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Theorem i1fmullem 23461

Description: Decompose the preimage of a product. (Contributed by Mario Carneiro, 19-Jun-2014.)

**Hypotheses**
Ref	Expression
i1fadd.1	⊢ (𝜑 → 𝐹 ∈ dom ∫₁)
i1fadd.2	⊢ (𝜑 → 𝐺 ∈ dom ∫₁)

**Assertion**
Ref	Expression
i1fmullem	⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (^◡(𝐹 ∘_𝑓 · 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.

Step	Hyp	Ref	Expression
1		i1fadd.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ dom ∫₁)
2		i1ff 23443	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫₁ → 𝐹:ℝ⟶ℝ)
3	1, 2	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
4		ffn 6045	. . . . . . . 8 ⊢ (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ)
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ)
6		i1fadd.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ dom ∫₁)
7		i1ff 23443	. . . . . . . . 9 ⊢ (𝐺 ∈ dom ∫₁ → 𝐺:ℝ⟶ℝ)
8	6, 7	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
9		ffn 6045	. . . . . . . 8 ⊢ (𝐺:ℝ⟶ℝ → 𝐺 Fn ℝ)
10	8, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ)
11		reex 10027	. . . . . . . 8 ⊢ ℝ ∈ V
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ V)
13		inidm 3822	. . . . . . 7 ⊢ (ℝ ∩ ℝ) = ℝ
14	5, 10, 12, 12, 13	offn 6908	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘_𝑓 · 𝐺) Fn ℝ)
15	14	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝐹 ∘_𝑓 · 𝐺) Fn ℝ)
16		fniniseg 6338	. . . . 5 ⊢ ((𝐹 ∘_𝑓 · 𝐺) Fn ℝ → (𝑧 ∈ (^◡(𝐹 ∘_𝑓 · 𝐺) “ {𝐴}) ↔ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 · 𝐺)‘𝑧) = 𝐴)))
17	15, 16	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (^◡(𝐹 ∘_𝑓 · 𝐺) “ {𝐴}) ↔ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 · 𝐺)‘𝑧) = 𝐴)))
18	5	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → 𝐹 Fn ℝ)
19	10	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → 𝐺 Fn ℝ)
20	11	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → ℝ ∈ V)
21		eqidd 2623	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) = (𝐹‘𝑧))
22		eqidd 2623	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) = (𝐺‘𝑧))
23	18, 19, 20, 20, 13, 21, 22	ofval 6906	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → ((𝐹 ∘_𝑓 · 𝐺)‘𝑧) = ((𝐹‘𝑧) · (𝐺‘𝑧)))
24	23	eqeq1d 2624	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (((𝐹 ∘_𝑓 · 𝐺)‘𝑧) = 𝐴 ↔ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴))
25	24	pm5.32da 673	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 · 𝐺)‘𝑧) = 𝐴) ↔ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)))
26	10	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐺 Fn ℝ)
27		simprl 794	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝑧 ∈ ℝ)
28		fnfvelrn 6356	. . . . . . . . 9 ⊢ ((𝐺 Fn ℝ ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ∈ ran 𝐺)
29	26, 27, 28	syl2anc 693	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) ∈ ran 𝐺)
30		eldifsni 4320	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (ℂ ∖ {0}) → 𝐴 ≠ 0)
31	30	ad2antlr 763	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐴 ≠ 0)
32		simprr 796	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)
33	3	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐹:ℝ⟶ℝ)
34	33, 27	ffvelrnd 6360	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐹‘𝑧) ∈ ℝ)
35	34	recnd 10068	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐹‘𝑧) ∈ ℂ)
36	35	mul01d 10235	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → ((𝐹‘𝑧) · 0) = 0)
37	31, 32, 36	3netr4d 2871	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → ((𝐹‘𝑧) · (𝐺‘𝑧)) ≠ ((𝐹‘𝑧) · 0))
38		oveq2 6658	. . . . . . . . . 10 ⊢ ((𝐺‘𝑧) = 0 → ((𝐹‘𝑧) · (𝐺‘𝑧)) = ((𝐹‘𝑧) · 0))
39	38	necon3i 2826	. . . . . . . . 9 ⊢ (((𝐹‘𝑧) · (𝐺‘𝑧)) ≠ ((𝐹‘𝑧) · 0) → (𝐺‘𝑧) ≠ 0)
40	37, 39	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) ≠ 0)
41		eldifsn 4317	. . . . . . . 8 ⊢ ((𝐺‘𝑧) ∈ (ran 𝐺 ∖ {0}) ↔ ((𝐺‘𝑧) ∈ ran 𝐺 ∧ (𝐺‘𝑧) ≠ 0))
42	29, 40, 41	sylanbrc 698	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) ∈ (ran 𝐺 ∖ {0}))
43	8	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐺:ℝ⟶ℝ)
44	43, 27	ffvelrnd 6360	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) ∈ ℝ)
45	44	recnd 10068	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) ∈ ℂ)
46	35, 45, 40	divcan4d 10807	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (((𝐹‘𝑧) · (𝐺‘𝑧)) / (𝐺‘𝑧)) = (𝐹‘𝑧))
