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Theorem i1fmullem 25742
Description: Decompose the preimage of a product. (Contributed by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
i1fadd.1 (𝜑𝐹 ∈ dom ∫1)
i1fadd.2 (𝜑𝐺 ∈ dom ∫1)
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 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 ∖ {0})) → (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 ∖ {0})) → (𝑧 ∈ ((𝐹f · 𝐺) “ {𝐴}) ↔ (𝑧 ∈ ℝ ∧ ((𝐹f · 𝐺)‘𝑧) = 𝐴)))
164adantr 480 . . . . . . 7 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → 𝐹 Fn ℝ)
178adantr 480 . . . . . . 7 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → 𝐺 Fn ℝ)
189a1i 11 . . . . . . 7 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ℝ ∈ V)
19 eqidd 2735 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) = (𝐹𝑧))
20 eqidd 2735 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) = (𝐺𝑧))
2116, 17, 18, 18, 11, 19, 20ofval 7707 . . . . . 6 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → ((𝐹f · 𝐺)‘𝑧) = ((𝐹𝑧) · (𝐺𝑧)))
2221eqeq1d 2736 . . . . 5 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (((𝐹f · 𝐺)‘𝑧) = 𝐴 ↔ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴))
2322pm5.32da 579 . . . 4 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹f · 𝐺)‘𝑧) = 𝐴) ↔ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)))
248ad2antrr 726 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝐺 Fn ℝ)
25 simprl 771 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝑧 ∈ ℝ)
26 fnfvelrn 7099 . . . . . . . . 9 ((𝐺 Fn ℝ ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) ∈ ran 𝐺)
2724, 25, 26syl2anc 584 . . . . . . . 8 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) ∈ ran 𝐺)
28 eldifsni 4794 . . . . . . . . . . 11 (𝐴 ∈ (ℂ ∖ {0}) → 𝐴 ≠ 0)
2928ad2antlr 727 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝐴 ≠ 0)
30 simprr 773 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)
313ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝐹:ℝ⟶ℝ)
3231, 25ffvelcdmd 7104 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐹𝑧) ∈ ℝ)
3332recnd 11286 . . . . . . . . . . 11 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐹𝑧) ∈ ℂ)
3433mul01d 11457 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → ((𝐹𝑧) · 0) = 0)
3529, 30, 343netr4d 3015 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → ((𝐹𝑧) · (𝐺𝑧)) ≠ ((𝐹𝑧) · 0))
36 oveq2 7438 . . . . . . . . . 10 ((𝐺𝑧) = 0 → ((𝐹𝑧) · (𝐺𝑧)) = ((𝐹𝑧) · 0))
3736necon3i 2970 . . . . . . . . 9 (((𝐹𝑧) · (𝐺𝑧)) ≠ ((𝐹𝑧) · 0) → (𝐺𝑧) ≠ 0)
3835, 37syl 17 . . . . . . . 8 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) ≠ 0)
39 eldifsn 4790 . . . . . . . 8 ((𝐺𝑧) ∈ (ran 𝐺 ∖ {0}) ↔ ((𝐺𝑧) ∈ ran 𝐺 ∧ (𝐺𝑧) ≠ 0))
4027, 38, 39sylanbrc 583 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) ∈ (ran 𝐺 ∖ {0}))
417ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝐺:ℝ⟶ℝ)
4241, 25ffvelcdmd 7104 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) ∈ ℝ)
4342recnd 11286 . . . . . . . . . . 11 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) ∈ ℂ)
4433, 43, 38divcan4d 12046 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (((𝐹𝑧) · (𝐺𝑧)) / (𝐺𝑧)) = (𝐹𝑧))
4530oveq1d 7445 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (((𝐹𝑧) · (𝐺𝑧)) / (𝐺𝑧)) = (𝐴 / (𝐺𝑧)))
4644, 45eqtr3d 2776 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐹𝑧) = (𝐴 / (𝐺𝑧)))
4731ffnd 6737 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝐹 Fn ℝ)
48 fniniseg 7079 . . . . . . . . . 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 713 . . . . . . . 8 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝑧 ∈ (𝐹 “ {(𝐴 / (𝐺𝑧))}))
51 eqidd 2735 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) = (𝐺𝑧))
52 fniniseg 7079 . . . . . . . . . 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 713 . . . . . . . 8 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝑧 ∈ (𝐺 “ {(𝐺𝑧)}))
5550, 54elind 4209 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝑧 ∈ ((𝐹 “ {(𝐴 / (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)})))
56 oveq2 7438 . . . . . . . . . . . 12 (𝑦 = (𝐺𝑧) → (𝐴 / 𝑦) = (𝐴 / (𝐺𝑧)))
5756sneqd 4642 . . . . . . . . . . 11 (𝑦 = (𝐺𝑧) → {(𝐴 / 𝑦)} = {(𝐴 / (𝐺𝑧))})
5857imaeq2d 6079 . . . . . . . . . 10 (𝑦 = (𝐺𝑧) → (𝐹 “ {(𝐴 / 𝑦)}) = (𝐹 “ {(𝐴 / (𝐺𝑧))}))
59 sneq 4640 . . . . . . . . . . 11 (𝑦 = (𝐺𝑧) → {𝑦} = {(𝐺𝑧)})
6059imaeq2d 6079 . . . . . . . . . 10 (𝑦 = (𝐺𝑧) → (𝐺 “ {𝑦}) = (𝐺 “ {(𝐺𝑧)}))
6158, 60ineq12d 4228 . . . . . . . . 9 (𝑦 = (𝐺𝑧) → ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) = ((𝐹 “ {(𝐴 / (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)})))
