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Theorem i1faddlem 24211
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 24194 . . . . . . . . 9 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
31, 2syl 17 . . . . . . . 8 (𝜑𝐹:ℝ⟶ℝ)
43ffnd 6511 . . . . . . 7 (𝜑𝐹 Fn ℝ)
5 i1fadd.2 . . . . . . . . 9 (𝜑𝐺 ∈ dom ∫1)
6 i1ff 24194 . . . . . . . . 9 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
75, 6syl 17 . . . . . . . 8 (𝜑𝐺:ℝ⟶ℝ)
87ffnd 6511 . . . . . . 7 (𝜑𝐺 Fn ℝ)
9 reex 10620 . . . . . . . 8 ℝ ∈ V
109a1i 11 . . . . . . 7 (𝜑 → ℝ ∈ V)
11 inidm 4198 . . . . . . 7 (ℝ ∩ ℝ) = ℝ
124, 8, 10, 10, 11offn 7413 . . . . . 6 (𝜑 → (𝐹f + 𝐺) Fn ℝ)
1312adantr 481 . . . . 5 ((𝜑𝐴 ∈ ℂ) → (𝐹f + 𝐺) Fn ℝ)
14 fniniseg 6825 . . . . 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 722 . . . . . . . 8 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝐺 Fn ℝ)
17 simprl 767 . . . . . . . 8 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ ℝ)
18 fnfvelrn 6843 . . . . . . . 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 769 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → ((𝐹f + 𝐺)‘𝑧) = 𝐴)
21 eqidd 2826 . . . . . . . . . . . . . 14 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) = (𝐹𝑧))
22 eqidd 2826 . . . . . . . . . . . . . 14 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) = (𝐺𝑧))
234, 8, 10, 10, 11, 21, 22ofval 7411 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ℝ) → ((𝐹f + 𝐺)‘𝑧) = ((𝐹𝑧) + (𝐺𝑧)))
2423ad2ant2r 743 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → ((𝐹f + 𝐺)‘𝑧) = ((𝐹𝑧) + (𝐺𝑧)))
2520, 24eqtr3d 2862 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝐴 = ((𝐹𝑧) + (𝐺𝑧)))
2625oveq1d 7166 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐴 − (𝐺𝑧)) = (((𝐹𝑧) + (𝐺𝑧)) − (𝐺𝑧)))
27 ax-resscn 10586 . . . . . . . . . . . . . 14 ℝ ⊆ ℂ
28 fss 6523 . . . . . . . . . . . . . 14 ((𝐹:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:ℝ⟶ℂ)
293, 27, 28sylancl 586 . . . . . . . . . . . . 13 (𝜑𝐹:ℝ⟶ℂ)
3029ad2antrr 722 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝐹:ℝ⟶ℂ)
3130, 17ffvelrnd 6847 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐹𝑧) ∈ ℂ)
32 fss 6523 . . . . . . . . . . . . . 14 ((𝐺:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐺:ℝ⟶ℂ)
337, 27, 32sylancl 586 . . . . . . . . . . . . 13 (𝜑𝐺:ℝ⟶ℂ)
3433ad2antrr 722 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝐺:ℝ⟶ℂ)
3534, 17ffvelrnd 6847 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐺𝑧) ∈ ℂ)
3631, 35pncand 10990 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (((𝐹𝑧) + (𝐺𝑧)) − (𝐺𝑧)) = (𝐹𝑧))
3726, 36eqtr2d 2861 . . . . . . . . 9 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐹𝑧) = (𝐴 − (𝐺𝑧)))
384ad2antrr 722 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝐹 Fn ℝ)
39 fniniseg 6825 . . . . . . . . . 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 709 . . . . . . . 8 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ (𝐹 “ {(𝐴 − (𝐺𝑧))}))
42 eqidd 2826 . . . . . . . . 9 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐺𝑧) = (𝐺𝑧))
43 fniniseg 6825 . . . . . . . . . 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 709 . . . . . . . 8 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ (𝐺 “ {(𝐺𝑧)}))
4641, 45elind 4174 . . . . . . 7 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ ((𝐹 “ {(𝐴 − (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)})))
47 oveq2 7159 . . . . . . . . . . . 12 (𝑦 = (𝐺𝑧) → (𝐴𝑦) = (𝐴 − (𝐺𝑧)))
4847sneqd 4575 . . . . . . . . . . 11 (𝑦 = (𝐺𝑧) → {(𝐴𝑦)} = {(𝐴 − (𝐺𝑧))})
4948imaeq2d 5926 . . . . . . . . . 10 (𝑦 = (𝐺𝑧) → (𝐹 “ {(𝐴𝑦)}) = (𝐹 “ {(𝐴 − (𝐺𝑧))}))
50 sneq 4573 . . . . . . . . . . 11 (𝑦 = (𝐺𝑧) → {𝑦} = {(𝐺𝑧)})
5150imaeq2d 5926 . . . . . . . . . 10 (𝑦 = (𝐺𝑧) → (𝐺 “ {𝑦}) = (𝐺 “ {(𝐺𝑧)}))
5249, 51ineq12d 4193 . . . . . . . . 9 (𝑦 = (𝐺𝑧) → ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) = ((𝐹 “ {(𝐴 − (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)})))
