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Theorem i1faddlem 25057
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 25040 . . . . . . . . 9 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
31, 2syl 17 . . . . . . . 8 (𝜑𝐹:ℝ⟶ℝ)
43ffnd 6669 . . . . . . 7 (𝜑𝐹 Fn ℝ)
5 i1fadd.2 . . . . . . . . 9 (𝜑𝐺 ∈ dom ∫1)
6 i1ff 25040 . . . . . . . . 9 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
75, 6syl 17 . . . . . . . 8 (𝜑𝐺:ℝ⟶ℝ)
87ffnd 6669 . . . . . . 7 (𝜑𝐺 Fn ℝ)
9 reex 11142 . . . . . . . 8 ℝ ∈ V
109a1i 11 . . . . . . 7 (𝜑 → ℝ ∈ V)
11 inidm 4178 . . . . . . 7 (ℝ ∩ ℝ) = ℝ
124, 8, 10, 10, 11offn 7630 . . . . . 6 (𝜑 → (𝐹f + 𝐺) Fn ℝ)
1312adantr 481 . . . . 5 ((𝜑𝐴 ∈ ℂ) → (𝐹f + 𝐺) Fn ℝ)
14 fniniseg 7010 . . . . 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 724 . . . . . . . 8 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝐺 Fn ℝ)
17 simprl 769 . . . . . . . 8 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ ℝ)
18 fnfvelrn 7031 . . . . . . . 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 771 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → ((𝐹f + 𝐺)‘𝑧) = 𝐴)
21 eqidd 2737 . . . . . . . . . . . . . 14 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) = (𝐹𝑧))
22 eqidd 2737 . . . . . . . . . . . . . 14 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) = (𝐺𝑧))
234, 8, 10, 10, 11, 21, 22ofval 7628 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ℝ) → ((𝐹f + 𝐺)‘𝑧) = ((𝐹𝑧) + (𝐺𝑧)))
2423ad2ant2r 745 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → ((𝐹f + 𝐺)‘𝑧) = ((𝐹𝑧) + (𝐺𝑧)))
2520, 24eqtr3d 2778 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝐴 = ((𝐹𝑧) + (𝐺𝑧)))
2625oveq1d 7372 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐴 − (𝐺𝑧)) = (((𝐹𝑧) + (𝐺𝑧)) − (𝐺𝑧)))
27 ax-resscn 11108 . . . . . . . . . . . . . 14 ℝ ⊆ ℂ
28 fss 6685 . . . . . . . . . . . . . 14 ((𝐹:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:ℝ⟶ℂ)
293, 27, 28sylancl 586 . . . . . . . . . . . . 13 (𝜑𝐹:ℝ⟶ℂ)
3029ad2antrr 724 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝐹:ℝ⟶ℂ)
3130, 17ffvelcdmd 7036 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐹𝑧) ∈ ℂ)
32 fss 6685 . . . . . . . . . . . . . 14 ((𝐺:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐺:ℝ⟶ℂ)
337, 27, 32sylancl 586 . . . . . . . . . . . . 13 (𝜑𝐺:ℝ⟶ℂ)
3433ad2antrr 724 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝐺:ℝ⟶ℂ)
3534, 17ffvelcdmd 7036 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐺𝑧) ∈ ℂ)
3631, 35pncand 11513 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (((𝐹𝑧) + (𝐺𝑧)) − (𝐺𝑧)) = (𝐹𝑧))
3726, 36eqtr2d 2777 . . . . . . . . 9 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐹𝑧) = (𝐴 − (𝐺𝑧)))
384ad2antrr 724 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝐹 Fn ℝ)
39 fniniseg 7010 . . . . . . . . . 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 711 . . . . . . . 8 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ (𝐹 “ {(𝐴 − (𝐺𝑧))}))
42 eqidd 2737 . . . . . . . . 9 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → (𝐺𝑧) = (𝐺𝑧))
43 fniniseg 7010 . . . . . . . . . 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 711 . . . . . . . 8 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ (𝐺 “ {(𝐺𝑧)}))
4641, 45elind 4154 . . . . . . 7 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ ((𝐹 “ {(𝐴 − (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)})))
47 oveq2 7365 . . . . . . . . . . . 12 (𝑦 = (𝐺𝑧) → (𝐴𝑦) = (𝐴 − (𝐺𝑧)))
4847sneqd 4598 . . . . . . . . . . 11 (𝑦 = (𝐺𝑧) → {(𝐴𝑦)} = {(𝐴 − (𝐺𝑧))})
4948imaeq2d 6013 . . . . . . . . . 10 (𝑦 = (𝐺𝑧) → (𝐹 “ {(𝐴𝑦)}) = (𝐹 “ {(𝐴 − (𝐺𝑧))}))
