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Theorem i1fmullem 24312
 Description: Decompose the preimage of a product. (Contributed by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
Assertion
Ref Expression
i1fmullem ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ((𝐹f · 𝐺) “ {𝐴}) = 𝑦 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹   𝑦,𝐺   𝜑,𝑦

Proof of Theorem i1fmullem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 i1fadd.1 . . . . . . . . 9 (𝜑𝐹 ∈ dom ∫1)
2 i1ff 24294 . . . . . . . . 9 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
31, 2syl 17 . . . . . . . 8 (𝜑𝐹:ℝ⟶ℝ)
43ffnd 6491 . . . . . . 7 (𝜑𝐹 Fn ℝ)
5 i1fadd.2 . . . . . . . . 9 (𝜑𝐺 ∈ dom ∫1)
6 i1ff 24294 . . . . . . . . 9 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
75, 6syl 17 . . . . . . . 8 (𝜑𝐺:ℝ⟶ℝ)
87ffnd 6491 . . . . . . 7 (𝜑𝐺 Fn ℝ)
9 reex 10624 . . . . . . . 8 ℝ ∈ V
109a1i 11 . . . . . . 7 (𝜑 → ℝ ∈ V)
11 inidm 4145 . . . . . . 7 (ℝ ∩ ℝ) = ℝ
124, 8, 10, 10, 11offn 7406 . . . . . 6 (𝜑 → (𝐹f · 𝐺) Fn ℝ)
1312adantr 484 . . . . 5 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (𝐹f · 𝐺) Fn ℝ)
14 fniniseg 6812 . . . . 5 ((𝐹f · 𝐺) Fn ℝ → (𝑧 ∈ ((𝐹f · 𝐺) “ {𝐴}) ↔ (𝑧 ∈ ℝ ∧ ((𝐹f · 𝐺)‘𝑧) = 𝐴)))
1513, 14syl 17 . . . 4 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ ((𝐹f · 𝐺) “ {𝐴}) ↔ (𝑧 ∈ ℝ ∧ ((𝐹f · 𝐺)‘𝑧) = 𝐴)))
164adantr 484 . . . . . . 7 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → 𝐹 Fn ℝ)
178adantr 484 . . . . . . 7 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → 𝐺 Fn ℝ)
189a1i 11 . . . . . . 7 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ℝ ∈ V)
19 eqidd 2799 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) = (𝐹𝑧))
20 eqidd 2799 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) = (𝐺𝑧))
2116, 17, 18, 18, 11, 19, 20ofval 7404 . . . . . 6 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → ((𝐹f · 𝐺)‘𝑧) = ((𝐹𝑧) · (𝐺𝑧)))
2221eqeq1d 2800 . . . . 5 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (((𝐹f · 𝐺)‘𝑧) = 𝐴 ↔ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴))
2322pm5.32da 582 . . . 4 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹f · 𝐺)‘𝑧) = 𝐴) ↔ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)))
248ad2antrr 725 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝐺 Fn ℝ)
25 simprl 770 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝑧 ∈ ℝ)
26 fnfvelrn 6830 . . . . . . . . 9 ((𝐺 Fn ℝ ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) ∈ ran 𝐺)
2724, 25, 26syl2anc 587 . . . . . . . 8 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) ∈ ran 𝐺)
28 eldifsni 4683 . . . . . . . . . . 11 (𝐴 ∈ (ℂ ∖ {0}) → 𝐴 ≠ 0)
2928ad2antlr 726 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝐴 ≠ 0)
30 simprr 772 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)
313ad2antrr 725 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝐹:ℝ⟶ℝ)
3231, 25ffvelrnd 6834 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐹𝑧) ∈ ℝ)
3332recnd 10665 . . . . . . . . . . 11 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐹𝑧) ∈ ℂ)
3433mul01d 10835 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → ((𝐹𝑧) · 0) = 0)
3529, 30, 343netr4d 3064 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → ((𝐹𝑧) · (𝐺𝑧)) ≠ ((𝐹𝑧) · 0))
36 oveq2 7148 . . . . . . . . . 10 ((𝐺𝑧) = 0 → ((𝐹𝑧) · (𝐺𝑧)) = ((𝐹𝑧) · 0))
3736necon3i 3019 . . . . . . . . 9 (((𝐹𝑧) · (𝐺𝑧)) ≠ ((𝐹𝑧) · 0) → (𝐺𝑧) ≠ 0)
3835, 37syl 17 . . . . . . . 8 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) ≠ 0)
39 eldifsn 4680 . . . . . . . 8 ((𝐺𝑧) ∈ (ran 𝐺 ∖ {0}) ↔ ((𝐺𝑧) ∈ ran 𝐺 ∧ (𝐺𝑧) ≠ 0))
4027, 38, 39sylanbrc 586 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) ∈ (ran 𝐺 ∖ {0}))
417ad2antrr 725 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝐺:ℝ⟶ℝ)
4241, 25ffvelrnd 6834 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) ∈ ℝ)
4342recnd 10665 . . . . . . . . . . 11 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) ∈ ℂ)