47	32	oveq1d 6665	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (((𝐹‘𝑧) · (𝐺‘𝑧)) / (𝐺‘𝑧)) = (𝐴 / (𝐺‘𝑧)))
48	46, 47	eqtr3d 2658	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐹‘𝑧) = (𝐴 / (𝐺‘𝑧)))
49	33, 4	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐹 Fn ℝ)
50		fniniseg 6338	. . . . . . . . . 10 ⊢ (𝐹 Fn ℝ → (𝑧 ∈ (^◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ↔ (𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 / (𝐺‘𝑧)))))
51	49, 50	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝑧 ∈ (^◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ↔ (𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 / (𝐺‘𝑧)))))
52	27, 48, 51	mpbir2and 957	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝑧 ∈ (^◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}))
53		eqidd 2623	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) = (𝐺‘𝑧))
54		fniniseg 6338	. . . . . . . . . 10 ⊢ (𝐺 Fn ℝ → (𝑧 ∈ (^◡𝐺 “ {(𝐺‘𝑧)}) ↔ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = (𝐺‘𝑧))))
55	26, 54	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝑧 ∈ (^◡𝐺 “ {(𝐺‘𝑧)}) ↔ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = (𝐺‘𝑧))))
56	27, 53, 55	mpbir2and 957	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝑧 ∈ (^◡𝐺 “ {(𝐺‘𝑧)}))
57		elin 3796	. . . . . . . 8 ⊢ (𝑧 ∈ ((^◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ∩ (^◡𝐺 “ {(𝐺‘𝑧)})) ↔ (𝑧 ∈ (^◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ∧ 𝑧 ∈ (^◡𝐺 “ {(𝐺‘𝑧)})))
58	52, 56, 57	sylanbrc 698	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝑧 ∈ ((^◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ∩ (^◡𝐺 “ {(𝐺‘𝑧)})))
59		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐺‘𝑧) → (𝐴 / 𝑦) = (𝐴 / (𝐺‘𝑧)))
60	59	sneqd 4189	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐺‘𝑧) → {(𝐴 / 𝑦)} = {(𝐴 / (𝐺‘𝑧))})
61	60	imaeq2d 5466	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑧) → (^◡𝐹 “ {(𝐴 / 𝑦)}) = (^◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}))
62		sneq 4187	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐺‘𝑧) → {𝑦} = {(𝐺‘𝑧)})
63	62	imaeq2d 5466	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑧) → (^◡𝐺 “ {𝑦}) = (^◡𝐺 “ {(𝐺‘𝑧)}))
64	61, 63	ineq12d 3815	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑧) → ((^◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (^◡𝐺 “ {𝑦})) = ((^◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ∩ (^◡𝐺 “ {(𝐺‘𝑧)})))
65	64	eleq2d 2687	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑧 ∈ ((^◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (^◡𝐺 “ {𝑦})) ↔ 𝑧 ∈ ((^◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ∩ (^◡𝐺 “ {(𝐺‘𝑧)}))))
66	65	rspcev 3309	. . . . . . 7 ⊢ (((𝐺‘𝑧) ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ((^◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ∩ (^◡𝐺 “ {(𝐺‘𝑧)}))) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((^◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (^◡𝐺 “ {𝑦})))
67	42, 58, 66	syl2anc 693	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((^◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (^◡𝐺 “ {𝑦})))
68	67	ex 450	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((^◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (^◡𝐺 “ {𝑦}))))
69		fniniseg 6338	. . . . . . . . . . 11 ⊢ (𝐹 Fn ℝ → (𝑧 ∈ (^◡𝐹 “ {(𝐴 / 𝑦)}) ↔ (𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 / 𝑦))))
70	18, 69	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (^◡𝐹 “ {(𝐴 / 𝑦)}) ↔ (𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 / 𝑦))))
71		fniniseg 6338	. . . . . . . . . . 11 ⊢ (𝐺 Fn ℝ → (𝑧 ∈ (^◡𝐺 “ {𝑦}) ↔ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦)))
72	19, 71	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (^◡𝐺 “ {𝑦}) ↔ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦)))
73	70, 72	anbi12d 747	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ (^◡𝐹 “ {(𝐴 / 𝑦)}) ∧ 𝑧 ∈ (^◡𝐺 “ {𝑦})) ↔ ((𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 / 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦))))
74		elin 3796	. . . . . . . . 9 ⊢ (𝑧 ∈ ((^◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (^◡𝐺 “ {𝑦})) ↔ (𝑧 ∈ (^◡𝐹 “ {(𝐴 / 𝑦)}) ∧ 𝑧 ∈ (^◡𝐺 “ {𝑦})))
