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Theorem i1faddlem 23366
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 23349 . . . . . . . . 9 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
31, 2syl 17 . . . . . . . 8 (𝜑𝐹:ℝ⟶ℝ)
4 ffn 6002 . . . . . . . 8 (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ)
53, 4syl 17 . . . . . . 7 (𝜑𝐹 Fn ℝ)
6 i1fadd.2 . . . . . . . . 9 (𝜑𝐺 ∈ dom ∫1)
7 i1ff 23349 . . . . . . . . 9 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
86, 7syl 17 . . . . . . . 8 (𝜑𝐺:ℝ⟶ℝ)
9 ffn 6002 . . . . . . . 8 (𝐺:ℝ⟶ℝ → 𝐺 Fn ℝ)
108, 9syl 17 . . . . . . 7 (𝜑𝐺 Fn ℝ)
11 reex 9971 . . . . . . . 8 ℝ ∈ V
1211a1i 11 . . . . . . 7 (𝜑 → ℝ ∈ V)
13 inidm 3800 . . . . . . 7 (ℝ ∩ ℝ) = ℝ
145, 10, 12, 12, 13offn 6861 . . . . . 6 (𝜑 → (𝐹𝑓 + 𝐺) Fn ℝ)
1514adantr 481 . . . . 5 ((𝜑𝐴 ∈ ℂ) → (𝐹𝑓 + 𝐺) Fn ℝ)
16 fniniseg 6294 . . . . 5 ((𝐹𝑓 + 𝐺) Fn ℝ → (𝑧 ∈ ((𝐹𝑓 + 𝐺) “ {𝐴}) ↔ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)))
1715, 16syl 17 . . . 4 ((𝜑𝐴 ∈ ℂ) → (𝑧 ∈ ((𝐹𝑓 + 𝐺) “ {𝐴}) ↔ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)))
1810ad2antrr 761 . . . . . . . 8 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝐺 Fn ℝ)
19 simprl 793 . . . . . . . 8 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ ℝ)
20 fnfvelrn 6312 . . . . . . . 8 ((𝐺 Fn ℝ ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) ∈ ran 𝐺)
2118, 19, 20syl2anc 692 . . . . . . 7 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝐺𝑧) ∈ ran 𝐺)
22 simprr 795 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)
23 eqidd 2622 . . . . . . . . . . . . . 14 ((𝜑𝑧 ∈ ℝ) → (𝐹𝑧) = (𝐹𝑧))
24 eqidd 2622 . . . . . . . . . . . . . 14 ((𝜑𝑧 ∈ ℝ) → (𝐺𝑧) = (𝐺𝑧))
255, 10, 12, 12, 13, 23, 24ofval 6859 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ℝ) → ((𝐹𝑓 + 𝐺)‘𝑧) = ((𝐹𝑧) + (𝐺𝑧)))
2625ad2ant2r 782 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → ((𝐹𝑓 + 𝐺)‘𝑧) = ((𝐹𝑧) + (𝐺𝑧)))
2722, 26eqtr3d 2657 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝐴 = ((𝐹𝑧) + (𝐺𝑧)))
2827oveq1d 6619 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝐴 − (𝐺𝑧)) = (((𝐹𝑧) + (𝐺𝑧)) − (𝐺𝑧)))
29 ax-resscn 9937 . . . . . . . . . . . . . 14 ℝ ⊆ ℂ
30 fss 6013 . . . . . . . . . . . . . 14 ((𝐹:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:ℝ⟶ℂ)
313, 29, 30sylancl 693 . . . . . . . . . . . . 13 (𝜑𝐹:ℝ⟶ℂ)
3231ad2antrr 761 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝐹:ℝ⟶ℂ)
3332, 19ffvelrnd 6316 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝐹𝑧) ∈ ℂ)
34 fss 6013 . . . . . . . . . . . . . 14 ((𝐺:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐺:ℝ⟶ℂ)
358, 29, 34sylancl 693 . . . . . . . . . . . . 13 (𝜑𝐺:ℝ⟶ℂ)
3635ad2antrr 761 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝐺:ℝ⟶ℂ)
3736, 19ffvelrnd 6316 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝐺𝑧) ∈ ℂ)
3833, 37pncand 10337 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → (((𝐹𝑧) + (𝐺𝑧)) − (𝐺𝑧)) = (𝐹𝑧))
3928, 38eqtr2d 2656 . . . . . . . . 9 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝐹𝑧) = (𝐴 − (𝐺𝑧)))
405ad2antrr 761 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝐹 Fn ℝ)
