Theorem i1faddlem 25650

Description: Decompose the preimage of a sum. (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
i1faddlem	⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (◡(𝐹 ∘f + 𝐺) “ {𝐴}) = ∪ 𝑦 ∈ ran 𝐺((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐹   𝑦,𝐺   𝜑,𝑦

Proof of Theorem i1faddlem

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		i1fadd.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ dom ∫1)
2		i1ff 25633	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ)
3	1, 2	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
4	3	ffnd 6663	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ)
5		i1fadd.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ dom ∫1)
6		i1ff 25633	. . . . . . . . 9 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ)
7	5, 6	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
8	7	ffnd 6663	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ)
9		reex 11117	. . . . . . . 8 ⊢ ℝ ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ V)
11		inidm 4179	. . . . . . 7 ⊢ (ℝ ∩ ℝ) = ℝ
12	4, 8, 10, 10, 11	offn 7635	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) Fn ℝ)
13	12	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝐹 ∘f + 𝐺) Fn ℝ)
14		fniniseg 7005	. . . . 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) → (𝑧 ∈ (◡(𝐹 ∘f + 𝐺) “ {𝐴}) ↔ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)))
16	8	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → 𝐺 Fn ℝ)
17		simprl 770	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ ℝ)
18		fnfvelrn 7025	. . . . . . . 8 ⊢ ((𝐺 Fn ℝ ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ∈ ran 𝐺)
19	16, 17, 18	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → (𝐺‘𝑧) ∈ ran 𝐺)
20		simprr 772	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)
21		eqidd 2737	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) = (𝐹‘𝑧))
22		eqidd 2737	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) = (𝐺‘𝑧))
23	4, 8, 10, 10, 11, 21, 22	ofval 7633	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ((𝐹 ∘f + 𝐺)‘𝑧) = ((𝐹‘𝑧) + (𝐺‘𝑧)))
24	23	ad2ant2r 747	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → ((𝐹 ∘f + 𝐺)‘𝑧) = ((𝐹‘𝑧) + (𝐺‘𝑧)))
25	20, 24	eqtr3d 2773	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → 𝐴 = ((𝐹‘𝑧) + (𝐺‘𝑧)))
26	25	oveq1d 7373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → (𝐴 − (𝐺‘𝑧)) = (((𝐹‘𝑧) + (𝐺‘𝑧)) − (𝐺‘𝑧)))
27		ax-resscn 11083	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℂ
28		fss 6678	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:ℝ⟶ℂ)
29	3, 27, 28	sylancl 586	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
30	29	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → 𝐹:ℝ⟶ℂ)
31	30, 17	ffvelcdmd 7030	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → (𝐹‘𝑧) ∈ ℂ)
32		fss 6678	. . . . . . . . . . . . . 14 ⊢ ((𝐺:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐺:ℝ⟶ℂ)
33	7, 27, 32	sylancl 586	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:ℝ⟶ℂ)
34	33	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → 𝐺:ℝ⟶ℂ)
35	34, 17	ffvelcdmd 7030	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → (𝐺‘𝑧) ∈ ℂ)
36	31, 35	pncand 11493	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → (((𝐹‘𝑧) + (𝐺‘𝑧)) − (𝐺‘𝑧)) = (𝐹‘𝑧))
37	26, 36	eqtr2d 2772	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → (𝐹‘𝑧) = (𝐴 − (𝐺‘𝑧)))
38	4	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → 𝐹 Fn ℝ)
39		fniniseg 7005	. . . . . . . . . 10 ⊢ (𝐹 Fn ℝ → (𝑧 ∈ (◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}) ↔ (𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 − (𝐺‘𝑧)))))
40	38, 39	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → (𝑧 ∈ (◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}) ↔ (𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 − (𝐺‘𝑧)))))
41	17, 37, 40	mpbir2and 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ (◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}))
42		eqidd 2737	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → (𝐺‘𝑧) = (𝐺‘𝑧))
43		fniniseg 7005	. . . . . . . . . 10 ⊢ (𝐺 Fn ℝ → (𝑧 ∈ (◡𝐺 “ {(𝐺‘𝑧)}) ↔ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = (𝐺‘𝑧))))
44	16, 43	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → (𝑧 ∈ (◡𝐺 “ {(𝐺‘𝑧)}) ↔ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = (𝐺‘𝑧))))
45	17, 42, 44	mpbir2and 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ (◡𝐺 “ {(𝐺‘𝑧)}))
46	41, 45	elind 4152	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ ((◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}) ∩ (◡𝐺 “ {(𝐺‘𝑧)})))
47		oveq2 7366	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐺‘𝑧) → (𝐴 − 𝑦) = (𝐴 − (𝐺‘𝑧)))
48	47	sneqd 4592	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐺‘𝑧) → {(𝐴 − 𝑦)} = {(𝐴 − (𝐺‘𝑧))})
49	48	imaeq2d 6019	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑧) → (◡𝐹 “ {(𝐴 − 𝑦)}) = (◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}))
50		sneq 4590	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐺‘𝑧) → {𝑦} = {(𝐺‘𝑧)})
51	50	imaeq2d 6019	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑧) → (◡𝐺 “ {𝑦}) = (◡𝐺 “ {(𝐺‘𝑧)}))
