Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  i1faddlem Structured version   Visualization version   GIF version

 Description: Decompose the preimage of a sum. (Contributed by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
Assertion
Ref Expression
i1faddlem ((𝜑𝐴 ∈ ℂ) → ((𝐹f + 𝐺) “ {𝐴}) = 𝑦 ∈ ran 𝐺((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹   𝑦,𝐺   𝜑,𝑦

Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 i1fadd.1 . . . . . . . . 9 (𝜑𝐹 ∈ dom ∫1)
2 i1ff 24390 . . . . . . . . 9 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
31, 2syl 17 . . . . . . . 8 (𝜑𝐹:ℝ⟶ℝ)
43ffnd 6504 . . . . . . 7 (𝜑𝐹 Fn ℝ)
5 i1fadd.2 . . . . . . . . 9 (𝜑𝐺 ∈ dom ∫1)
6 i1ff 24390 . . . . . . . . 9 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
75, 6syl 17 . . . . . . . 8 (𝜑𝐺:ℝ⟶ℝ)
87ffnd 6504 . . . . . . 7 (𝜑𝐺 Fn ℝ)
9 reex 10679 . . . . . . . 8 ℝ ∈ V
109a1i 11 . . . . . . 7 (𝜑 → ℝ ∈ V)
11 inidm 4125 . . . . . . 7 (ℝ ∩ ℝ) = ℝ
124, 8, 10, 10, 11offn 7423 . . . . . 6 (𝜑 → (𝐹f + 𝐺) Fn ℝ)
1312adantr 484 . . . . 5 ((𝜑𝐴 ∈ ℂ) → (𝐹f + 𝐺) Fn ℝ)
14 fniniseg 6826 . . . . 5 ((𝐹f + 𝐺) Fn ℝ → (𝑧 ∈ ((𝐹f + 𝐺) “ {𝐴}) ↔ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)))
1513, 14syl 17 . . . 4 ((𝜑𝐴 ∈ ℂ) → (𝑧 ∈ ((𝐹f + 𝐺) “ {𝐴}) ↔ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)))
168ad2antrr 725 . . . . . . . 8 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝐺 Fn ℝ)
17 simprl 770 . . . . . . . 8 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ ℝ)
18 fnfvelrn 6845 . . . . . . . 8 ((𝐺 Fn ℝ ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) ∈ ran 𝐺)
1916, 17, 18syl2anc 587 . . . . . . 7 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐺𝑧) ∈ ran 𝐺)
20 simprr 772 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → ((𝐹f + 𝐺)‘𝑧) = 𝐴)
21 eqidd 2759 . . . . . . . . . . . . . 14 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) = (𝐹𝑧))
22 eqidd 2759 . . . . . . . . . . . . . 14 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) = (𝐺𝑧))
234, 8, 10, 10, 11, 21, 22ofval 7421 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ℝ) → ((𝐹f + 𝐺)‘𝑧) = ((𝐹𝑧) + (𝐺𝑧)))
2423ad2ant2r 746 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → ((𝐹f + 𝐺)‘𝑧) = ((𝐹𝑧) + (𝐺𝑧)))
2520, 24eqtr3d 2795 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝐴 = ((𝐹𝑧) + (𝐺𝑧)))
2625oveq1d 7171 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐴 − (𝐺𝑧)) = (((𝐹𝑧) + (𝐺𝑧)) − (𝐺𝑧)))
27 ax-resscn 10645 . . . . . . . . . . . . . 14 ℝ ⊆ ℂ
28 fss 6517 . . . . . . . . . . . . . 14 ((𝐹:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:ℝ⟶ℂ)
293, 27, 28sylancl 589 . . . . . . . . . . . . 13 (𝜑𝐹:ℝ⟶ℂ)
3029ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝐹:ℝ⟶ℂ)
3130, 17ffvelrnd 6849 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐹𝑧) ∈ ℂ)
32 fss 6517 . . . . . . . . . . . . . 14 ((𝐺:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐺:ℝ⟶ℂ)
337, 27, 32sylancl 589 . . . . . . . . . . . . 13 (𝜑𝐺:ℝ⟶ℂ)
3433ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝐺:ℝ⟶ℂ)
3534, 17ffvelrnd 6849 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐺𝑧) ∈ ℂ)
3631, 35pncand 11049 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (((𝐹𝑧) + (𝐺𝑧)) − (𝐺𝑧)) = (𝐹𝑧))