41 fniniseg 6294 . . . . . . . . . 10 (𝐹 Fn ℝ → (𝑧 ∈ (𝐹 “ {(𝐴 − (𝐺𝑧))}) ↔ (𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴 − (𝐺𝑧)))))
4240, 41syl 17 . . . . . . . . 9 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝑧 ∈ (𝐹 “ {(𝐴 − (𝐺𝑧))}) ↔ (𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴 − (𝐺𝑧)))))
4319, 39, 42mpbir2and 956 . . . . . . . 8 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ (𝐹 “ {(𝐴 − (𝐺𝑧))}))
44 eqidd 2622 . . . . . . . . 9 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝐺𝑧) = (𝐺𝑧))
45 fniniseg 6294 . . . . . . . . . 10 (𝐺 Fn ℝ → (𝑧 ∈ (𝐺 “ {(𝐺𝑧)}) ↔ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = (𝐺𝑧))))
4618, 45syl 17 . . . . . . . . 9 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝑧 ∈ (𝐺 “ {(𝐺𝑧)}) ↔ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = (𝐺𝑧))))
4719, 44, 46mpbir2and 956 . . . . . . . 8 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ (𝐺 “ {(𝐺𝑧)}))
4843, 47elind 3776 . . . . . . 7 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ ((𝐹 “ {(𝐴 − (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)})))
49 oveq2 6612 . . . . . . . . . . . 12 (𝑦 = (𝐺𝑧) → (𝐴𝑦) = (𝐴 − (𝐺𝑧)))
5049sneqd 4160 . . . . . . . . . . 11 (𝑦 = (𝐺𝑧) → {(𝐴𝑦)} = {(𝐴 − (𝐺𝑧))})
5150imaeq2d 5425 . . . . . . . . . 10 (𝑦 = (𝐺𝑧) → (𝐹 “ {(𝐴𝑦)}) = (𝐹 “ {(𝐴 − (𝐺𝑧))}))
52 sneq 4158 . . . . . . . . . . 11 (𝑦 = (𝐺𝑧) → {𝑦} = {(𝐺𝑧)})
5352imaeq2d 5425 . . . . . . . . . 10 (𝑦 = (𝐺𝑧) → (𝐺 “ {𝑦}) = (𝐺 “ {(𝐺𝑧)}))
5451, 53ineq12d 3793 . . . . . . . . 9 (𝑦 = (𝐺𝑧) → ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) = ((𝐹 “ {(𝐴 − (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)})))
5554eleq2d 2684 . . . . . . . 8 (𝑦 = (𝐺𝑧) → (𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ 𝑧 ∈ ((𝐹 “ {(𝐴 − (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)}))))
5655rspcev 3295 . . . . . . 7 (((𝐺𝑧) ∈ ran 𝐺𝑧 ∈ ((𝐹 “ {(𝐴 − (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)}))) → ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})))
5721, 48, 56syl2anc 692 . . . . . 6 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)) → ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})))
5857ex 450 . . . . 5 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴) → ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦}))))
59 elin 3774 . . . . . . 7 (𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ (𝑧 ∈ (𝐹 “ {(𝐴𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})))
605adantr 481 . . . . . . . . . 10 ((𝜑𝐴 ∈ ℂ) → 𝐹 Fn ℝ)
61 fniniseg 6294 . . . . . . . . . 10 (𝐹 Fn ℝ → (𝑧 ∈ (𝐹 “ {(𝐴𝑦)}) ↔ (𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴𝑦))))
6260, 61syl 17 . . . . . . . . 9 ((𝜑𝐴 ∈ ℂ) → (𝑧 ∈ (𝐹 “ {(𝐴𝑦)}) ↔ (𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴𝑦))))
6310adantr 481 . . . . . . . . . 10 ((𝜑𝐴 ∈ ℂ) → 𝐺 Fn ℝ)
64 fniniseg 6294 . . . . . . . . . 10 (𝐺 Fn ℝ → (𝑧 ∈ (𝐺 “ {𝑦}) ↔ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)))
6563, 64syl 17 . . . . . . . . 9 ((𝜑𝐴 ∈ ℂ) → (𝑧 ∈ (𝐺 “ {𝑦}) ↔ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)))
