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Theorem ftc1anclem2 37701
Description: Lemma for ftc1anc 37708- 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 4649 . . 3 (𝐺 ∈ {ℜ, ℑ} → (𝐺 = ℜ ∨ 𝐺 = ℑ))
2 fveq1 6905 . . . . . . . . . 10 (𝐺 = ℜ → (𝐺‘(𝐹𝑡)) = (ℜ‘(𝐹𝑡)))
32fveq2d 6910 . . . . . . . . 9 (𝐺 = ℜ → (abs‘(𝐺‘(𝐹𝑡))) = (abs‘(ℜ‘(𝐹𝑡))))
43ifeq1d 4545 . . . . . . . 8 (𝐺 = ℜ → if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0) = if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0))
54mpteq2dv 5244 . . . . . . 7 (𝐺 = ℜ → (𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0)))
65fveq2d 6910 . . . . . 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 ffvelcdm 7101 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (𝐹𝑡) ∈ ℂ)
98recld 15233 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (ℜ‘(𝐹𝑡)) ∈ ℝ)
109adantlr 715 . . . . . . . . 9 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝑡𝐴) → (ℜ‘(𝐹𝑡)) ∈ ℝ)
11 simpl 482 . . . . . . . . . . . . 13 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → 𝐹:𝐴⟶ℂ)
1211feqmptd 6977 . . . . . . . . . . . 12 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → 𝐹 = (𝑡𝐴 ↦ (𝐹𝑡)))
13 simpr 484 . . . . . . . . . . . 12 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → 𝐹 ∈ 𝐿1)
1412, 13eqeltrrd 2842 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (𝐹𝑡)) ∈ 𝐿1)
158iblcn 25834 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → ((𝑡𝐴 ↦ (𝐹𝑡)) ∈ 𝐿1 ↔ ((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ 𝐿1 ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ 𝐿1)))
1615biimpa 476 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ (𝑡𝐴 ↦ (𝐹𝑡)) ∈ 𝐿1) → ((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ 𝐿1 ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ 𝐿1))
1714, 16syldan 591 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → ((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ 𝐿1 ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ 𝐿1))
1817simpld 494 . . . . . . . . 9 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ 𝐿1)
199recnd 11289 . . . . . . . . . . . 12 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (ℜ‘(𝐹𝑡)) ∈ ℂ)
20 eqidd 2738 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) = (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))))
21 absf 15376 . . . . . . . . . . . . . 14 abs:ℂ⟶ℝ
2221a1i 11 . . . . . . . . . . . . 13 (𝐹:𝐴⟶ℂ → abs:ℂ⟶ℝ)
2322feqmptd 6977 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → abs = (𝑥 ∈ ℂ ↦ (abs‘𝑥)))
24 fveq2 6906 . . . . . . . . . . . 12 (𝑥 = (ℜ‘(𝐹𝑡)) → (abs‘𝑥) = (abs‘(ℜ‘(𝐹𝑡))))
2519, 20, 23, 24fmptco 7149 . . . . . . . . . . 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)))))
279fmpttd 7135 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))):𝐴⟶ℝ)
2827adantr 480 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))):𝐴⟶ℝ)
29 iblmbf 25802 . . . . . . . . . . . . . . 15 (𝐹 ∈ 𝐿1𝐹 ∈ MblFn)
3029adantl 481 . . . . . . . . . . . . . 14 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → 𝐹 ∈ MblFn)
3112, 30eqeltrrd 2842 . . . . . . . . . . . . 13 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (𝐹𝑡)) ∈ MblFn)
328ismbfcn2 25673 . . . . . . . . . . . . . 14 (𝐹:𝐴⟶ℂ → ((𝑡𝐴 ↦ (𝐹𝑡)) ∈ MblFn ↔ ((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn)))
3332biimpa 476 . . . . . . . . . . . . 13 ((𝐹:𝐴⟶ℂ ∧ (𝑡𝐴 ↦ (𝐹𝑡)) ∈ MblFn) → ((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn))
3431, 33syldan 591 . . . . . . . . . . . 12 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → ((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn))
3534simpld 494 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn)
36 ftc1anclem1 37700 . . . . . . . . . . 11 (((𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))):𝐴⟶ℝ ∧ (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn) → (abs ∘ (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡)))) ∈ MblFn)
3728, 35, 36syl2anc 584 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (abs ∘ (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡)))) ∈ MblFn)
3826, 37eqeltrrd 2842 . . . . . . . . 9 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))) ∈ MblFn)
3910, 18, 38iblabsnc 37691 . . . . . . . 8 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))) ∈ 𝐿1)
4019abscld 15475 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (abs‘(ℜ‘(𝐹𝑡))) ∈ ℝ)
4119absge0d 15483 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → 0 ≤ (abs‘(ℜ‘(𝐹𝑡))))
4240, 41iblpos 25828 . . . . . . . . 9 (𝐹:𝐴⟶ℂ → ((𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))) ∈ 𝐿1 ↔ ((𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))) ∈ MblFn ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0))) ∈ ℝ)))
4342adantr 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)))
4439, 43mpbid 232 . . . . . . 7 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → ((𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))) ∈ MblFn ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0))) ∈ ℝ))
