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Theorem i1fmullem 23384
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 23366 . . . . . . . . 9 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
31, 2syl 17 . . . . . . . 8 (𝜑𝐹:ℝ⟶ℝ)
4 ffn 6007 . . . . . . . 8 (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ)
53, 4syl 17 . . . . . . 7 (𝜑𝐹 Fn ℝ)
6 i1fadd.2 . . . . . . . . 9 (𝜑𝐺 ∈ dom ∫1)
7 i1ff 23366 . . . . . . . . 9 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
86, 7syl 17 . . . . . . . 8 (𝜑𝐺:ℝ⟶ℝ)
9 ffn 6007 . . . . . . . 8 (𝐺:ℝ⟶ℝ → 𝐺 Fn ℝ)
108, 9syl 17 . . . . . . 7 (𝜑𝐺 Fn ℝ)
11 reex 9979 . . . . . . . 8 ℝ ∈ V
1211a1i 11 . . . . . . 7 (𝜑 → ℝ ∈ V)
13 inidm 3805 . . . . . . 7 (ℝ ∩ ℝ) = ℝ
145, 10, 12, 12, 13offn 6868 . . . . . 6 (𝜑 → (𝐹𝑓 · 𝐺) Fn ℝ)
1514adantr 481 . . . . 5 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (𝐹𝑓 · 𝐺) Fn ℝ)
16 fniniseg 6299 . . . . 5 ((𝐹𝑓 · 𝐺) Fn ℝ → (𝑧 ∈ ((𝐹𝑓 · 𝐺) “ {𝐴}) ↔ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 · 𝐺)‘𝑧) = 𝐴)))
1715, 16syl 17 . . . 4 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ ((𝐹𝑓 · 𝐺) “ {𝐴}) ↔ (𝑧 ∈ ℝ ∧ ((𝐹𝑓 · 𝐺)‘𝑧) = 𝐴)))
185adantr 481 . . . . . . 7 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → 𝐹 Fn ℝ)
1910adantr 481 . . . . . . 7 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → 𝐺 Fn ℝ)
2011a1i 11 . . . . . . 7 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ℝ ∈ V)
21 eqidd 2622 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (𝐹𝑧) = (𝐹𝑧))
22 eqidd 2622 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) = (𝐺𝑧))
2318, 19, 20, 20, 13, 21, 22ofval 6866 . . . . . 6 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → ((𝐹𝑓 · 𝐺)‘𝑧) = ((𝐹𝑧) · (𝐺𝑧)))
2423eqeq1d 2623 . . . . 5 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (((𝐹𝑓 · 𝐺)‘𝑧) = 𝐴 ↔ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴))
2524pm5.32da 672 . . . 4 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹𝑓 · 𝐺)‘𝑧) = 𝐴) ↔ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)))
2610ad2antrr 761 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝐺 Fn ℝ)
27 simprl 793 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝑧 ∈ ℝ)
28 fnfvelrn 6317 . . . . . . . . 9 ((𝐺 Fn ℝ ∧ 𝑧 ∈ ℝ) → (𝐺𝑧) ∈ ran 𝐺)
2926, 27, 28syl2anc 692 . . . . . . . 8 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) ∈ ran 𝐺)
30 eldifsni 4294 . . . . . . . . . . 11 (𝐴 ∈ (ℂ ∖ {0}) → 𝐴 ≠ 0)
3130ad2antlr 762 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝐴 ≠ 0)
32 simprr 795 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)
333ad2antrr 761 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝐹:ℝ⟶ℝ)
3433, 27ffvelrnd 6321 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐹𝑧) ∈ ℝ)
3534recnd 10020 . . . . . . . . . . 11 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐹𝑧) ∈ ℂ)
3635mul01d 10187 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → ((𝐹𝑧) · 0) = 0)
3731, 32, 363netr4d 2867 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → ((𝐹𝑧) · (𝐺𝑧)) ≠ ((𝐹𝑧) · 0))
38 oveq2 6618 . . . . . . . . . 10 ((𝐺𝑧) = 0 → ((𝐹𝑧) · (𝐺𝑧)) = ((𝐹𝑧) · 0))
3938necon3i 2822 . . . . . . . . 9 (((𝐹𝑧) · (𝐺𝑧)) ≠ ((𝐹𝑧) · 0) → (𝐺𝑧) ≠ 0)
4037, 39syl 17 . . . . . . . 8 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) ≠ 0)
41 eldifsn 4292 . . . . . . . 8 ((𝐺𝑧) ∈ (ran 𝐺 ∖ {0}) ↔ ((𝐺𝑧) ∈ ran 𝐺 ∧ (𝐺𝑧) ≠ 0))
4229, 40, 41sylanbrc 697 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) ∈ (ran 𝐺 ∖ {0}))
438ad2antrr 761 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝐺:ℝ⟶ℝ)
4443, 27ffvelrnd 6321 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) ∈ ℝ)
4544recnd 10020 . . . . . . . . . . 11 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) ∈ ℂ)
4635, 45, 40divcan4d 10759 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (((𝐹𝑧) · (𝐺𝑧)) / (𝐺𝑧)) = (𝐹𝑧))
