Theorem i1fmullem 25004

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 24986	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ)
3	1, 2	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
4	3	ffnd 6666	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ)
5		i1fadd.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ dom ∫1)
6		i1ff 24986	. . . . . . . . 9 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ)
7	5, 6	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
8	7	ffnd 6666	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ)
9		reex 11100	. . . . . . . 8 ⊢ ℝ ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ V)
11		inidm 4176	. . . . . . 7 ⊢ (ℝ ∩ ℝ) = ℝ
12	4, 8, 10, 10, 11	offn 7622	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘f · 𝐺) Fn ℝ)
13	12	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝐹 ∘f · 𝐺) Fn ℝ)
14		fniniseg 7007	. . . . 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 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → 𝐹 Fn ℝ)
17	8	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → 𝐺 Fn ℝ)
18	9	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → ℝ ∈ V)
19		eqidd 2737	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) = (𝐹‘𝑧))
20		eqidd 2737	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) = (𝐺‘𝑧))
21	16, 17, 18, 18, 11, 19, 20	ofval 7620	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → ((𝐹 ∘f · 𝐺)‘𝑧) = ((𝐹‘𝑧) · (𝐺‘𝑧)))
22	21	eqeq1d 2738	. . . . 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 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐺 Fn ℝ)
25		simprl 769	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝑧 ∈ ℝ)
26		fnfvelrn 7028	. . . . . . . . 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 4748	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (ℂ ∖ {0}) → 𝐴 ≠ 0)
29	28	ad2antlr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐴 ≠ 0)
30		simprr 771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)
31	3	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐹:ℝ⟶ℝ)
32	31, 25	ffvelcdmd 7032	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐹‘𝑧) ∈ ℝ)
33	32	recnd 11141	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐹‘𝑧) ∈ ℂ)
34	33	mul01d 11312	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → ((𝐹‘𝑧) · 0) = 0)
35	29, 30, 34	3netr4d 3019	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → ((𝐹‘𝑧) · (𝐺‘𝑧)) ≠ ((𝐹‘𝑧) · 0))
36		oveq2 7359	. . . . . . . . . 10 ⊢ ((𝐺‘𝑧) = 0 → ((𝐹‘𝑧) · (𝐺‘𝑧)) = ((𝐹‘𝑧) · 0))
37	36	necon3i 2974	. . . . . . . . 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 4745	. . . . . . . 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 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐺:ℝ⟶ℝ)
42	41, 25	ffvelcdmd 7032	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) ∈ ℝ)
43	42	recnd 11141	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) ∈ ℂ)
44	33, 43, 38	divcan4d 11895	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (((𝐹‘𝑧) · (𝐺‘𝑧)) / (𝐺‘𝑧)) = (𝐹‘𝑧))
45	30	oveq1d 7366	. . . . . . . . . 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 2778	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐹‘𝑧) = (𝐴 / (𝐺‘𝑧)))
47	31	ffnd 6666	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐹 Fn ℝ)
48		fniniseg 7007	. . . . . . . . . 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 711	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝑧 ∈ (◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}))
51		eqidd 2737	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) = (𝐺‘𝑧))
52		fniniseg 7007	. . . . . . . . . 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 711	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝑧 ∈ (◡𝐺 “ {(𝐺‘𝑧)}))
55	50, 54	elind 4152	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝑧 ∈ ((◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ∩ (◡𝐺 “ {(𝐺‘𝑧)})))
56		oveq2 7359	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐺‘𝑧) → (𝐴 / 𝑦) = (𝐴 / (𝐺‘𝑧)))
57	56	sneqd 4596	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐺‘𝑧) → {(𝐴 / 𝑦)} = {(𝐴 / (𝐺‘𝑧))})
58	57	imaeq2d 6011	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑧) → (◡𝐹 “ {(𝐴 / 𝑦)}) = (◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}))
59		sneq 4594	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐺‘𝑧) → {𝑦} = {(𝐺‘𝑧)})
60	59	imaeq2d 6011	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑧) → (◡𝐺 “ {𝑦}) = (◡𝐺 “ {(𝐺‘𝑧)}))
61	58, 60	ineq12d 4171	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑧) → ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) = ((◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ∩ (◡𝐺 “ {(𝐺‘𝑧)})))
62	61	eleq2d 2823	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) ↔ 𝑧 ∈ ((◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ∩ (◡𝐺 “ {(𝐺‘𝑧)}))))
63	62	rspcev 3579	. . . . . . 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 413	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦}))))
66		fniniseg 7007	. . . . . . . . . . 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 7007	. . . . . . . . . . 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 631	. . . . . . . . 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 3924	. . . . . . . . 9 ⊢ (𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) ↔ (𝑧 ∈ (◡𝐹 “ {(𝐴 / 𝑦)}) ∧ 𝑧 ∈ (◡𝐺 “ {𝑦})))
72		anandi 674	. . . . . . . . 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 313	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) ↔ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦))))
74	73	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))))
75		eldifi 4084	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (ℂ ∖ {0}) → 𝐴 ∈ ℂ)
76	75	ad2antlr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝐴 ∈ ℂ)
77	7	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝐺:ℝ⟶ℝ)
78	77	frnd 6673	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → ran 𝐺 ⊆ ℝ)
79		simprl 769	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ (ran 𝐺 ∖ {0}))
80		eldifsn 4745	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑦 ∈ ran 𝐺 ∧ 𝑦 ≠ 0))
81	79, 80	sylib 217	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → (𝑦 ∈ ran 𝐺 ∧ 𝑦 ≠ 0))
82	81	simpld 495	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ran 𝐺)
83	78, 82	sseldd 3943	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ℝ)
84	83	recnd 11141	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ℂ)
85	81	simprd 496	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ≠ 0)
86	76, 84, 85	divcan1d 11890	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → ((𝐴 / 𝑦) · 𝑦) = 𝐴)
87		oveq12 7360	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦) → ((𝐹‘𝑧) · (𝐺‘𝑧)) = ((𝐴 / 𝑦) · 𝑦))
88	87	eqeq1d 2738	. . . . . . . . . 10 ⊢ (((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦) → (((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴 ↔ ((𝐴 / 𝑦) · 𝑦) = 𝐴))
89	86, 88	syl5ibrcom 246	. . . . . . . . 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 468	. . . . . . . 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 572	. . . . . . 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 239	. . . . . 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 3150	. . . . 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 211	. . . 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 304	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (◡(𝐹 ∘f · 𝐺) “ {𝐴}) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦}))))
96		eliun 4956	. . 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 288	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (◡(𝐹 ∘f · 𝐺) “ {𝐴}) ↔ 𝑧 ∈ ∪ 𝑦 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦}))))
98	97	eqrdv 2734	1 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (◡(𝐹 ∘f · 𝐺) “ {𝐴}) = ∪ 𝑦 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2941  ∃wrex 3071  Vcvv 3443   ∖ cdif 3905   ∩ cin 3907  {csn 4584  ∪ ciun 4952  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   “ cima 5634   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7351   ∘_f cof 7607  ℂcc 11007  ℝcr 11008  0cc0 11009   · cmul 11014   / cdiv 11770  ∫₁citg1 24925

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 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

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 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-po 5543  df-so 5544  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-of 7609  df-er 8606  df-en 8842  df-dom 8843  df-sdom 8844  df-pnf 11149  df-mnf 11150  df-xr 11151  df-ltxr 11152  df-le 11153  df-sub 11345  df-neg 11346  df-div 11771  df-sum 15525  df-itg1 24930

This theorem is referenced by:  i1fmul  25006