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Theorem i1faddlem 25741
Description: Decompose the preimage of a sum. (Contributed by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
i1fadd.1 (𝜑𝐹 ∈ dom ∫1)
i1fadd.2 (𝜑𝐺 ∈ dom ∫1)
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

Proof of Theorem i1faddlem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 i1fadd.1 . . . . . . . . 9 (𝜑𝐹 ∈ dom ∫1)
2 i1ff 25724 . . . . . . . . 9 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
31, 2syl 17 . . . . . . . 8 (𝜑𝐹:ℝ⟶ℝ)
43ffnd 6737 . . . . . . 7 (𝜑𝐹 Fn ℝ)
5 i1fadd.2 . . . . . . . . 9 (𝜑𝐺 ∈ dom ∫1)
6 i1ff 25724 . . . . . . . . 9 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
75, 6syl 17 . . . . . . . 8 (𝜑𝐺:ℝ⟶ℝ)
87ffnd 6737 . . . . . . 7 (𝜑𝐺 Fn ℝ)
9 reex 11243 . . . . . . . 8 ℝ ∈ V
109a1i 11 . . . . . . 7 (𝜑 → ℝ ∈ V)
11 inidm 4234 . . . . . . 7 (ℝ ∩ ℝ) = ℝ
124, 8, 10, 10, 11offn 7709 . . . . . 6 (𝜑 → (𝐹f + 𝐺) Fn ℝ)
1312adantr 480 . . . . 5 ((𝜑𝐴 ∈ ℂ) → (𝐹f + 𝐺) Fn ℝ)
14 fniniseg 7079 . . . . 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 726 . . . . . . . 8 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝐺 Fn ℝ)
17 simprl 771 . . . . . . . 8 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ ℝ)
18 fnfvelrn 7099 . . . . . . . 8 ((𝐺 Fn ℝ ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) ∈ ran 𝐺)
1916, 17, 18syl2anc 584 . . . . . . 7 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐺𝑧) ∈ ran 𝐺)
20 simprr 773 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → ((𝐹f + 𝐺)‘𝑧) = 𝐴)
21 eqidd 2735 . . . . . . . . . . . . . 14 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) = (𝐹𝑧))
22 eqidd 2735 . . . . . . . . . . . . . 14 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) = (𝐺𝑧))
234, 8, 10, 10, 11, 21, 22ofval 7707 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ℝ) → ((𝐹f + 𝐺)‘𝑧) = ((𝐹𝑧) + (𝐺𝑧)))
2423ad2ant2r 747 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → ((𝐹f + 𝐺)‘𝑧) = ((𝐹𝑧) + (𝐺𝑧)))
2520, 24eqtr3d 2776 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝐴 = ((𝐹𝑧) + (𝐺𝑧)))
2625oveq1d 7445 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐴 − (𝐺𝑧)) = (((𝐹𝑧) + (𝐺𝑧)) − (𝐺𝑧)))
27 ax-resscn 11209 . . . . . . . . . . . . . 14 ℝ ⊆ ℂ
28 fss 6752 . . . . . . . . . . . . . 14 ((𝐹:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:ℝ⟶ℂ)
293, 27, 28sylancl 586 . . . . . . . . . . . . 13 (𝜑𝐹:ℝ⟶ℂ)
3029ad2antrr 726 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝐹:ℝ⟶ℂ)
3130, 17ffvelcdmd 7104 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐹𝑧) ∈ ℂ)
32 fss 6752 . . . . . . . . . . . . . 14 ((𝐺:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐺:ℝ⟶ℂ)
337, 27, 32sylancl 586 . . . . . . . . . . . . 13 (𝜑𝐺:ℝ⟶ℂ)
3433ad2antrr 726 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝐺:ℝ⟶ℂ)
3534, 17ffvelcdmd 7104 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐺𝑧) ∈ ℂ)
3631, 35pncand 11618 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (((𝐹𝑧) + (𝐺𝑧)) − (𝐺𝑧)) = (𝐹𝑧))
3726, 36eqtr2d 2775 . . . . . . . . 9 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐹𝑧) = (𝐴 − (𝐺𝑧)))
384ad2antrr 726 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝐹 Fn ℝ)
39 fniniseg 7079 . . . . . . . . . 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 713 . . . . . . . 8 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ (𝐹 “ {(𝐴 − (𝐺𝑧))}))
42 eqidd 2735 . . . . . . . . 9 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐺𝑧) = (𝐺𝑧))
43 fniniseg 7079 . . . . . . . . . 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 713 . . . . . . . 8 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ (𝐺 “ {(𝐺𝑧)}))
4641, 45elind 4209 . . . . . . 7 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ ((𝐹 “ {(𝐴 − (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)})))
47 oveq2 7438 . . . . . . . . . . . 12 (𝑦 = (𝐺𝑧) → (𝐴𝑦) = (𝐴 − (𝐺𝑧)))
4847sneqd 4642 . . . . . . . . . . 11 (𝑦 = (𝐺𝑧) → {(𝐴𝑦)} = {(𝐴 − (𝐺𝑧))})
4948imaeq2d 6079 . . . . . . . . . 10 (𝑦 = (𝐺𝑧) → (𝐹 “ {(𝐴𝑦)}) = (𝐹 “ {(𝐴 − (𝐺𝑧))}))
50 sneq 4640 . . . . . . . . . . 11 (𝑦 = (𝐺𝑧) → {𝑦} = {(𝐺𝑧)})
5150imaeq2d 6079 . . . . . . . . . 10 (𝑦 = (𝐺𝑧) → (𝐺 “ {𝑦}) = (𝐺 “ {(𝐺𝑧)}))
5249, 51ineq12d 4228 . . . . . . . . 9 (𝑦 = (𝐺𝑧) → ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) = ((𝐹 “ {(𝐴 − (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)})))