4732oveq1d 6625 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (((𝐹𝑧) · (𝐺𝑧)) / (𝐺𝑧)) = (𝐴 / (𝐺𝑧)))
4846, 47eqtr3d 2657 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐹𝑧) = (𝐴 / (𝐺𝑧)))
4933, 4syl 17 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝐹 Fn ℝ)
50 fniniseg 6299 . . . . . . . . . 10 (𝐹 Fn ℝ → (𝑧 ∈ (𝐹 “ {(𝐴 / (𝐺𝑧))}) ↔ (𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴 / (𝐺𝑧)))))
5149, 50syl 17 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝑧 ∈ (𝐹 “ {(𝐴 / (𝐺𝑧))}) ↔ (𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴 / (𝐺𝑧)))))
5227, 48, 51mpbir2and 956 . . . . . . . 8 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝑧 ∈ (𝐹 “ {(𝐴 / (𝐺𝑧))}))
53 eqidd 2622 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝐺𝑧) = (𝐺𝑧))
54 fniniseg 6299 . . . . . . . . . 10 (𝐺 Fn ℝ → (𝑧 ∈ (𝐺 “ {(𝐺𝑧)}) ↔ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = (𝐺𝑧))))
5526, 54syl 17 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → (𝑧 ∈ (𝐺 “ {(𝐺𝑧)}) ↔ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = (𝐺𝑧))))
5627, 53, 55mpbir2and 956 . . . . . . . 8 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝑧 ∈ (𝐺 “ {(𝐺𝑧)}))
57 elin 3779 . . . . . . . 8 (𝑧 ∈ ((𝐹 “ {(𝐴 / (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)})) ↔ (𝑧 ∈ (𝐹 “ {(𝐴 / (𝐺𝑧))}) ∧ 𝑧 ∈ (𝐺 “ {(𝐺𝑧)})))
5852, 56, 57sylanbrc 697 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → 𝑧 ∈ ((𝐹 “ {(𝐴 / (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)})))
59 oveq2 6618 . . . . . . . . . . . 12 (𝑦 = (𝐺𝑧) → (𝐴 / 𝑦) = (𝐴 / (𝐺𝑧)))
6059sneqd 4165 . . . . . . . . . . 11 (𝑦 = (𝐺𝑧) → {(𝐴 / 𝑦)} = {(𝐴 / (𝐺𝑧))})
6160imaeq2d 5430 . . . . . . . . . 10 (𝑦 = (𝐺𝑧) → (𝐹 “ {(𝐴 / 𝑦)}) = (𝐹 “ {(𝐴 / (𝐺𝑧))}))
62 sneq 4163 . . . . . . . . . . 11 (𝑦 = (𝐺𝑧) → {𝑦} = {(𝐺𝑧)})
6362imaeq2d 5430 . . . . . . . . . 10 (𝑦 = (𝐺𝑧) → (𝐺 “ {𝑦}) = (𝐺 “ {(𝐺𝑧)}))
6461, 63ineq12d 3798 . . . . . . . . 9 (𝑦 = (𝐺𝑧) → ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) = ((𝐹 “ {(𝐴 / (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)})))
6564eleq2d 2684 . . . . . . . 8 (𝑦 = (𝐺𝑧) → (𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ 𝑧 ∈ ((𝐹 “ {(𝐴 / (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)}))))
6665rspcev 3298 . . . . . . 7 (((𝐺𝑧) ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ((𝐹 “ {(𝐴 / (𝐺𝑧))}) ∩ (𝐺 “ {(𝐺𝑧)}))) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})))
6742, 58, 66syl2anc 692 . . . . . 6 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})))
6867ex 450 . . . . 5 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦}))))
69 fniniseg 6299 . . . . . . . . . . 11 (𝐹 Fn ℝ → (𝑧 ∈ (𝐹 “ {(𝐴 / 𝑦)}) ↔ (𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴 / 𝑦))))
7018, 69syl 17 . . . . . . . . . 10 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (𝐹 “ {(𝐴 / 𝑦)}) ↔ (𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴 / 𝑦))))
71 fniniseg 6299 . . . . . . . . . . 11 (𝐺 Fn ℝ → (𝑧 ∈ (𝐺 “ {𝑦}) ↔ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)))
7219, 71syl 17 . . . . . . . . . 10 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (𝐺 “ {𝑦}) ↔ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)))
7370, 72anbi12d 746 . . . . . . . . 9 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ (𝐹 “ {(𝐴 / 𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})) ↔ ((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴 / 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦))))
74 elin 3779 . . . . . . . . 9 (𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ (𝑧 ∈ (𝐹 “ {(𝐴 / 𝑦)}) ∧ 𝑧 ∈ (𝐺 “ {𝑦})))
75 anandi 870 . . . . . . . . 9 ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦)) ↔ ((𝑧 ∈ ℝ ∧ (𝐹𝑧) = (𝐴 / 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺𝑧) = 𝑦)))
