Theorem i1faddlem 24407

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 24390	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ)
3	1, 2	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
4	3	ffnd 6504	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ)
5		i1fadd.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ dom ∫1)
6		i1ff 24390	. . . . . . . . 9 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ)
7	5, 6	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
8	7	ffnd 6504	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ)
9		reex 10679	. . . . . . . 8 ⊢ ℝ ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ V)
11		inidm 4125	. . . . . . 7 ⊢ (ℝ ∩ ℝ) = ℝ
12	4, 8, 10, 10, 11	offn 7423	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) Fn ℝ)
13	12	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝐹 ∘f + 𝐺) Fn ℝ)
14		fniniseg 6826	. . . . 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 725	. . . . . . . 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 6845	. . . . . . . 8 ⊢ ((𝐺 Fn ℝ ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ∈ ran 𝐺)
19	16, 17, 18	syl2anc 587	. . . . . . 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 2759	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) = (𝐹‘𝑧))
22		eqidd 2759	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) = (𝐺‘𝑧))
23	4, 8, 10, 10, 11, 21, 22	ofval 7421	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ((𝐹 ∘f + 𝐺)‘𝑧) = ((𝐹‘𝑧) + (𝐺‘𝑧)))
24	23	ad2ant2r 746	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → ((𝐹 ∘f + 𝐺)‘𝑧) = ((𝐹‘𝑧) + (𝐺‘𝑧)))
25	20, 24	eqtr3d 2795	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → 𝐴 = ((𝐹‘𝑧) + (𝐺‘𝑧)))
26	25	oveq1d 7171	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → (𝐴 − (𝐺‘𝑧)) = (((𝐹‘𝑧) + (𝐺‘𝑧)) − (𝐺‘𝑧)))
27		ax-resscn 10645	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℂ
28		fss 6517	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:ℝ⟶ℂ)
29	3, 27, 28	sylancl 589	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
30	29	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → 𝐹:ℝ⟶ℂ)
31	30, 17	ffvelrnd 6849	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → (𝐹‘𝑧) ∈ ℂ)
32		fss 6517	. . . . . . . . . . . . . 14 ⊢ ((𝐺:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐺:ℝ⟶ℂ)
33	7, 27, 32	sylancl 589	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:ℝ⟶ℂ)
34	33	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → 𝐺:ℝ⟶ℂ)
35	34, 17	ffvelrnd 6849	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → (𝐺‘𝑧) ∈ ℂ)
36	31, 35	pncand 11049	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → (((𝐹‘𝑧) + (𝐺‘𝑧)) − (𝐺‘𝑧)) = (𝐹‘𝑧))
37	26, 36	eqtr2d 2794	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → (𝐹‘𝑧) = (𝐴 − (𝐺‘𝑧)))
38	4	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → 𝐹 Fn ℝ)
39		fniniseg 6826	. . . . . . . . . 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 712	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ (◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}))
42		eqidd 2759	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → (𝐺‘𝑧) = (𝐺‘𝑧))
43		fniniseg 6826	. . . . . . . . . 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 712	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ (◡𝐺 “ {(𝐺‘𝑧)}))
46	41, 45	elind 4101	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ ((◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}) ∩ (◡𝐺 “ {(𝐺‘𝑧)})))
47		oveq2 7164	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐺‘𝑧) → (𝐴 − 𝑦) = (𝐴 − (𝐺‘𝑧)))
48	47	sneqd 4537	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐺‘𝑧) → {(𝐴 − 𝑦)} = {(𝐴 − (𝐺‘𝑧))})
49	48	imaeq2d 5906	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑧) → (◡𝐹 “ {(𝐴 − 𝑦)}) = (◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}))
50		sneq 4535	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐺‘𝑧) → {𝑦} = {(𝐺‘𝑧)})
51	50	imaeq2d 5906	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑧) → (◡𝐺 “ {𝑦}) = (◡𝐺 “ {(𝐺‘𝑧)}))
52	49, 51	ineq12d 4120	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑧) → ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})) = ((◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}) ∩ (◡𝐺 “ {(𝐺‘𝑧)})))
53	52	eleq2d 2837	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑧 ∈ ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})) ↔ 𝑧 ∈ ((◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}) ∩ (◡𝐺 “ {(𝐺‘𝑧)}))))
54	53	rspcev 3543	. . . . . . 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 587	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)) → ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})))
56	55	ex 416	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴) → ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦}))))
57		elin 3876	. . . . . . 7 ⊢ (𝑧 ∈ ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})) ↔ (𝑧 ∈ (◡𝐹 “ {(𝐴 − 𝑦)}) ∧ 𝑧 ∈ (◡𝐺 “ {𝑦})))
58	4	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → 𝐹 Fn ℝ)
59		fniniseg 6826	. . . . . . . . . 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 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → 𝐺 Fn ℝ)
62		fniniseg 6826	. . . . . . . . . 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 633	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → ((𝑧 ∈ (◡𝐹 “ {(𝐴 − 𝑦)}) ∧ 𝑧 ∈ (◡𝐺 “ {𝑦})) ↔ ((𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 − 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦))))
65		anandi 675	. . . . . . . . 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 746	. . . . . . . . . . . 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 7174	. . . . . . . . . . . 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 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → 𝐺:ℝ⟶ℂ)
73	72, 66	ffvelrnd 6849	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → (𝐺‘𝑧) ∈ ℂ)
74	69, 73	eqeltrrd 2853	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → 𝑦 ∈ ℂ)
75	71, 74	npcand 11052	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → ((𝐴 − 𝑦) + 𝑦) = 𝐴)
76	67, 70, 75	3eqtrd 2797	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)
77	66, 76	jca 515	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴))
78	77	ex 416	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)))
79	65, 78	syl5bir 246	. . . . . . . 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 243	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → ((𝑧 ∈ (◡𝐹 “ {(𝐴 − 𝑦)}) ∧ 𝑧 ∈ (◡𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)))
81	57, 80	syl5bi 245	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝑧 ∈ ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)))
82	81	rexlimdvw 3214	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴)))
83	56, 82	impbid 215	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹 ∘f + 𝐺)‘𝑧) = 𝐴) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦}))))
84	15, 83	bitrd 282	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝑧 ∈ (◡(𝐹 ∘f + 𝐺) “ {𝐴}) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦}))))
85		eliun 4890	. . 3 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ ran 𝐺((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})))
86	84, 85	bitr4di 292	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝑧 ∈ (◡(𝐹 ∘f + 𝐺) “ {𝐴}) ↔ 𝑧 ∈ ∪ 𝑦 ∈ ran 𝐺((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦}))))
87	86	eqrdv 2756	1 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (◡(𝐹 ∘f + 𝐺) “ {𝐴}) = ∪ 𝑦 ∈ ran 𝐺((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3071  Vcvv 3409   ∩ cin 3859   ⊆ wss 3860  {csn 4525  ∪ ciun 4886  ^◡ccnv 5527  dom cdm 5528  ran crn 5529   “ cima 5531   Fn wfn 6335  ⟶wf 6336  ‘cfv 6340  (class class class)co 7156   ∘_f cof 7409  ℂcc 10586  ℝcr 10587   + caddc 10591   − cmin 10921  ∫₁citg1 24329

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-po 5447  df-so 5448  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7411  df-er 8305  df-en 8541  df-dom 8542  df-sdom 8543  df-pnf 10728  df-mnf 10729  df-ltxr 10731  df-sub 10923  df-sum 15104  df-itg1 24334

This theorem is referenced by:  i1fadd  24409  itg1addlem4  24413