50 sneq 4596 . . . . . . . . . . 11 (𝑦 = (𝐺𝑧) → {𝑦} = {(𝐺𝑧)})
5150imaeq2d 6013 . . . . . . . . . 10 (𝑦 = (𝐺𝑧) → (𝐺 “ {𝑦}) = (𝐺 “ {(𝐺𝑧)}))
5249, 51ineq12d 4173 . . . . . . . . 9 (𝑦 = (𝐺𝑧) → ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) = ((𝐹 “ {(𝐴 − (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)})))
5352eleq2d 2823 . . . . . . . 8 (𝑦 = (𝐺𝑧) → (𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ 𝑧 ∈ ((𝐹 “ {(𝐴 − (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)}))))
5453rspcev 3581 . . . . . . 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 3926 . . . . . . 7 (𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ (𝑧 ∈ (𝐹 “ {(𝐴𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})))
584adantr 481 . . . . . . . . . 10 ((𝜑𝐴 ∈ ℂ) → 𝐹 Fn ℝ)
59 fniniseg 7010 . . . . . . . . . 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 7010 . . . . . . . . . 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 631 . . . . . . . 8 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ (𝐹 “ {(𝐴𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})) ↔ ((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦))))
65 anandi 674 . . . . . . . . 9 ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦)) ↔ ((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)))
66 simprl 769 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → 𝑧 ∈ ℝ)
6723ad2ant2r 745 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → ((𝐹f + 𝐺)‘𝑧) = ((𝐹𝑧) + (𝐺𝑧)))
68 simprrl 779 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → (𝐹𝑧) = (𝐴𝑦))
69 simprrr 780 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → (𝐺𝑧) = 𝑦)
7068, 69oveq12d 7375 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → ((𝐹𝑧) + (𝐺𝑧)) = ((𝐴𝑦) + 𝑦))
71 simplr 767 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → 𝐴 ∈ ℂ)
7233ad2antrr 724 . . . . . . . . . . . . . . 15 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → 𝐺:ℝ⟶ℂ)
7372, 66ffvelcdmd 7036 . . . . . . . . . . . . . 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 11516 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → ((𝐴𝑦) + 𝑦) = 𝐴)
7667, 70, 753eqtrd 2780 . . . . . . . . . . 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, 78biimtrrid 242 . . . . . . . 8 ((𝜑𝐴 ∈ ℂ) → (((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)))
8064, 79sylbid 239 . . . . . . 7 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ (𝐹 “ {(𝐴𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴)))
8157, 80biimtrid 241 . . . . . 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 211 . . . 4 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹f + 𝐺)‘𝑧) = 𝐴) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦}))))
8415, 83bitrd 278 . . 3 ((𝜑𝐴 ∈ ℂ) → (𝑧 ∈ ((𝐹f + 𝐺) “ {𝐴}) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦}))))
85 eliun 4958 . . 3 (𝑧 𝑦 ∈ ran 𝐺((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})))
8684, 85bitr4di 288 . 2 ((𝜑𝐴 ∈ ℂ) → (𝑧 ∈ ((𝐹f + 𝐺) “ {𝐴}) ↔ 𝑧 𝑦 ∈ ran 𝐺((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦}))))
8786eqrdv 2734 1 ((𝜑𝐴 ∈ ℂ) → ((𝐹f + 𝐺) “ {𝐴}) = 𝑦 ∈ ran 𝐺((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3073  Vcvv 3445  cin 3909  wss 3910  {csn 4586   ciun 4954  ccnv 5632  dom cdm 5633  ran crn 5634  cima 5636   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  f cof 7615  cc 11049  cr 11050   + caddc 11054  cmin 11385  1citg1 24979
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-ltxr 11194  df-sub 11387  df-sum 15571  df-itg1 24984
This theorem is referenced by:  i1fadd  25059  itg1addlem4  25063  itg1addlem4OLD  25064
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