7673, 74, 753bitr4g 303 . . . . . . . 8 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦))))
7776adantr 481 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) → (𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ (𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦))))
78 eldifi 3715 . . . . . . . . . . . 12 (𝐴 ∈ (ℂ ∖ {0}) → 𝐴 ∈ ℂ)
7978ad2antlr 762 . . . . . . . . . . 11 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝐴 ∈ ℂ)
808ad2antrr 761 . . . . . . . . . . . . . 14 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝐺:ℝ⟶ℝ)
81 frn 6015 . . . . . . . . . . . . . 14 (𝐺:ℝ⟶ℝ → ran 𝐺 ⊆ ℝ)
8280, 81syl 17 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → ran 𝐺 ⊆ ℝ)
83 simprl 793 . . . . . . . . . . . . . . 15 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ (ran 𝐺 ∖ {0}))
84 eldifsn 4292 . . . . . . . . . . . . . . 15 (𝑦 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑦 ∈ ran 𝐺𝑦 ≠ 0))
8583, 84sylib 208 . . . . . . . . . . . . . 14 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → (𝑦 ∈ ran 𝐺𝑦 ≠ 0))
8685simpld 475 . . . . . . . . . . . . 13 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ran 𝐺)
8782, 86sseldd 3588 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ℝ)
8887recnd 10020 . . . . . . . . . . 11 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ℂ)
8985simprd 479 . . . . . . . . . . 11 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ≠ 0)
9079, 88, 89divcan1d 10754 . . . . . . . . . 10 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → ((𝐴 / 𝑦) · 𝑦) = 𝐴)
91 oveq12 6619 . . . . . . . . . . 11 (((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦) → ((𝐹𝑧) · (𝐺𝑧)) = ((𝐴 / 𝑦) · 𝑦))
9291eqeq1d 2623 . . . . . . . . . 10 (((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦) → (((𝐹𝑧) · (𝐺𝑧)) = 𝐴 ↔ ((𝐴 / 𝑦) · 𝑦) = 𝐴))
9390, 92syl5ibrcom 237 . . . . . . . . 9 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → (((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦) → ((𝐹𝑧) · (𝐺𝑧)) = 𝐴))
9493anassrs 679 . . . . . . . 8 ((((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑧 ∈ ℝ) → (((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦) → ((𝐹𝑧) · (𝐺𝑧)) = 𝐴))
9594imdistanda 728 . . . . . . 7 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) = (𝐴 / 𝑦) ∧ (𝐺𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)))
9677, 95sylbid 230 . . . . . 6 (((𝜑𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) → (𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)))
9796rexlimdva 3025 . . . . 5 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴)))
9868, 97impbid 202 . . . 4 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹𝑧) · (𝐺𝑧)) = 𝐴) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦}))))
9917, 25, 983bitrd 294 . . 3 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ ((𝐹𝑓 · 𝐺) “ {𝐴}) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦}))))
100 eliun 4495 . . 3 (𝑧 𝑦 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})))
10199, 100syl6bbr 278 . 2 ((𝜑𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ ((𝐹𝑓 · 𝐺) “ {𝐴}) ↔ 𝑧 𝑦 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦}))))
102101eqrdv 2619 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 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wrex 2908  Vcvv 3189  cdif 3556  cin 3558  wss 3559  {csn 4153   ciun 4490  ccnv 5078  dom cdm 5079  ran crn 5080  cima 5082   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  𝑓 cof 6855  cc 9886  cr 9887  0cc0 9888   · cmul 9893   / cdiv 10636  1citg1 23307
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
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-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-sum 14359  df-itg1 23312
This theorem is referenced by:  i1fmul  23386
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