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Theorem i1faddlem 23266

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 23249	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫₁ → 𝐹:ℝ⟶ℝ)
3	1, 2	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
4		ffn 5958	. . . . . . . 8 ⊢ (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ)
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ)
6		i1fadd.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ dom ∫₁)
7		i1ff 23249	. . . . . . . . 9 ⊢ (𝐺 ∈ dom ∫₁ → 𝐺:ℝ⟶ℝ)
8	6, 7	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
9		ffn 5958	. . . . . . . 8 ⊢ (𝐺:ℝ⟶ℝ → 𝐺 Fn ℝ)
10	8, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ)
11		reex 9906	. . . . . . . 8 ⊢ ℝ ∈ V
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ V)
13		inidm 3784	. . . . . . 7 ⊢ (ℝ ∩ ℝ) = ℝ
14	5, 10, 12, 12, 13	offn 6806	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐺) Fn ℝ)
15	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝐹 ∘_𝑓 + 𝐺) Fn ℝ)
16		fniniseg 6246	. . . . 5 ⊢ ((𝐹 ∘_𝑓 + 𝐺) Fn ℝ → (𝑧 ∈ (^◡(𝐹 ∘_𝑓 + 𝐺) “ {𝐴}) ↔ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)))
17	15, 16	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝑧 ∈ (^◡(𝐹 ∘_𝑓 + 𝐺) “ {𝐴}) ↔ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)))
18	10	ad2antrr 758	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝐺 Fn ℝ)
19		simprl 790	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ ℝ)
20		fnfvelrn 6264	. . . . . . . 8 ⊢ ((𝐺 Fn ℝ ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ∈ ran 𝐺)
21	18, 19, 20	syl2anc 691	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝐺‘𝑧) ∈ ran 𝐺)
22		simprr 792	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)
23		eqidd 2611	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) = (𝐹‘𝑧))
24		eqidd 2611	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) = (𝐺‘𝑧))
25	5, 10, 12, 12, 13, 23, 24	ofval 6804	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = ((𝐹‘𝑧) + (𝐺‘𝑧)))
26	25	ad2ant2r 779	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = ((𝐹‘𝑧) + (𝐺‘𝑧)))
27	22, 26	eqtr3d 2646	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝐴 = ((𝐹‘𝑧) + (𝐺‘𝑧)))
28	27	oveq1d 6564	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝐴 − (𝐺‘𝑧)) = (((𝐹‘𝑧) + (𝐺‘𝑧)) − (𝐺‘𝑧)))
29		ax-resscn 9872	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℂ
30		fss 5969	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:ℝ⟶ℂ)
31	3, 29, 30	sylancl 693	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
32	31	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝐹:ℝ⟶ℂ)
33	32, 19	ffvelrnd 6268	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝐹‘𝑧) ∈ ℂ)
34		fss 5969	. . . . . . . . . . . . . 14 ⊢ ((𝐺:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐺:ℝ⟶ℂ)
35	8, 29, 34	sylancl 693	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:ℝ⟶ℂ)
36	35	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝐺:ℝ⟶ℂ)
37	36, 19	ffvelrnd 6268	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝐺‘𝑧) ∈ ℂ)
38	33, 37	pncand 10272	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → (((𝐹‘𝑧) + (𝐺‘𝑧)) − (𝐺‘𝑧)) = (𝐹‘𝑧))
39	28, 38	eqtr2d 2645	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝐹‘𝑧) = (𝐴 − (𝐺‘𝑧)))
40	5	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝐹 Fn ℝ)
41		fniniseg 6246	. . . . . . . . . 10 ⊢ (𝐹 Fn ℝ → (𝑧 ∈ (^◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}) ↔ (𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 − (𝐺‘𝑧)))))
42	40, 41	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)))))
43	19, 39, 42	mpbir2and 959	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ (^◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}))
44		eqidd 2611	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝐺‘𝑧) = (𝐺‘𝑧))
45		fniniseg 6246	. . . . . . . . . 10 ⊢ (𝐺 Fn ℝ → (𝑧 ∈ (^◡𝐺 “ {(𝐺‘𝑧)}) ↔ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = (𝐺‘𝑧))))
46	18, 45	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝑧 ∈ (^◡𝐺 “ {(𝐺‘𝑧)}) ↔ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = (𝐺‘𝑧))))
47	19, 44, 46	mpbir2and 959	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ (^◡𝐺 “ {(𝐺‘𝑧)}))
48	43, 47	elind 3760	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ ((^◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}) ∩ (^◡𝐺 “ {(𝐺‘𝑧)})))
49		oveq2 6557	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐺‘𝑧) → (𝐴 − 𝑦) = (𝐴 − (𝐺‘𝑧)))
50	49	sneqd 4137	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐺‘𝑧) → {(𝐴 − 𝑦)} = {(𝐴 − (𝐺‘𝑧))})
51	50	imaeq2d 5385	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑧) → (^◡𝐹 “ {(𝐴 − 𝑦)}) = (^◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}))
52		sneq 4135	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐺‘𝑧) → {𝑦} = {(𝐺‘𝑧)})
53	52	imaeq2d 5385	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑧) → (^◡𝐺 “ {𝑦}) = (^◡𝐺 “ {(𝐺‘𝑧)}))
