MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  i1fmullem Structured version   Visualization version   GIF version

Theorem i1fmullem 25058
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 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 ∖ {0})) → (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 ∖ {0})) → (𝑧 ∈ ((𝐹f · 𝐺) “ {𝐴}) ↔ (𝑧 ∈ ℝ ∧ ((𝐹f · 𝐺)‘𝑧) = 𝐴)))
164adantr 481 . . . . . . 7 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → 𝐹 Fn ℝ)
178adantr 481 . . . . . . 7 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → 𝐺 Fn ℝ)
189a1i 11 . . . . . . 7 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ℝ ∈ V)
19 eqidd 2737 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) = (𝐹𝑧))
20 eqidd 2737 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) = (𝐺𝑧))
2116, 17, 18, 18, 11, 19, 20ofval 7628 . . . . . 6 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → ((𝐹f · 𝐺)‘𝑧) = ((𝐹𝑧) · (𝐺𝑧)))
2221eqeq1d 2738 . . . . 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 724 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝐺 Fn ℝ)
25 simprl 769 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝑧 ∈ ℝ)
26 fnfvelrn 7031 . . . . . . . . 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 4750 . . . . . . . . . . 11 (𝐴 ∈ (ℂ ∖ {0}) → 𝐴 ≠ 0)
2928ad2antlr 725 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝐴 ≠ 0)
30 simprr 771 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)
313ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝐹:ℝ⟶ℝ)
3231, 25ffvelcdmd 7036 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐹𝑧) ∈ ℝ)
3332recnd 11183 . . . . . . . . . . 11 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐹𝑧) ∈ ℂ)
3433mul01d 11354 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → ((𝐹𝑧) · 0) = 0)
3529, 30, 343netr4d 3021 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → ((𝐹𝑧) · (𝐺𝑧)) ≠ ((𝐹𝑧) · 0))
36 oveq2 7365 . . . . . . . . . 10 ((𝐺𝑧) = 0 → ((𝐹𝑧) · (𝐺𝑧)) = ((𝐹𝑧) · 0))
3736necon3i 2976 . . . . . . . . 9 (((𝐹𝑧) · (𝐺𝑧)) ≠ ((𝐹𝑧) · 0) → (𝐺𝑧) ≠ 0)
3835, 37syl 17 . . . . . . . 8 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) ≠ 0)
39 eldifsn 4747 . . . . . . . 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 724 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝐺:ℝ⟶ℝ)
4241, 25ffvelcdmd 7036 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) ∈ ℝ)
4342recnd 11183 . . . . . . . . . . 11 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) ∈ ℂ)
4433, 43, 38divcan4d 11937 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (((𝐹𝑧) · (𝐺𝑧)) / (𝐺𝑧)) = (𝐹𝑧))
4530oveq1d 7372 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (((𝐹𝑧) · (𝐺𝑧)) / (𝐺𝑧)) = (𝐴 / (𝐺𝑧)))
4644, 45eqtr3d 2778 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐹𝑧) = (𝐴 / (𝐺𝑧)))
4731ffnd 6669 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝐹 Fn ℝ)
48 fniniseg 7010 . . . . . . . . . 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 711 . . . . . . . 8 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝑧 ∈ (𝐹 “ {(𝐴 / (𝐺𝑧))}))
51 eqidd 2737 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) = (𝐺𝑧))
52 fniniseg 7010 . . . . . . . . . 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 711 . . . . . . . 8 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝑧 ∈ (𝐺 “ {(𝐺𝑧)}))
5550, 54elind 4154 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝑧 ∈ ((𝐹 “ {(𝐴 / (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)})))
56 oveq2 7365 . . . . . . . . . . . 12 (𝑦 = (𝐺𝑧) → (𝐴 / 𝑦) = (𝐴 / (𝐺𝑧)))
5756sneqd 4598 . . . . . . . . . . 11 (𝑦 = (𝐺𝑧) → {(𝐴 / 𝑦)} = {(𝐴 / (𝐺𝑧))})
5857imaeq2d 6013 . . . . . . . . . 10 (𝑦 = (𝐺𝑧) → (𝐹 “ {(𝐴 / 𝑦)}) = (𝐹 “ {(𝐴 / (𝐺𝑧))}))
59 sneq 4596 . . . . . . . . . . 11 (𝑦 = (𝐺𝑧) → {𝑦} = {(𝐺𝑧)})
6059imaeq2d 6013 . . . . . . . . . 10 (𝑦 = (𝐺𝑧) → (𝐺 “ {𝑦}) = (𝐺 “ {(𝐺𝑧)}))
6158, 60ineq12d 4173 . . . . . . . . 9 (𝑦 = (𝐺𝑧) → ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) = ((𝐹 “ {(𝐴 / (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)})))
6261eleq2d 2823 . . . . . . . 8 (𝑦 = (𝐺𝑧) → (𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ 𝑧 ∈ ((𝐹 “ {(𝐴 / (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)}))))