4544simprd 495 . . . . . 6 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0))) ∈ ℝ)
4645adantr 480 . . . . 5 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝐺 = ℜ) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0))) ∈ ℝ)
477, 46eqeltrd 2841 . . . 4 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝐺 = ℜ) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) ∈ ℝ)
48 fveq1 6905 . . . . . . . . . 10 (𝐺 = ℑ → (𝐺‘(𝐹𝑡)) = (ℑ‘(𝐹𝑡)))
4948fveq2d 6910 . . . . . . . . 9 (𝐺 = ℑ → (abs‘(𝐺‘(𝐹𝑡))) = (abs‘(ℑ‘(𝐹𝑡))))
5049ifeq1d 4545 . . . . . . . 8 (𝐺 = ℑ → if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0) = if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0))
5150mpteq2dv 5244 . . . . . . 7 (𝐺 = ℑ → (𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0)))
5251fveq2d 6910 . . . . . 6 (𝐺 = ℑ → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) = (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0))))
5352adantl 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))))
548imcld 15234 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (ℑ‘(𝐹𝑡)) ∈ ℝ)
5554recnd 11289 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (ℑ‘(𝐹𝑡)) ∈ ℂ)
5655adantlr 715 . . . . . . . . 9 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝑡𝐴) → (ℑ‘(𝐹𝑡)) ∈ ℂ)
5717simprd 495 . . . . . . . . 9 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ 𝐿1)
58 eqidd 2738 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) = (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))))
59 fveq2 6906 . . . . . . . . . . . 12 (𝑥 = (ℑ‘(𝐹𝑡)) → (abs‘𝑥) = (abs‘(ℑ‘(𝐹𝑡))))
6055, 58, 23, 59fmptco 7149 . . . . . . . . . . 11 (𝐹:𝐴⟶ℂ → (abs ∘ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡)))) = (𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))))
6160adantr 480 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (abs ∘ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡)))) = (𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))))
6254fmpttd 7135 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))):𝐴⟶ℝ)
6362adantr 480 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))):𝐴⟶ℝ)
6434simprd 495 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn)
65 ftc1anclem1 37700 . . . . . . . . . . 11 (((𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))):𝐴⟶ℝ ∧ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn) → (abs ∘ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡)))) ∈ MblFn)
6663, 64, 65syl2anc 584 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (abs ∘ (𝑡𝐴 ↦ (ℑ‘(𝐹𝑡)))) ∈ MblFn)
6761, 66eqeltrrd 2842 . . . . . . . . 9 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))) ∈ MblFn)
6856, 57, 67iblabsnc 37691 . . . . . . . 8 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))) ∈ 𝐿1)
6955abscld 15475 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (abs‘(ℑ‘(𝐹𝑡))) ∈ ℝ)
7055absge0d 15483 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → 0 ≤ (abs‘(ℑ‘(𝐹𝑡))))
7169, 70iblpos 25828 . . . . . . . . 9 (𝐹:𝐴⟶ℂ → ((𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))) ∈ 𝐿1 ↔ ((𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))) ∈ MblFn ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0))) ∈ ℝ)))
7271adantr 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)))
7368, 72mpbid 232 . . . . . . 7 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → ((𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))) ∈ MblFn ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0))) ∈ ℝ))
7473simprd 495 . . . . . 6 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0))) ∈ ℝ)
7574adantr 480 . . . . 5 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝐺 = ℑ) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0))) ∈ ℝ)
7653, 75eqeltrd 2841 . . . 4 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝐺 = ℑ) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) ∈ ℝ)
7747, 76jaodan 960 . . 3 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ (𝐺 = ℜ ∨ 𝐺 = ℑ)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) ∈ ℝ)
781, 77sylan2 593 . 2 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝐺 ∈ {ℜ, ℑ}) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) ∈ ℝ)
79783impa 1110 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 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  ifcif 4525  {cpr 4628  cmpt 5225  ccom 5689  wf 6557  cfv 6561  cc 11153  cr 11154  0cc0 11155  cre 15136  cim 15137  abscabs 15273  MblFncmbf 25649  2citg2 25651  𝐿1cibl 25652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-rest 17467  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cmp 23395  df-ovol 25499  df-vol 25500  df-mbf 25654  df-itg1 25655  df-itg2 25656  df-ibl 25657  df-0p 25705
This theorem is referenced by:  ftc1anclem8  37707
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