478 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn)
67 ftc1anclem1 32655 . . . . . . . . . . 11 (((𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))):𝐴⟶ℝ ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn) → (abs ∘ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡)))) ∈ MblFn)
6865, 66, 67syl2anc 691 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (abs ∘ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡)))) ∈ MblFn)
6962, 68eqeltrrd 2689 . . . . . . . . 9 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))) ∈ MblFn)
7057, 58, 69iblabsnc 32644 . . . . . . . 8 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))) ∈ 𝐿1)
7156abscld 14023 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (abs‘(ℑ‘(𝐹𝑡))) ∈ ℝ)
7256absge0d 14031 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → 0 ≤ (abs‘(ℑ‘(𝐹𝑡))))
7371, 72iblpos 23365 . . . . . . . . 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 480 . . . . . . . 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 221 . . . . . . 7 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → ((𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))) ∈ MblFn ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0))) ∈ ℝ))
7675simprd 478 . . . . . 6 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0))) ∈ ℝ)
7776adantr 480 . . . . 5 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝐺 = ℑ) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0))) ∈ ℝ)
7854, 77eqeltrd 2688 . . . 4 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝐺 = ℑ) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) ∈ ℝ)
7948, 78jaodan 822 . . 3 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ (𝐺 = ℜ ∨ 𝐺 = ℑ)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) ∈ ℝ)
801, 79sylan2 490 . 2 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝐺 ∈ {ℜ, ℑ}) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) ∈ ℝ)
81803impa 1251 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 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ifcif 4036  {cpr 4127   ↦ cmpt 4643   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  ℂcc 9813  ℝcr 9814  0cc0 9815  ℜcre 13685  ℑcim 13686  abscabs 13822  MblFncmbf 23189  ∫2citg2 23191  𝐿1cibl 23192 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041  df-mbf 23194  df-itg1 23195  df-itg2 23196  df-ibl 23197  df-0p 23243 This theorem is referenced by:  ftc1anclem8  32662
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