52	49, 51	ineq12d 4173	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑧) → ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})) = ((◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}) ∩ (◡𝐺 “ {(𝐺‘𝑧)})))
53	52	eleq2d 2822	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑧 ∈ ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})) ↔ 𝑧 ∈ ((◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}) ∩ (◡𝐺 “ {(𝐺‘𝑧)}))))
54	53	rspcev 3576	. . . . . . 7 ⊢ (((𝐺‘𝑧) ∈ ran 𝐺 ∧ 𝑧 ∈ ((◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}) ∩ (◡𝐺 “ {(𝐺‘𝑧)}))) → ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})))
55	19, 46, 54	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})))
56	55	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴) → ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦}))))
57		elin 3917	. . . . . . 7 ⊢ (𝑧 ∈ ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})) ↔ (𝑧 ∈ (◡𝐹 “ {(𝐴 − 𝑦)}) ∧ 𝑧 ∈ (◡𝐺 “ {𝑦})))
58	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → 𝐹 Fn ℝ)
59		fniniseg 7005	. . . . . . . . . 10 ⊢ (𝐹 Fn ℝ → (𝑧 ∈ (◡𝐹 “ {(𝐴 − 𝑦)}) ↔ (𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 − 𝑦))))
60	58, 59	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝑧 ∈ (◡𝐹 “ {(𝐴 − 𝑦)}) ↔ (𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 − 𝑦))))
61	8	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → 𝐺 Fn ℝ)
62		fniniseg 7005	. . . . . . . . . 10 ⊢ (𝐺 Fn ℝ → (𝑧 ∈ (◡𝐺 “ {𝑦}) ↔ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦)))
63	61, 62	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝑧 ∈ (◡𝐺 “ {𝑦}) ↔ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦)))
64	60, 63	anbi12d 632	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → ((𝑧 ∈ (◡𝐹 “ {(𝐴 − 𝑦)}) ∧ 𝑧 ∈ (◡𝐺 “ {𝑦})) ↔ ((𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 − 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦))))
65		anandi 676	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦)) ↔ ((𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 − 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦)))
66		simprl 770	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → 𝑧 ∈ ℝ)
67	23	ad2ant2r 747	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → ((𝐹 ∘f + 𝐺)‘𝑧) = ((𝐹‘𝑧) + (𝐺‘𝑧)))
68		simprrl 780	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → (𝐹‘𝑧) = (𝐴 − 𝑦))
69		simprrr 781	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → (𝐺‘𝑧) = 𝑦)
70	68, 69	oveq12d 7376	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → ((𝐹‘𝑧) + (𝐺‘𝑧)) = ((𝐴 − 𝑦) + 𝑦))
71		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → 𝐴 ∈ ℂ)
72	33	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → 𝐺:ℝ⟶ℂ)
73	72, 66	ffvelcdmd 7030	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → (𝐺‘𝑧) ∈ ℂ)
74	69, 73	eqeltrrd 2837	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → 𝑦 ∈ ℂ)
75	71, 74	npcand 11496	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → ((𝐴 − 𝑦) + 𝑦) = 𝐴)
76	67, 70, 75	3eqtrd 2775	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)
77	66, 76	jca 511	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴))
78	77	ex 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)))
79	65, 78	biimtrrid 243	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (((𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 − 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)))
80	64, 79	sylbid 240	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → ((𝑧 ∈ (◡𝐹 “ {(𝐴 − 𝑦)}) ∧ 𝑧 ∈ (◡𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)))
81	57, 80	biimtrid 242	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝑧 ∈ ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)))
82	81	rexlimdvw 3142	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)))
83	56, 82	impbid 212	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦}))))
84	15, 83	bitrd 279	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝑧 ∈ (◡(𝐹 ∘f + 𝐺) “ {𝐴}) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦}))))
85		eliun 4950	. . 3 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ ran 𝐺((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})))
86	84, 85	bitr4di 289	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝑧 ∈ (◡(𝐹 ∘f + 𝐺) “ {𝐴}) ↔ 𝑧 ∈ ∪ 𝑦 ∈ ran 𝐺((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦}))))
87	86	eqrdv 2734	1 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (◡(𝐹 ∘f + 𝐺) “ {𝐴}) = ∪ 𝑦 ∈ ran 𝐺((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3060  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901  {csn 4580  ∪ ciun 4946  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   “ cima 5627   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ∘_f cof 7620  ℂcc 11024  ℝcr 11025   + caddc 11029   − cmin 11364  ∫₁citg1 25572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-po 5532  df-so 5533  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-ltxr 11171  df-sub 11366  df-sum 15610  df-itg1 25577

This theorem is referenced by:  i1fadd  25652  itg1addlem4  25656