4433, 43, 38divcan4d 11418 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (((𝐹𝑧) · (𝐺𝑧)) / (𝐺𝑧)) = (𝐹𝑧))
4530oveq1d 7155 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (((𝐹𝑧) · (𝐺𝑧)) / (𝐺𝑧)) = (𝐴 / (𝐺𝑧)))
4644, 45eqtr3d 2835 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐹𝑧) = (𝐴 / (𝐺𝑧)))
4731ffnd 6491 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝐹 Fn ℝ)
48 fniniseg 6812 . . . . . . . . . 10 (𝐹 Fn ℝ → (𝑧 ∈ (𝐹 “ {(𝐴 / (𝐺𝑧))}) ↔ (𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴 / (𝐺𝑧)))))
4947, 48syl 17 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝑧 ∈ (𝐹 “ {(𝐴 / (𝐺𝑧))}) ↔ (𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴 / (𝐺𝑧)))))
5025, 46, 49mpbir2and 712 . . . . . . . 8 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝑧 ∈ (𝐹 “ {(𝐴 / (𝐺𝑧))}))
51 eqidd 2799 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) = (𝐺𝑧))
52 fniniseg 6812 . . . . . . . . . 10 (𝐺 Fn ℝ → (𝑧 ∈ (𝐺 “ {(𝐺𝑧)}) ↔ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = (𝐺𝑧))))
5324, 52syl 17 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝑧 ∈ (𝐺 “ {(𝐺𝑧)}) ↔ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = (𝐺𝑧))))
5425, 51, 53mpbir2and 712 . . . . . . . 8 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝑧 ∈ (𝐺 “ {(𝐺𝑧)}))
5550, 54elind 4121 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝑧 ∈ ((𝐹 “ {(𝐴 / (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)})))
56 oveq2 7148 . . . . . . . . . . . 12 (𝑦 = (𝐺𝑧) → (𝐴 / 𝑦) = (𝐴 / (𝐺𝑧)))
5756sneqd 4537 . . . . . . . . . . 11 (𝑦 = (𝐺𝑧) → {(𝐴 / 𝑦)} = {(𝐴 / (𝐺𝑧))})
5857imaeq2d 5897 . . . . . . . . . 10 (𝑦 = (𝐺𝑧) → (𝐹 “ {(𝐴 / 𝑦)}) = (𝐹 “ {(𝐴 / (𝐺𝑧))}))
59 sneq 4535 . . . . . . . . . . 11 (𝑦 = (𝐺𝑧) → {𝑦} = {(𝐺𝑧)})
6059imaeq2d 5897 . . . . . . . . . 10 (𝑦 = (𝐺𝑧) → (𝐺 “ {𝑦}) = (𝐺 “ {(𝐺𝑧)}))
6158, 60ineq12d 4140 . . . . . . . . 9 (𝑦 = (𝐺𝑧) → ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) = ((𝐹 “ {(𝐴 / (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)})))
6261eleq2d 2875 . . . . . . . 8 (𝑦 = (𝐺𝑧) → (𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ 𝑧 ∈ ((𝐹 “ {(𝐴 / (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)}))))
6362rspcev 3571 . . . . . . 7 (((𝐺𝑧) ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ((𝐹 “ {(𝐴 / (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)}))) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})))
6440, 55, 63syl2anc 587 . . . . . 6 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})))
6564ex 416 . . . . 5 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦}))))
66 fniniseg 6812 . . . . . . . . . . 11 (𝐹 Fn ℝ → (𝑧 ∈ (𝐹 “ {(𝐴 / 𝑦)}) ↔ (𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴 / 𝑦))))
6716, 66syl 17 . . . . . . . . . 10 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (𝐹 “ {(𝐴 / 𝑦)}) ↔ (𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴 / 𝑦))))
68 fniniseg 6812 . . . . . . . . . . 11 (𝐺 Fn ℝ → (𝑧 ∈ (𝐺 “ {𝑦}) ↔ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)))
6917, 68syl 17 . . . . . . . . . 10 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (𝐺 “ {𝑦}) ↔ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)))
7067, 69anbi12d 633 . . . . . . . . 9 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ (𝐹 “ {(𝐴 / 𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})) ↔ ((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴 / 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦))))
71 elin 3897 . . . . . . . . 9 (𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ (𝑧 ∈ (𝐹 “ {(𝐴 / 𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})))
72 anandi 675 . . . . . . . . 9 ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦)) ↔ ((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴 / 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)))
7370, 71, 723bitr4g 317 . . . . . . . 8 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦))))