5352eleq2d 2824 . . . . . . . 8 (𝑦 = (𝐺𝑧) → (𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ 𝑧 ∈ ((𝐹 “ {(𝐴 − (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)}))))
5453rspcev 3621 . . . . . . 7 (((𝐺𝑧) ∈ ran 𝐺𝑧 ∈ ((𝐹 “ {(𝐴 − (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)}))) → ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})))
5519, 46, 54syl2anc 584 . . . . . 6 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})))
5655ex 412 . . . . 5 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴) → ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦}))))
57 elin 3978 . . . . . . 7 (𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ (𝑧 ∈ (𝐹 “ {(𝐴𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})))
584adantr 480 . . . . . . . . . 10 ((𝜑𝐴 ∈ ℂ) → 𝐹 Fn ℝ)
59 fniniseg 7079 . . . . . . . . . 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 480 . . . . . . . . . 10 ((𝜑𝐴 ∈ ℂ) → 𝐺 Fn ℝ)
62 fniniseg 7079 . . . . . . . . . 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 632 . . . . . . . 8 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ (𝐹 “ {(𝐴𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})) ↔ ((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦))))
65 anandi 676 . . . . . . . . 9 ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦)) ↔ ((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)))
66 simprl 771 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → 𝑧 ∈ ℝ)
6723ad2ant2r 747 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → ((𝐹f + 𝐺)‘𝑧) = ((𝐹𝑧) + (𝐺𝑧)))
68 simprrl 781 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → (𝐹𝑧) = (𝐴𝑦))
69 simprrr 782 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → (𝐺𝑧) = 𝑦)
7068, 69oveq12d 7448 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → ((𝐹𝑧) + (𝐺𝑧)) = ((𝐴𝑦) + 𝑦))
71 simplr 769 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → 𝐴 ∈ ℂ)
7233ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → 𝐺:ℝ⟶ℂ)
7372, 66ffvelcdmd 7104 . . . . . . . . . . . . . 14 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → (𝐺𝑧) ∈ ℂ)
7469, 73eqeltrrd 2839 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → 𝑦 ∈ ℂ)
7571, 74npcand 11621 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → ((𝐴𝑦) + 𝑦) = 𝐴)
7667, 70, 753eqtrd 2778 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → ((𝐹f + 𝐺)‘𝑧) = 𝐴)
7766, 76jca 511 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴))
7877ex 412 . . . . . . . . 9 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)))
7965, 78biimtrrid 243 . . . . . . . 8 ((𝜑𝐴 ∈ ℂ) → (((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)))
8064, 79sylbid 240 . . . . . . 7 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ (𝐹 “ {(𝐴𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)))
8157, 80biimtrid 242 . . . . . 6 ((𝜑𝐴 ∈ ℂ) → (𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)))
8281rexlimdvw 3157 . . . . 5 ((𝜑𝐴 ∈ ℂ) → (∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)))
8356, 82impbid 212 . . . 4 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦}))))
8415, 83bitrd 279 . . 3 ((𝜑𝐴 ∈ ℂ) → (𝑧 ∈ ((𝐹f + 𝐺) “ {𝐴}) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦}))))
85 eliun 4999 . . 3 (𝑧 𝑦 ∈ ran 𝐺((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})))
8684, 85bitr4di 289 . 2 ((𝜑𝐴 ∈ ℂ) → (𝑧 ∈ ((𝐹f + 𝐺) “ {𝐴}) ↔ 𝑧 𝑦 ∈ ran 𝐺((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦}))))
8786eqrdv 2732 1 ((𝜑𝐴 ∈ ℂ) → ((𝐹f + 𝐺) “ {𝐴}) = 𝑦 ∈ ran 𝐺((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wrex 3067  Vcvv 3477  cin 3961  wss 3962  {csn 4630   ciun 4995  ccnv 5687  dom cdm 5688  ran crn 5689  cima 5691   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  f cof 7694  cc 11150  cr 11151   + caddc 11155  cmin 11489  1citg1 25663
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-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
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-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-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-po 5596  df-so 5597  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-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-ltxr 11297  df-sub 11491  df-sum 15719  df-itg1 25668
This theorem is referenced by:  i1fadd  25743  itg1addlem4  25747  itg1addlem4OLD  25748
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