6662, 65anbi12d 746 . . . . . . . 8 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ (𝐹 “ {(𝐴𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})) ↔ ((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦))))
67 anandi 870 . . . . . . . . 9 ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦)) ↔ ((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)))
68 simprl 793 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → 𝑧 ∈ ℝ)
6925ad2ant2r 782 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → ((𝐹𝑓 + 𝐺)‘𝑧) = ((𝐹𝑧) + (𝐺𝑧)))
70 simprrl 803 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → (𝐹𝑧) = (𝐴𝑦))
71 simprrr 804 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → (𝐺𝑧) = 𝑦)
7270, 71oveq12d 6622 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → ((𝐹𝑧) + (𝐺𝑧)) = ((𝐴𝑦) + 𝑦))
73 simplr 791 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → 𝐴 ∈ ℂ)
7435ad2antrr 761 . . . . . . . . . . . . . . 15 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → 𝐺:ℝ⟶ℂ)
7574, 68ffvelrnd 6316 . . . . . . . . . . . . . 14 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → (𝐺𝑧) ∈ ℂ)
7671, 75eqeltrrd 2699 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → 𝑦 ∈ ℂ)
7773, 76npcand 10340 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → ((𝐴𝑦) + 𝑦) = 𝐴)
7869, 72, 773eqtrd 2659 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)
7968, 78jca 554 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦))) → (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴))
8079ex 450 . . . . . . . . 9 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴𝑦) ∧ (𝐺𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)))
8167, 80syl5bir 233 . . . . . . . 8 ((𝜑𝐴 ∈ ℂ) → (((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)))
8266, 81sylbid 230 . . . . . . 7 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ (𝐹 “ {(𝐴𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)))
8359, 82syl5bi 232 . . . . . 6 ((𝜑𝐴 ∈ ℂ) → (𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)))
8483rexlimdvw 3027 . . . . 5 ((𝜑𝐴 ∈ ℂ) → (∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴)))
8558, 84impbid 202 . . . 4 ((𝜑𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹𝑓 + 𝐺)‘𝑧) = 𝐴) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦}))))
8617, 85bitrd 268 . . 3 ((𝜑𝐴 ∈ ℂ) → (𝑧 ∈ ((𝐹𝑓 + 𝐺) “ {𝐴}) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦}))))
87 eliun 4490 . . 3 (𝑧 𝑦 ∈ ran 𝐺((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})))
8886, 87syl6bbr 278 . 2 ((𝜑𝐴 ∈ ℂ) → (𝑧 ∈ ((𝐹𝑓 + 𝐺) “ {𝐴}) ↔ 𝑧 𝑦 ∈ ran 𝐺((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦}))))
8988eqrdv 2619 1 ((𝜑𝐴 ∈ ℂ) → ((𝐹𝑓 + 𝐺) “ {𝐴}) = 𝑦 ∈ ran 𝐺((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3186  cin 3554  wss 3555  {csn 4148   ciun 4485  ccnv 5073  dom cdm 5074  ran crn 5075  cima 5077   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  𝑓 cof 6848  cc 9878  cr 9879   + caddc 9883  cmin 10210  1citg1 23290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-ltxr 10023  df-sub 10212  df-sum 14351  df-itg1 23295
This theorem is referenced by:  i1fadd  23368  itg1addlem4  23372
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