Theorem i1fmullem 25729

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 25711	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ)
3	1, 2	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
4	3	ffnd 6737	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ)
5		i1fadd.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ dom ∫1)
6		i1ff 25711	. . . . . . . . 9 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ)
7	5, 6	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
8	7	ffnd 6737	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ)
9		reex 11246	. . . . . . . 8 ⊢ ℝ ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ V)
11		inidm 4227	. . . . . . 7 ⊢ (ℝ ∩ ℝ) = ℝ
12	4, 8, 10, 10, 11	offn 7710	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘f · 𝐺) Fn ℝ)
13	12	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝐹 ∘f · 𝐺) Fn ℝ)
14		fniniseg 7080	. . . . 5 ⊢ ((𝐹 ∘f · 𝐺) Fn ℝ → (𝑧 ∈ (◡(𝐹 ∘f · 𝐺) “ {𝐴}) ↔ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f · 𝐺)‘𝑧) = 𝐴)))
15	13, 14	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (◡(𝐹 ∘f · 𝐺) “ {𝐴}) ↔ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f · 𝐺)‘𝑧) = 𝐴)))
16	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → 𝐹 Fn ℝ)
17	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → 𝐺 Fn ℝ)
18	9	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → ℝ ∈ V)
19		eqidd 2738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) = (𝐹‘𝑧))
20		eqidd 2738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) = (𝐺‘𝑧))
21	16, 17, 18, 18, 11, 19, 20	ofval 7708	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → ((𝐹 ∘f · 𝐺)‘𝑧) = ((𝐹‘𝑧) · (𝐺‘𝑧)))
22	21	eqeq1d 2739	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (((𝐹 ∘f · 𝐺)‘𝑧) = 𝐴 ↔ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴))
23	22	pm5.32da 579	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹 ∘f · 𝐺)‘𝑧) = 𝐴) ↔ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)))
24	8	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐺 Fn ℝ)
25		simprl 771	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝑧 ∈ ℝ)
26		fnfvelrn 7100	. . . . . . . . 9 ⊢ ((𝐺 Fn ℝ ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ∈ ran 𝐺)
27	24, 25, 26	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) ∈ ran 𝐺)
28		eldifsni 4790	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (ℂ ∖ {0}) → 𝐴 ≠ 0)
29	28	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐴 ≠ 0)
30		simprr 773	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)
31	3	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐹:ℝ⟶ℝ)
32	31, 25	ffvelcdmd 7105	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐹‘𝑧) ∈ ℝ)
33	32	recnd 11289	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐹‘𝑧) ∈ ℂ)
34	33	mul01d 11460	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → ((𝐹‘𝑧) · 0) = 0)
35	29, 30, 34	3netr4d 3018	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → ((𝐹‘𝑧) · (𝐺‘𝑧)) ≠ ((𝐹‘𝑧) · 0))
36		oveq2 7439	. . . . . . . . . 10 ⊢ ((𝐺‘𝑧) = 0 → ((𝐹‘𝑧) · (𝐺‘𝑧)) = ((𝐹‘𝑧) · 0))
37	36	necon3i 2973	. . . . . . . . 9 ⊢ (((𝐹‘𝑧) · (𝐺‘𝑧)) ≠ ((𝐹‘𝑧) · 0) → (𝐺‘𝑧) ≠ 0)
38	35, 37	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) ≠ 0)
39		eldifsn 4786	. . . . . . . 8 ⊢ ((𝐺‘𝑧) ∈ (ran 𝐺 ∖ {0}) ↔ ((𝐺‘𝑧) ∈ ran 𝐺 ∧ (𝐺‘𝑧) ≠ 0))
40	27, 38, 39	sylanbrc 583	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) ∈ (ran 𝐺 ∖ {0}))
41	7	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐺:ℝ⟶ℝ)
42	41, 25	ffvelcdmd 7105	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) ∈ ℝ)
43	42	recnd 11289	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) ∈ ℂ)
44	33, 43, 38	divcan4d 12049	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (((𝐹‘𝑧) · (𝐺‘𝑧)) / (𝐺‘𝑧)) = (𝐹‘𝑧))
45	30	oveq1d 7446	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (((𝐹‘𝑧) · (𝐺‘𝑧)) / (𝐺‘𝑧)) = (𝐴 / (𝐺‘𝑧)))
46	44, 45	eqtr3d 2779	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐹‘𝑧) = (𝐴 / (𝐺‘𝑧)))
47	31	ffnd 6737	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐹 Fn ℝ)
48		fniniseg 7080	. . . . . . . . . 10 ⊢ (𝐹 Fn ℝ → (𝑧 ∈ (◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ↔ (𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 / (𝐺‘𝑧)))))
49	47, 48	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)))))
50	25, 46, 49	mpbir2and 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝑧 ∈ (◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}))
51		eqidd 2738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) = (𝐺‘𝑧))
52		fniniseg 7080	. . . . . . . . . 10 ⊢ (𝐺 Fn ℝ → (𝑧 ∈ (◡𝐺 “ {(𝐺‘𝑧)}) ↔ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = (𝐺‘𝑧))))
53	24, 52	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝑧 ∈ (◡𝐺 “ {(𝐺‘𝑧)}) ↔ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = (𝐺‘𝑧))))
54	25, 51, 53	mpbir2and 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝑧 ∈ (◡𝐺 “ {(𝐺‘𝑧)}))
55	50, 54	elind 4200	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝑧 ∈ ((◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ∩ (◡𝐺 “ {(𝐺‘𝑧)})))
56		oveq2 7439	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐺‘𝑧) → (𝐴 / 𝑦) = (𝐴 / (𝐺‘𝑧)))
57	56	sneqd 4638	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐺‘𝑧) → {(𝐴 / 𝑦)} = {(𝐴 / (𝐺‘𝑧))})
58	57	imaeq2d 6078	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑧) → (◡𝐹 “ {(𝐴 / 𝑦)}) = (◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}))