7473adantr 484 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) → (𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦))))
75 eldifi 4054 . . . . . . . . . . . 12 (𝐴 ∈ (ℂ ∖ {0}) → 𝐴 ∈ ℂ)
7675ad2antlr 726 . . . . . . . . . . 11 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝐴 ∈ ℂ)
777ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝐺:ℝ⟶ℝ)
7877frnd 6497 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → ran 𝐺 ⊆ ℝ)
79 simprl 770 . . . . . . . . . . . . . . 15 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ (ran 𝐺 ∖ {0}))
80 eldifsn 4680 . . . . . . . . . . . . . . 15 (𝑦 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑦 ∈ ran 𝐺𝑦 ≠ 0))
8179, 80sylib 221 . . . . . . . . . . . . . 14 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → (𝑦 ∈ ran 𝐺𝑦 ≠ 0))
8281simpld 498 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ran 𝐺)
8378, 82sseldd 3916 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ℝ)
8483recnd 10665 . . . . . . . . . . 11 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ℂ)
8581simprd 499 . . . . . . . . . . 11 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ≠ 0)
8676, 84, 85divcan1d 11413 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → ((𝐴 / 𝑦) · 𝑦) = 𝐴)
87 oveq12 7149 . . . . . . . . . . 11 (((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦) → ((𝐹𝑧) · (𝐺𝑧)) = ((𝐴 / 𝑦) · 𝑦))
8887eqeq1d 2800 . . . . . . . . . 10 (((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦) → (((𝐹𝑧) · (𝐺𝑧)) = 𝐴 ↔ ((𝐴 / 𝑦) · 𝑦) = 𝐴))
8986, 88syl5ibrcom 250 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → (((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦) → ((𝐹𝑧) · (𝐺𝑧)) = 𝐴))
9089anassrs 471 . . . . . . . 8 ((((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑧 ∈ ℝ) → (((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦) → ((𝐹𝑧) · (𝐺𝑧)) = 𝐴))
9190imdistanda 575 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)))
9274, 91sylbid 243 . . . . . 6 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) → (𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)))
9392rexlimdva 3243 . . . . 5 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)))
9465, 93impbid 215 . . . 4 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦}))))
9515, 23, 943bitrd 308 . . 3 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ ((𝐹f · 𝐺) “ {𝐴}) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦}))))
96 eliun 4886 . . 3 (𝑧 𝑦 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})))
9795, 96bitr4di 292 . 2 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ ((𝐹f · 𝐺) “ {𝐴}) ↔ 𝑧 𝑦 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦}))))
9897eqrdv 2796 1 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ((𝐹f · 𝐺) “ {𝐴}) = 𝑦 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∃wrex 3107  Vcvv 3441   ∖ cdif 3878   ∩ cin 3880  {csn 4525  ∪ ciun 4882  ◡ccnv 5519  dom cdm 5520  ran crn 5521   “ cima 5523   Fn wfn 6322  ⟶wf 6323  ‘cfv 6327  (class class class)co 7140   ∘f cof 7393  ℂcc 10531  ℝcr 10532  0cc0 10533   · cmul 10538   / cdiv 11293  ∫1citg1 24233 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 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448  ax-cnex 10589  ax-resscn 10590  ax-1cn 10591  ax-icn 10592  ax-addcl 10593  ax-addrcl 10594  ax-mulcl 10595  ax-mulrcl 10596  ax-mulcom 10597  ax-addass 10598  ax-mulass 10599  ax-distr 10600  ax-i2m1 10601  ax-1ne0 10602  ax-1rid 10603  ax-rnegex 10604  ax-rrecex 10605  ax-cnre 10606  ax-pre-lttri 10607  ax-pre-lttrn 10608  ax-pre-ltadd 10609  ax-pre-mulgt0 10610 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-po 5439  df-so 5440  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-of 7395  df-er 8279  df-en 8500  df-dom 8501  df-sdom 8502  df-pnf 10673  df-mnf 10674  df-xr 10675  df-ltxr 10676  df-le 10677  df-sub 10868  df-neg 10869  df-div 11294  df-sum 15042  df-itg1 24238 This theorem is referenced by:  i1fmul  24314
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