5352eleq2d 2902 . . . . . . . 8 (𝑦 = (𝐺𝑧) → (𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ 𝑧 ∈ ((𝐹 “ {(𝐴 − (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)}))))
5453rspcev 3626 . . . . . . 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 413 . . . . 5 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴) → ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦}))))
57 elin 4172 . . . . . . 7 (𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ (𝑧 ∈ (𝐹 “ {(𝐴𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})))
584adantr 481 . . . . . . . . . 10 ((𝜑𝐴 ∈ ℂ) → 𝐹 Fn ℝ)
59 fniniseg 6825 . . . . . . . . . 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 481 . . . . . . . . . 10 ((𝜑𝐴 ∈ ℂ) → 𝐺 Fn ℝ)
62 fniniseg 6825 . . . . . . . . . 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 630 . . . . . . . 8 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ (𝐹 “ {(𝐴𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})) ↔ ((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦))))
65 anandi 672 . . . . . . . . 9 ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦)) ↔ ((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)))
66 simprl 767 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → 𝑧 ∈ ℝ)
6723ad2ant2r 743 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → ((𝐹f + 𝐺)‘𝑧) = ((𝐹𝑧) + (𝐺𝑧)))
68 simprrl 777 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → (𝐹𝑧) = (𝐴𝑦))
69 simprrr 778 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → (𝐺𝑧) = 𝑦)
7068, 69oveq12d 7169 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → ((𝐹𝑧) + (𝐺𝑧)) = ((𝐴𝑦) + 𝑦))
71 simplr 765 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → 𝐴 ∈ ℂ)
7233ad2antrr 722 . . . . . . . . . . . . . . 15 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → 𝐺:ℝ⟶ℂ)
7372, 66ffvelrnd 6847 . . . . . . . . . . . . . 14 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → (𝐺𝑧) ∈ ℂ)
7469, 73eqeltrrd 2918 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → 𝑦 ∈ ℂ)
7571, 74npcand 10993 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → ((𝐴𝑦) + 𝑦) = 𝐴)
7667, 70, 753eqtrd 2864 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → ((𝐹f + 𝐺)‘𝑧) = 𝐴)
7766, 76jca 512 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴))
7877ex 413 . . . . . . . . 9 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)))
7965, 78syl5bir 244 . . . . . . . 8 ((𝜑𝐴 ∈ ℂ) → (((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)))
8064, 79sylbid 241 . . . . . . 7 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ (𝐹 “ {(𝐴𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)))
8157, 80syl5bi 243 . . . . . 6 ((𝜑𝐴 ∈ ℂ) → (𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)))
8281rexlimdvw 3294 . . . . 5 ((𝜑𝐴 ∈ ℂ) → (∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)))
8356, 82impbid 213 . . . 4 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦}))))
8415, 83bitrd 280 . . 3 ((𝜑𝐴 ∈ ℂ) → (𝑧 ∈ ((𝐹f + 𝐺) “ {𝐴}) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦}))))
85 eliun 4920 . . 3 (𝑧 𝑦 ∈ ran 𝐺((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})))
8684, 85syl6bbr 290 . 2 ((𝜑𝐴 ∈ ℂ) → (𝑧 ∈ ((𝐹f + 𝐺) “ {𝐴}) ↔ 𝑧 𝑦 ∈ ran 𝐺((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦}))))
8786eqrdv 2823 1 ((𝜑𝐴 ∈ ℂ) → ((𝐹f + 𝐺) “ {𝐴}) = 𝑦 ∈ ran 𝐺((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wrex 3143  Vcvv 3499  cin 3938  wss 3939  {csn 4563   ciun 4916  ccnv 5552  dom cdm 5553  ran crn 5554  cima 5556   Fn wfn 6346  wf 6347  cfv 6351  (class class class)co 7151  f cof 7400  cc 10527  cr 10528   + caddc 10532  cmin 10862  1citg1 24133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-po 5472  df-so 5473  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-ltxr 10672  df-sub 10864  df-sum 15036  df-itg1 24138
This theorem is referenced by:  i1fadd  24213  itg1addlem4  24217
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