6362rspcev 3581 . . . . . . 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 413 . . . . 5 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦}))))
66 fniniseg 7010 . . . . . . . . . . 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 7010 . . . . . . . . . . 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 631 . . . . . . . . 9 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ (𝐹 “ {(𝐴 / 𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})) ↔ ((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴 / 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦))))
71 elin 3926 . . . . . . . . 9 (𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ (𝑧 ∈ (𝐹 “ {(𝐴 / 𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})))
72 anandi 674 . . . . . . . . 9 ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦)) ↔ ((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴 / 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)))
7370, 71, 723bitr4g 313 . . . . . . . 8 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦))))
7473adantr 481 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) → (𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦))))
75 eldifi 4086 . . . . . . . . . . . 12 (𝐴 ∈ (ℂ ∖ {0}) → 𝐴 ∈ ℂ)
7675ad2antlr 725 . . . . . . . . . . 11 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝐴 ∈ ℂ)
777ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝐺:ℝ⟶ℝ)
7877frnd 6676 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → ran 𝐺 ⊆ ℝ)
79 simprl 769 . . . . . . . . . . . . . . 15 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ (ran 𝐺 ∖ {0}))
80 eldifsn 4747 . . . . . . . . . . . . . . 15 (𝑦 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑦 ∈ ran 𝐺𝑦 ≠ 0))
8179, 80sylib 217 . . . . . . . . . . . . . 14 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → (𝑦 ∈ ran 𝐺𝑦 ≠ 0))
8281simpld 495 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ran 𝐺)
8378, 82sseldd 3945 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ℝ)
8483recnd 11183 . . . . . . . . . . 11 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ℂ)
8581simprd 496 . . . . . . . . . . 11 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ≠ 0)
8676, 84, 85divcan1d 11932 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → ((𝐴 / 𝑦) · 𝑦) = 𝐴)
87 oveq12 7366 . . . . . . . . . . 11 (((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦) → ((𝐹𝑧) · (𝐺𝑧)) = ((𝐴 / 𝑦) · 𝑦))
8887eqeq1d 2738 . . . . . . . . . 10 (((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦) → (((𝐹𝑧) · (𝐺𝑧)) = 𝐴 ↔ ((𝐴 / 𝑦) · 𝑦) = 𝐴))
8986, 88syl5ibrcom 246 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → (((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦) → ((𝐹𝑧) · (𝐺𝑧)) = 𝐴))
9089anassrs 468 . . . . . . . 8 ((((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑧 ∈ ℝ) → (((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦) → ((𝐹𝑧) · (𝐺𝑧)) = 𝐴))
9190imdistanda 572 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)))
9274, 91sylbid 239 . . . . . 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 211 . . . 4 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦}))))
9515, 23, 943bitrd 304 . . 3 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ ((𝐹f · 𝐺) “ {𝐴}) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦}))))
96 eliun 4958 . . 3 (𝑧 𝑦 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})))
9795, 96bitr4di 288 . 2 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ ((𝐹f · 𝐺) “ {𝐴}) ↔ 𝑧 𝑦 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦}))))
9897eqrdv 2734 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 205  wa 396   = wceq 1541  wcel 2106  wne 2943  wrex 3073  Vcvv 3445  cdif 3907  cin 3909  {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  0cc0 11051   · cmul 11056   / cdiv 11812  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  ax-pre-mulgt0 11128
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-rmo 3353  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-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-sum 15571  df-itg1 24984
This theorem is referenced by:  i1fmul  25060
  Copyright terms: Public domain W3C validator