3726, 36eqtr2d 2794 . . . . . . . . 9 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐹𝑧) = (𝐴 − (𝐺𝑧)))
384ad2antrr 725 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝐹 Fn ℝ)
39 fniniseg 6826 . . . . . . . . . 10 (𝐹 Fn ℝ → (𝑧 ∈ (𝐹 “ {(𝐴 − (𝐺𝑧))}) ↔ (𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴 − (𝐺𝑧)))))
4038, 39syl 17 . . . . . . . . 9 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝑧 ∈ (𝐹 “ {(𝐴 − (𝐺𝑧))}) ↔ (𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴 − (𝐺𝑧)))))
4117, 37, 40mpbir2and 712 . . . . . . . 8 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ (𝐹 “ {(𝐴 − (𝐺𝑧))}))
42 eqidd 2759 . . . . . . . . 9 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐺𝑧) = (𝐺𝑧))
43 fniniseg 6826 . . . . . . . . . 10 (𝐺 Fn ℝ → (𝑧 ∈ (𝐺 “ {(𝐺𝑧)}) ↔ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = (𝐺𝑧))))
4416, 43syl 17 . . . . . . . . 9 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝑧 ∈ (𝐺 “ {(𝐺𝑧)}) ↔ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = (𝐺𝑧))))
4517, 42, 44mpbir2and 712 . . . . . . . 8 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ (𝐺 “ {(𝐺𝑧)}))
4641, 45elind 4101 . . . . . . 7 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ ((𝐹 “ {(𝐴 − (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)})))
47 oveq2 7164 . . . . . . . . . . . 12 (𝑦 = (𝐺𝑧) → (𝐴𝑦) = (𝐴 − (𝐺𝑧)))
4847sneqd 4537 . . . . . . . . . . 11 (𝑦 = (𝐺𝑧) → {(𝐴𝑦)} = {(𝐴 − (𝐺𝑧))})
4948imaeq2d 5906 . . . . . . . . . 10 (𝑦 = (𝐺𝑧) → (𝐹 “ {(𝐴𝑦)}) = (𝐹 “ {(𝐴 − (𝐺𝑧))}))
50 sneq 4535 . . . . . . . . . . 11 (𝑦 = (𝐺𝑧) → {𝑦} = {(𝐺𝑧)})
5150imaeq2d 5906 . . . . . . . . . 10 (𝑦 = (𝐺𝑧) → (𝐺 “ {𝑦}) = (𝐺 “ {(𝐺𝑧)}))
5249, 51ineq12d 4120 . . . . . . . . 9 (𝑦 = (𝐺𝑧) → ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) = ((𝐹 “ {(𝐴 − (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)})))
5352eleq2d 2837 . . . . . . . 8 (𝑦 = (𝐺𝑧) → (𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ 𝑧 ∈ ((𝐹 “ {(𝐴 − (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)}))))
5453rspcev 3543 . . . . . . 7 (((𝐺𝑧) ∈ ran 𝐺𝑧 ∈ ((𝐹 “ {(𝐴 − (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)}))) → ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})))
5519, 46, 54syl2anc 587 . . . . . 6 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})))
5655ex 416 . . . . 5 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴) → ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦}))))
57 elin 3876 . . . . . . 7 (𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ (𝑧 ∈ (𝐹 “ {(𝐴𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})))
584adantr 484 . . . . . . . . . 10 ((𝜑𝐴 ∈ ℂ) → 𝐹 Fn ℝ)
59 fniniseg 6826 . . . . . . . . . 10 (𝐹 Fn ℝ → (𝑧 ∈ (𝐹 “ {(𝐴𝑦)}) ↔ (𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴𝑦))))
6058, 59syl 17 . . . . . . . . 9 ((𝜑𝐴 ∈ ℂ) → (𝑧 ∈ (𝐹 “ {(𝐴𝑦)}) ↔ (𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴𝑦))))
618adantr 484 . . . . . . . . . 10 ((𝜑𝐴 ∈ ℂ) → 𝐺 Fn ℝ)
62 fniniseg 6826 . . . . . . . . . 10 (𝐺 Fn ℝ → (𝑧 ∈ (𝐺 “ {𝑦}) ↔ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)))
6361, 62syl 17 . . . . . . . . 9 ((𝜑𝐴 ∈ ℂ) → (𝑧 ∈ (𝐺 “ {𝑦}) ↔ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)))
