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Theorem ftc1anclem2 37680
Description: Lemma for ftc1anc 37687- 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 4653 . . 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 4549 . . . . . . . 8 (𝐺 = ℜ → if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0) = if(𝑡𝐴, (abs‘(ℜ‘(𝐹𝑡))), 0))
54mpteq2dv 5249 . . . . . . 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 7100 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (𝐹𝑡) ∈ ℂ)
98recld 15229 . . . . . . . . . 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 6976 . . . . . . . . . . . 12 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → 𝐹 = (𝑡𝐴 ↦ (𝐹𝑡)))
13 simpr 484 . . . . . . . . . . . 12 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → 𝐹 ∈ 𝐿1)
1412, 13eqeltrrd 2839 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (𝐹𝑡)) ∈ 𝐿1)
158iblcn 25848 . . . . . . . . . . . 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 11286 . . . . . . . . . . . 12 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (ℜ‘(𝐹𝑡)) ∈ ℂ)
20 eqidd 2735 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))) = (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))))
21 absf 15372 . . . . . . . . . . . . . 14 abs:ℂ⟶ℝ
2221a1i 11 . . . . . . . . . . . . 13 (𝐹:𝐴⟶ℂ → abs:ℂ⟶ℝ)
2322feqmptd 6976 . . . . . . . . . . . 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 7148 . . . . . . . . . . 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 7134 . . . . . . . . . . . 12 (𝐹:𝐴⟶ℂ → (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))):𝐴⟶ℝ)
2827adantr 480 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (ℜ‘(𝐹𝑡))):𝐴⟶ℝ)
29 iblmbf 25816 . . . . . . . . . . . . . . 15 (𝐹 ∈ 𝐿1𝐹 ∈ MblFn)
3029adantl 481 . . . . . . . . . . . . . 14 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → 𝐹 ∈ MblFn)
3112, 30eqeltrrd 2839 . . . . . . . . . . . . 13 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (𝐹𝑡)) ∈ MblFn)
328ismbfcn2 25686 . . . . . . . . . . . . . 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 37679 . . . . . . . . . . 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 2839 . . . . . . . . 9 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))) ∈ MblFn)
3910, 18, 38iblabsnc 37670 . . . . . . . 8 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (abs‘(ℜ‘(𝐹𝑡)))) ∈ 𝐿1)
4019abscld 15471 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (abs‘(ℜ‘(𝐹𝑡))) ∈ ℝ)
4119absge0d 15479 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → 0 ≤ (abs‘(ℜ‘(𝐹𝑡))))
4240, 41iblpos 25842 . . . . . . . . 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 2838 . . . 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 4549 . . . . . . . 8 (𝐺 = ℑ → if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0) = if(𝑡𝐴, (abs‘(ℑ‘(𝐹𝑡))), 0))
5150mpteq2dv 5249 . . . . . . 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 15230 . . . . . . . . . . 11 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (ℑ‘(𝐹𝑡)) ∈ ℝ)
5554recnd 11286 . . . . . . . . . 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 2735 . . . . . . . . . . . 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 7148 . . . . . . . . . . 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 7134 . . . . . . . . . . . 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 37679 . . . . . . . . . . 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 2839 . . . . . . . . 9 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))) ∈ MblFn)
6856, 57, 67iblabsnc 37670 . . . . . . . 8 ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (𝑡𝐴 ↦ (abs‘(ℑ‘(𝐹𝑡)))) ∈ 𝐿1)
6955abscld 15471 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → (abs‘(ℑ‘(𝐹𝑡))) ∈ ℝ)
7055absge0d 15479 . . . . . . . . . 10 ((𝐹:𝐴⟶ℂ ∧ 𝑡𝐴) → 0 ≤ (abs‘(ℑ‘(𝐹𝑡))))
7169, 70iblpos 25842 . . . . . . . . 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 2838 . . . 4 (((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1) ∧ 𝐺 = ℑ) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐴, (abs‘(𝐺‘(𝐹𝑡))), 0))) ∈ ℝ)
7747, 76jaodan 959 . . 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 1109 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 847  w3a 1086   = wceq 1536  wcel 2105  ifcif 4530  {cpr 4632  cmpt 5230  ccom 5692  wf 6558  cfv 6562  cc 11150  cr 11151  0cc0 11152  cre 15132  cim 15133  abscabs 15269  MblFncmbf 25662  2citg2 25664  𝐿1cibl 25665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-addf 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-disj 5115  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-ofr 7697  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fi 9448  df-sup 9479  df-inf 9480  df-oi 9547  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-ioo 13387  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-fl 13828  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-sum 15719  df-rest 17468  df-topgen 17489  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-top 22915  df-topon 22932  df-bases 22968  df-cmp 23410  df-ovol 25512  df-vol 25513  df-mbf 25667  df-itg1 25668  df-itg2 25669  df-ibl 25670  df-0p 25718
This theorem is referenced by:  ftc1anclem8  37686
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