Theorem i1fmullem 24591

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 24573	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ)
3	1, 2	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
4	3	ffnd 6546	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ)
5		i1fadd.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ dom ∫1)
6		i1ff 24573	. . . . . . . . 9 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ)
7	5, 6	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
8	7	ffnd 6546	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ)
9		reex 10820	. . . . . . . 8 ⊢ ℝ ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ V)
11		inidm 4133	. . . . . . 7 ⊢ (ℝ ∩ ℝ) = ℝ
12	4, 8, 10, 10, 11	offn 7481	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘f · 𝐺) Fn ℝ)
13	12	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝐹 ∘f · 𝐺) Fn ℝ)
14		fniniseg 6880	. . . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → 𝐹 Fn ℝ)
17	8	adantr 484	. . . . . . 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 7479	. . . . . 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 582	. . . 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 6901	. . . . . . . . 9 ⊢ ((𝐺 Fn ℝ ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ∈ ran 𝐺)
27	24, 25, 26	syl2anc 587	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) ∈ ran 𝐺)
28		eldifsni 4703	. . . . . . . . . . 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	ffvelrnd 6905	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐹‘𝑧) ∈ ℝ)
33	32	recnd 10861	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐹‘𝑧) ∈ ℂ)
34	33	mul01d 11031	. . . . . . . . . 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 7221	. . . . . . . . . 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 4700	. . . . . . . 8 ⊢ ((𝐺‘𝑧) ∈ (ran 𝐺 ∖ {0}) ↔ ((𝐺‘𝑧) ∈ ran 𝐺 ∧ (𝐺‘𝑧) ≠ 0))
40	27, 38, 39	sylanbrc 586	. . . . . . 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	ffvelrnd 6905	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) ∈ ℝ)
43	42	recnd 10861	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) ∈ ℂ)
44	33, 43, 38	divcan4d 11614	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (((𝐹‘𝑧) · (𝐺‘𝑧)) / (𝐺‘𝑧)) = (𝐹‘𝑧))
45	30	oveq1d 7228	. . . . . . . . . 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 6546	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐹 Fn ℝ)
48		fniniseg 6880	. . . . . . . . . 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 6880	. . . . . . . . . 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 4108	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝑧 ∈ ((◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ∩ (◡𝐺 “ {(𝐺‘𝑧)})))
56		oveq2 7221	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐺‘𝑧) → (𝐴 / 𝑦) = (𝐴 / (𝐺‘𝑧)))
57	56	sneqd 4553	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐺‘𝑧) → {(𝐴 / 𝑦)} = {(𝐴 / (𝐺‘𝑧))})
58	57	imaeq2d 5929	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑧) → (◡𝐹 “ {(𝐴 / 𝑦)}) = (◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}))
59		sneq 4551	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐺‘𝑧) → {𝑦} = {(𝐺‘𝑧)})
60	59	imaeq2d 5929	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑧) → (◡𝐺 “ {𝑦}) = (◡𝐺 “ {(𝐺‘𝑧)}))
61	58, 60	ineq12d 4128	. . . . . . . . 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 3537	. . . . . . 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 587	. . . . . 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 416	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦}))))
66		fniniseg 6880	. . . . . . . . . . 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 6880	. . . . . . . . . . 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 634	. . . . . . . . 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 3882	. . . . . . . . 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 317	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) ↔ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦))))
74	73	adantr 484	. . . . . . 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 4041	. . . . . . . . . . . 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 6553	. . . . . . . . . . . . 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 4700	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑦 ∈ ran 𝐺 ∧ 𝑦 ≠ 0))
81	79, 80	sylib 221	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → (𝑦 ∈ ran 𝐺 ∧ 𝑦 ≠ 0))
82	81	simpld 498	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ran 𝐺)
83	78, 82	sseldd 3902	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ℝ)
84	83	recnd 10861	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ℂ)
85	81	simprd 499	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ≠ 0)
86	76, 84, 85	divcan1d 11609	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → ((𝐴 / 𝑦) · 𝑦) = 𝐴)
87		oveq12 7222	. . . . . . . . . . 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 250	. . . . . . . . 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 471	. . . . . . . 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 575	. . . . . . 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 243	. . . . . 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 3203	. . . . 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 215	. . . 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 308	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (◡(𝐹 ∘f · 𝐺) “ {𝐴}) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦}))))
96		eliun 4908	. . 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 292	. 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 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  ∃wrex 3062  Vcvv 3408   ∖ cdif 3863   ∩ cin 3865  {csn 4541  ∪ ciun 4904  ^◡ccnv 5550  dom cdm 5551  ran crn 5552   “ cima 5554   Fn wfn 6375  ⟶wf 6376  ‘cfv 6380  (class class class)co 7213   ∘_f cof 7467  ℂcc 10727  ℝcr 10728  0cc0 10729   · cmul 10734   / cdiv 11489  ∫₁citg1 24512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-po 5468  df-so 5469  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-sum 15250  df-itg1 24517

This theorem is referenced by:  i1fmul  24593