6460, 63anbi12d 633 . . . . . . . 8 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ (𝐹 “ {(𝐴𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})) ↔ ((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦))))
65 anandi 675 . . . . . . . . 9 ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦)) ↔ ((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)))
66 simprl 770 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → 𝑧 ∈ ℝ)
6723ad2ant2r 746 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → ((𝐹f + 𝐺)‘𝑧) = ((𝐹𝑧) + (𝐺𝑧)))
68 simprrl 780 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → (𝐹𝑧) = (𝐴𝑦))
69 simprrr 781 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → (𝐺𝑧) = 𝑦)
7068, 69oveq12d 7174 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → ((𝐹𝑧) + (𝐺𝑧)) = ((𝐴𝑦) + 𝑦))
71 simplr 768 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → 𝐴 ∈ ℂ)
7233ad2antrr 725 . . . . . . . . . . . . . . 15 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → 𝐺:ℝ⟶ℂ)
7372, 66ffvelrnd 6849 . . . . . . . . . . . . . 14 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → (𝐺𝑧) ∈ ℂ)
7469, 73eqeltrrd 2853 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → 𝑦 ∈ ℂ)
7571, 74npcand 11052 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → ((𝐴𝑦) + 𝑦) = 𝐴)
7667, 70, 753eqtrd 2797 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → ((𝐹f + 𝐺)‘𝑧) = 𝐴)
7766, 76jca 515 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴))
7877ex 416 . . . . . . . . 9 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)))
7965, 78syl5bir 246 . . . . . . . 8 ((𝜑𝐴 ∈ ℂ) → (((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)))
8064, 79sylbid 243 . . . . . . 7 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ (𝐹 “ {(𝐴𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)))
8157, 80syl5bi 245 . . . . . 6 ((𝜑𝐴 ∈ ℂ) → (𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)))
8281rexlimdvw 3214 . . . . 5 ((𝜑𝐴 ∈ ℂ) → (∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)))
8356, 82impbid 215 . . . 4 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦}))))
8415, 83bitrd 282 . . 3 ((𝜑𝐴 ∈ ℂ) → (𝑧 ∈ ((𝐹f + 𝐺) “ {𝐴}) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦}))))
85 eliun 4890 . . 3 (𝑧 𝑦 ∈ ran 𝐺((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})))
8684, 85bitr4di 292 . 2 ((𝜑𝐴 ∈ ℂ) → (𝑧 ∈ ((𝐹f + 𝐺) “ {𝐴}) ↔ 𝑧 𝑦 ∈ ran 𝐺((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦}))))
8786eqrdv 2756 1 ((𝜑𝐴 ∈ ℂ) → ((𝐹f + 𝐺) “ {𝐴}) = 𝑦 ∈ ran 𝐺((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3071  Vcvv 3409   ∩ cin 3859   ⊆ wss 3860  {csn 4525  ∪ ciun 4886  ◡ccnv 5527  dom cdm 5528  ran crn 5529   “ cima 5531   Fn wfn 6335  ⟶wf 6336  ‘cfv 6340  (class class class)co 7156   ∘f cof 7409  ℂcc 10586  ℝcr 10587   + caddc 10591   − cmin 10921  ∫1citg1 24329 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-po 5447  df-so 5448  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7411  df-er 8305  df-en 8541  df-dom 8542  df-sdom 8543  df-pnf 10728  df-mnf 10729  df-ltxr 10731  df-sub 10923  df-sum 15104  df-itg1 24334 This theorem is referenced by:  i1fadd  24409  itg1addlem4  24413
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