6261eleq2d 2824 . . . . . . . 8 (𝑦 = (𝐺𝑧) → (𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ 𝑧 ∈ ((𝐹 “ {(𝐴 / (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)}))))
6362rspcev 3621 . . . . . . 7 (((𝐺𝑧) ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ((𝐹 “ {(𝐴 / (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)}))) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})))
6440, 55, 63syl2anc 584 . . . . . 6 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})))
6564ex 412 . . . . 5 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦}))))
66 fniniseg 7079 . . . . . . . . . . 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 7079 . . . . . . . . . . 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 632 . . . . . . . . 9 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ (𝐹 “ {(𝐴 / 𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})) ↔ ((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴 / 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦))))
71 elin 3978 . . . . . . . . 9 (𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ (𝑧 ∈ (𝐹 “ {(𝐴 / 𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})))
72 anandi 676 . . . . . . . . 9 ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦)) ↔ ((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴 / 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)))
7370, 71, 723bitr4g 314 . . . . . . . 8 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦))))
7473adantr 480 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) → (𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦))))
75 eldifi 4140 . . . . . . . . . . . 12 (𝐴 ∈ (ℂ ∖ {0}) → 𝐴 ∈ ℂ)
7675ad2antlr 727 . . . . . . . . . . 11 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝐴 ∈ ℂ)
777ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝐺:ℝ⟶ℝ)
7877frnd 6744 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → ran 𝐺 ⊆ ℝ)
79 simprl 771 . . . . . . . . . . . . . . 15 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ (ran 𝐺 ∖ {0}))
80 eldifsn 4790 . . . . . . . . . . . . . . 15 (𝑦 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑦 ∈ ran 𝐺𝑦 ≠ 0))
8179, 80sylib 218 . . . . . . . . . . . . . 14 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → (𝑦 ∈ ran 𝐺𝑦 ≠ 0))
8281simpld 494 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ran 𝐺)
8378, 82sseldd 3995 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ℝ)
8483recnd 11286 . . . . . . . . . . 11 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ℂ)
8581simprd 495 . . . . . . . . . . 11 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ≠ 0)
8676, 84, 85divcan1d 12041 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → ((𝐴 / 𝑦) · 𝑦) = 𝐴)
87 oveq12 7439 . . . . . . . . . . 11 (((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦) → ((𝐹𝑧) · (𝐺𝑧)) = ((𝐴 / 𝑦) · 𝑦))
8887eqeq1d 2736 . . . . . . . . . 10 (((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦) → (((𝐹𝑧) · (𝐺𝑧)) = 𝐴 ↔ ((𝐴 / 𝑦) · 𝑦) = 𝐴))
8986, 88syl5ibrcom 247 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → (((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦) → ((𝐹𝑧) · (𝐺𝑧)) = 𝐴))
9089anassrs 467 . . . . . . . 8 ((((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑧 ∈ ℝ) → (((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦) → ((𝐹𝑧) · (𝐺𝑧)) = 𝐴))
9190imdistanda 571 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)))
9274, 91sylbid 240 . . . . . 6 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) → (𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)))
9392rexlimdva 3152 . . . . 5 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)))
9465, 93impbid 212 . . . 4 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦}))))
9515, 23, 943bitrd 305 . . 3 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ ((𝐹f · 𝐺) “ {𝐴}) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦}))))
96 eliun 4999 . . 3 (𝑧 𝑦 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})))
9795, 96bitr4di 289 . 2 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ ((𝐹f · 𝐺) “ {𝐴}) ↔ 𝑧 𝑦 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦}))))
9897eqrdv 2732 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 206  wa 395   = wceq 1536  wcel 2105  wne 2937  wrex 3067  Vcvv 3477  cdif 3959  cin 3961  {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  0cc0 11152   · cmul 11157   / cdiv 11917  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  ax-pre-mulgt0 11229
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-rmo 3377  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-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-sum 15719  df-itg1 25668
This theorem is referenced by:  i1fmul  25744
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