59		sneq 4636	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐺‘𝑧) → {𝑦} = {(𝐺‘𝑧)})
60	59	imaeq2d 6078	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑧) → (◡𝐺 “ {𝑦}) = (◡𝐺 “ {(𝐺‘𝑧)}))
61	58, 60	ineq12d 4221	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑧) → ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) = ((◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ∩ (◡𝐺 “ {(𝐺‘𝑧)})))
62	61	eleq2d 2827	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) ↔ 𝑧 ∈ ((◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ∩ (◡𝐺 “ {(𝐺‘𝑧)}))))
63	62	rspcev 3622	. . . . . . 7 ⊢ (((𝐺‘𝑧) ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ((◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ∩ (◡𝐺 “ {(𝐺‘𝑧)}))) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})))
64	40, 55, 63	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})))
65	64	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦}))))
66		fniniseg 7080	. . . . . . . . . . 11 ⊢ (𝐹 Fn ℝ → (𝑧 ∈ (◡𝐹 “ {(𝐴 / 𝑦)}) ↔ (𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 / 𝑦))))
67	16, 66	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (◡𝐹 “ {(𝐴 / 𝑦)}) ↔ (𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 / 𝑦))))
68		fniniseg 7080	. . . . . . . . . . 11 ⊢ (𝐺 Fn ℝ → (𝑧 ∈ (◡𝐺 “ {𝑦}) ↔ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦)))
69	17, 68	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (◡𝐺 “ {𝑦}) ↔ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦)))
70	67, 69	anbi12d 632	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ (◡𝐹 “ {(𝐴 / 𝑦)}) ∧ 𝑧 ∈ (◡𝐺 “ {𝑦})) ↔ ((𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 / 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦))))
71		elin 3967	. . . . . . . . 9 ⊢ (𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) ↔ (𝑧 ∈ (◡𝐹 “ {(𝐴 / 𝑦)}) ∧ 𝑧 ∈ (◡𝐺 “ {𝑦})))
72		anandi 676	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦)) ↔ ((𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 / 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦)))
73	70, 71, 72	3bitr4g 314	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) ↔ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦))))
74	73	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) → (𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) ↔ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦))))
75		eldifi 4131	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (ℂ ∖ {0}) → 𝐴 ∈ ℂ)
76	75	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝐴 ∈ ℂ)
77	7	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝐺:ℝ⟶ℝ)
78	77	frnd 6744	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → ran 𝐺 ⊆ ℝ)
79		simprl 771	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ (ran 𝐺 ∖ {0}))
80		eldifsn 4786	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑦 ∈ ran 𝐺 ∧ 𝑦 ≠ 0))
81	79, 80	sylib 218	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → (𝑦 ∈ ran 𝐺 ∧ 𝑦 ≠ 0))
82	81	simpld 494	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ran 𝐺)
83	78, 82	sseldd 3984	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ℝ)
84	83	recnd 11289	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ℂ)
85	81	simprd 495	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ≠ 0)
86	76, 84, 85	divcan1d 12044	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → ((𝐴 / 𝑦) · 𝑦) = 𝐴)
87		oveq12 7440	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦) → ((𝐹‘𝑧) · (𝐺‘𝑧)) = ((𝐴 / 𝑦) · 𝑦))
88	87	eqeq1d 2739	. . . . . . . . . 10 ⊢ (((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦) → (((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴 ↔ ((𝐴 / 𝑦) · 𝑦) = 𝐴))
89	86, 88	syl5ibrcom 247	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → (((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦) → ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴))
90	89	anassrs 467	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑧 ∈ ℝ) → (((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦) → ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴))
91	90	imdistanda 571	. . . . . . 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)))
92	74, 91	sylbid 240	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) → (𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)))
93	92	rexlimdva 3155	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)))
94	65, 93	impbid 212	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦}))))
95	15, 23, 94	3bitrd 305	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (◡(𝐹 ∘f · 𝐺) “ {𝐴}) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦}))))
96		eliun 4995	. . 3 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})))
97	95, 96	bitr4di 289	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (◡(𝐹 ∘f · 𝐺) “ {𝐴}) ↔ 𝑧 ∈ ∪ 𝑦 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦}))))
98	97	eqrdv 2735	1 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (◡(𝐹 ∘f · 𝐺) “ {𝐴}) = ∪ 𝑦 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∃wrex 3070  Vcvv 3480   ∖ cdif 3948   ∩ cin 3950  {csn 4626  ∪ ciun 4991  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   “ cima 5688   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∘_f cof 7695  ℂcc 11153  ℝcr 11154  0cc0 11155   · cmul 11160   / cdiv 11920  ∫₁citg1 25650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-sum 15723  df-itg1 25655

This theorem is referenced by:  i1fmul  25731