75		anandi 871	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦)) ↔ ((𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 / 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦)))
76	73, 74, 75	3bitr4g 303	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ ((^◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (^◡𝐺 “ {𝑦})) ↔ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦))))
77	76	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) → (𝑧 ∈ ((^◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (^◡𝐺 “ {𝑦})) ↔ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦))))
78		eldifi 3732	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (ℂ ∖ {0}) → 𝐴 ∈ ℂ)
79	78	ad2antlr 763	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝐴 ∈ ℂ)
80	8	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝐺:ℝ⟶ℝ)
81		frn 6053	. . . . . . . . . . . . . 14 ⊢ (𝐺:ℝ⟶ℝ → ran 𝐺 ⊆ ℝ)
82	80, 81	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → ran 𝐺 ⊆ ℝ)
83		simprl 794	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ (ran 𝐺 ∖ {0}))
84		eldifsn 4317	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑦 ∈ ran 𝐺 ∧ 𝑦 ≠ 0))
85	83, 84	sylib 208	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → (𝑦 ∈ ran 𝐺 ∧ 𝑦 ≠ 0))
86	85	simpld 475	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ran 𝐺)
87	82, 86	sseldd 3604	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ℝ)
88	87	recnd 10068	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ℂ)
89	85	simprd 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ≠ 0)
90	79, 88, 89	divcan1d 10802	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → ((𝐴 / 𝑦) · 𝑦) = 𝐴)
91		oveq12 6659	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦) → ((𝐹‘𝑧) · (𝐺‘𝑧)) = ((𝐴 / 𝑦) · 𝑦))
92	91	eqeq1d 2624	. . . . . . . . . 10 ⊢ (((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦) → (((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴 ↔ ((𝐴 / 𝑦) · 𝑦) = 𝐴))
93	90, 92	syl5ibrcom 237	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → (((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦) → ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴))
94	93	anassrs 680	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑧 ∈ ℝ) → (((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦) → ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴))
95	94	imdistanda 729	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)))
96	77, 95	sylbid 230	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) → (𝑧 ∈ ((^◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (^◡𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)))
97	96	rexlimdva 3031	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((^◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (^◡𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)))
98	68, 97	impbid 202	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((^◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (^◡𝐺 “ {𝑦}))))
99	17, 25, 98	3bitrd 294	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (^◡(𝐹 ∘_𝑓 · 𝐺) “ {𝐴}) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((^◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (^◡𝐺 “ {𝑦}))))
100		eliun 4524	. . 3 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ (ran 𝐺 ∖ {0})((^◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (^◡𝐺 “ {𝑦})) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((^◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (^◡𝐺 “ {𝑦})))
101	99, 100	syl6bbr 278	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (^◡(𝐹 ∘_𝑓 · 𝐺) “ {𝐴}) ↔ 𝑧 ∈ ∪ 𝑦 ∈ (ran 𝐺 ∖ {0})((^◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (^◡𝐺 “ {𝑦}))))
102	101	eqrdv 2620	1 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (^◡(𝐹 ∘_𝑓 · 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 1483 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 Vcvv 3200 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 {csn 4177 ∪ ciun 4520 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 “ cima 5117 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘_𝑓 cof 6895 ℂcc 9934 ℝcr 9935 0cc0 9936 · cmul 9941 / cdiv 10684 ∫₁citg1 23384

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-sum 14417 df-itg1 23389

This theorem is referenced by: i1fmul 23463

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