54	51, 53	ineq12d 3777	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑧) → ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦})) = ((^◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}) ∩ (^◡𝐺 “ {(𝐺‘𝑧)})))
55	54	eleq2d 2673	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑧 ∈ ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦})) ↔ 𝑧 ∈ ((^◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}) ∩ (^◡𝐺 “ {(𝐺‘𝑧)}))))
56	55	rspcev 3282	. . . . . . 7 ⊢ (((𝐺‘𝑧) ∈ ran 𝐺 ∧ 𝑧 ∈ ((^◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}) ∩ (^◡𝐺 “ {(𝐺‘𝑧)}))) → ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦})))
57	21, 48, 56	syl2anc 691	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦})))
58	57	ex 449	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴) → ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦}))))
59		elin 3758	. . . . . . 7 ⊢ (𝑧 ∈ ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦})) ↔ (𝑧 ∈ (^◡𝐹 “ {(𝐴 − 𝑦)}) ∧ 𝑧 ∈ (^◡𝐺 “ {𝑦})))
60	5	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → 𝐹 Fn ℝ)
61		fniniseg 6246	. . . . . . . . . 10 ⊢ (𝐹 Fn ℝ → (𝑧 ∈ (^◡𝐹 “ {(𝐴 − 𝑦)}) ↔ (𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 − 𝑦))))
62	60, 61	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝑧 ∈ (^◡𝐹 “ {(𝐴 − 𝑦)}) ↔ (𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 − 𝑦))))
63	10	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → 𝐺 Fn ℝ)
64		fniniseg 6246	. . . . . . . . . 10 ⊢ (𝐺 Fn ℝ → (𝑧 ∈ (^◡𝐺 “ {𝑦}) ↔ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦)))
65	63, 64	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝑧 ∈ (^◡𝐺 “ {𝑦}) ↔ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦)))
66	62, 65	anbi12d 743	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → ((𝑧 ∈ (^◡𝐹 “ {(𝐴 − 𝑦)}) ∧ 𝑧 ∈ (^◡𝐺 “ {𝑦})) ↔ ((𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 − 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦))))
67		anandi 867	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦)) ↔ ((𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 − 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦)))
68		simprl 790	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → 𝑧 ∈ ℝ)
69	25	ad2ant2r 779	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = ((𝐹‘𝑧) + (𝐺‘𝑧)))
70		simprrl 800	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → (𝐹‘𝑧) = (𝐴 − 𝑦))
71		simprrr 801	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → (𝐺‘𝑧) = 𝑦)
72	70, 71	oveq12d 6567	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → ((𝐹‘𝑧) + (𝐺‘𝑧)) = ((𝐴 − 𝑦) + 𝑦))
73		simplr 788	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → 𝐴 ∈ ℂ)
74	35	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → 𝐺:ℝ⟶ℂ)
75	74, 68	ffvelrnd 6268	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → (𝐺‘𝑧) ∈ ℂ)
76	71, 75	eqeltrrd 2689	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → 𝑦 ∈ ℂ)
77	73, 76	npcand 10275	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → ((𝐴 − 𝑦) + 𝑦) = 𝐴)
78	69, 72, 77	3eqtrd 2648	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)
79	68, 78	jca 553	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴))
80	79	ex 449	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)))
81	67, 80	syl5bir 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (((𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 − 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)))
82	66, 81	sylbid 229	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → ((𝑧 ∈ (^◡𝐹 “ {(𝐴 − 𝑦)}) ∧ 𝑧 ∈ (^◡𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)))
83	59, 82	syl5bi 231	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝑧 ∈ ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)))
84	83	rexlimdvw 3016	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)))
85	58, 84	impbid 201	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦}))))
86	17, 85	bitrd 267	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝑧 ∈ (^◡(𝐹 ∘_𝑓 + 𝐺) “ {𝐴}) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦}))))
87		eliun 4460	. . 3 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ ran 𝐺((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦})) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦})))
88	86, 87	syl6bbr 277	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝑧 ∈ (^◡(𝐹 ∘_𝑓 + 𝐺) “ {𝐴}) ↔ 𝑧 ∈ ∪ 𝑦 ∈ ran 𝐺((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦}))))
89	88	eqrdv 2608	1 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (^◡(𝐹 ∘_𝑓 + 𝐺) “ {𝐴}) = ∪ 𝑦 ∈ ran 𝐺((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 {csn 4125 ∪ ciun 4455 ^◡ccnv 5037 dom cdm 5038 ran crn 5039 “ cima 5041 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘_𝑓 cof 6793 ℂcc 9813 ℝcr 9814 + caddc 9818 − cmin 10145 ∫₁citg1 23190

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-sum 14265 df-itg1 23195

This theorem is referenced by: i1fadd 23268 itg1addlem4 23272

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