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Theorem i1faddlem 23460

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 23443	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫₁ → 𝐹:ℝ⟶ℝ)
3	1, 2	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
4		ffn 6045	. . . . . . . 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 23443	. . . . . . . . 9 ⊢ (𝐺 ∈ dom ∫₁ → 𝐺:ℝ⟶ℝ)
8	6, 7	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
9		ffn 6045	. . . . . . . 8 ⊢ (𝐺:ℝ⟶ℝ → 𝐺 Fn ℝ)
10	8, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ)
11		reex 10027	. . . . . . . 8 ⊢ ℝ ∈ V
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ V)
13		inidm 3822	. . . . . . 7 ⊢ (ℝ ∩ ℝ) = ℝ
14	5, 10, 12, 12, 13	offn 6908	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐺) Fn ℝ)
15	14	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝐹 ∘_𝑓 + 𝐺) Fn ℝ)
16		fniniseg 6338	. . . . 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 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝐺 Fn ℝ)
19		simprl 794	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ ℝ)
20		fnfvelrn 6356	. . . . . . . 8 ⊢ ((𝐺 Fn ℝ ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ∈ ran 𝐺)
21	18, 19, 20	syl2anc 693	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝐺‘𝑧) ∈ ran 𝐺)
22		simprr 796	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)
23		eqidd 2623	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) = (𝐹‘𝑧))
24		eqidd 2623	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) = (𝐺‘𝑧))
25	5, 10, 12, 12, 13, 23, 24	ofval 6906	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = ((𝐹‘𝑧) + (𝐺‘𝑧)))
26	25	ad2ant2r 783	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = ((𝐹‘𝑧) + (𝐺‘𝑧)))
27	22, 26	eqtr3d 2658	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝐴 = ((𝐹‘𝑧) + (𝐺‘𝑧)))
28	27	oveq1d 6665	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝐴 − (𝐺‘𝑧)) = (((𝐹‘𝑧) + (𝐺‘𝑧)) − (𝐺‘𝑧)))
29		ax-resscn 9993	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℂ
30		fss 6056	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:ℝ⟶ℂ)
31	3, 29, 30	sylancl 694	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
32	31	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝐹:ℝ⟶ℂ)
33	32, 19	ffvelrnd 6360	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝐹‘𝑧) ∈ ℂ)
34		fss 6056	. . . . . . . . . . . . . 14 ⊢ ((𝐺:ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐺:ℝ⟶ℂ)
35	8, 29, 34	sylancl 694	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:ℝ⟶ℂ)
36	35	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝐺:ℝ⟶ℂ)
37	36, 19	ffvelrnd 6360	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝐺‘𝑧) ∈ ℂ)
38	33, 37	pncand 10393	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → (((𝐹‘𝑧) + (𝐺‘𝑧)) − (𝐺‘𝑧)) = (𝐹‘𝑧))
39	28, 38	eqtr2d 2657	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝐹‘𝑧) = (𝐴 − (𝐺‘𝑧)))
40	5	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝐹 Fn ℝ)
41		fniniseg 6338	. . . . . . . . . 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 957	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ (^◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}))
44		eqidd 2623	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → (𝐺‘𝑧) = (𝐺‘𝑧))
45		fniniseg 6338	. . . . . . . . . 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 957	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ (^◡𝐺 “ {(𝐺‘𝑧)}))
48	43, 47	elind 3798	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → 𝑧 ∈ ((^◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}) ∩ (^◡𝐺 “ {(𝐺‘𝑧)})))
49		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐺‘𝑧) → (𝐴 − 𝑦) = (𝐴 − (𝐺‘𝑧)))
50	49	sneqd 4189	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐺‘𝑧) → {(𝐴 − 𝑦)} = {(𝐴 − (𝐺‘𝑧))})
51	50	imaeq2d 5466	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑧) → (^◡𝐹 “ {(𝐴 − 𝑦)}) = (^◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}))
52		sneq 4187	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐺‘𝑧) → {𝑦} = {(𝐺‘𝑧)})
53	52	imaeq2d 5466	. . . . . . . . . 10 ⊢ (𝑦 = (𝐺‘𝑧) → (^◡𝐺 “ {𝑦}) = (^◡𝐺 “ {(𝐺‘𝑧)}))
54	51, 53	ineq12d 3815	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑧) → ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦})) = ((^◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}) ∩ (^◡𝐺 “ {(𝐺‘𝑧)})))
55	54	eleq2d 2687	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑧 ∈ ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦})) ↔ 𝑧 ∈ ((^◡𝐹 “ {(𝐴 − (𝐺‘𝑧))}) ∩ (^◡𝐺 “ {(𝐺‘𝑧)}))))
56	55	rspcev 3309	. . . . . . 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 693	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)) → ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦})))
58	57	ex 450	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴) → ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦}))))
59		elin 3796	. . . . . . 7 ⊢ (𝑧 ∈ ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦})) ↔ (𝑧 ∈ (^◡𝐹 “ {(𝐴 − 𝑦)}) ∧ 𝑧 ∈ (^◡𝐺 “ {𝑦})))
60	5	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → 𝐹 Fn ℝ)
61		fniniseg 6338	. . . . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → 𝐺 Fn ℝ)
64		fniniseg 6338	. . . . . . . . . 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 747	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → ((𝑧 ∈ (^◡𝐹 “ {(𝐴 − 𝑦)}) ∧ 𝑧 ∈ (^◡𝐺 “ {𝑦})) ↔ ((𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 − 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦))))
67		anandi 871	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦)) ↔ ((𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 − 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦)))
68		simprl 794	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → 𝑧 ∈ ℝ)
69	25	ad2ant2r 783	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = ((𝐹‘𝑧) + (𝐺‘𝑧)))
70		simprrl 804	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → (𝐹‘𝑧) = (𝐴 − 𝑦))
71		simprrr 805	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → (𝐺‘𝑧) = 𝑦)
72	70, 71	oveq12d 6668	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → ((𝐹‘𝑧) + (𝐺‘𝑧)) = ((𝐴 − 𝑦) + 𝑦))
73		simplr 792	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → 𝐴 ∈ ℂ)
74	35	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → 𝐺:ℝ⟶ℂ)
75	74, 68	ffvelrnd 6360	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → (𝐺‘𝑧) ∈ ℂ)
76	71, 75	eqeltrrd 2702	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → 𝑦 ∈ ℂ)
77	73, 76	npcand 10396	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → ((𝐴 − 𝑦) + 𝑦) = 𝐴)
78	69, 72, 77	3eqtrd 2660	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)
79	68, 78	jca 554	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ ℂ) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦))) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴))
80	79	ex 450	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 − 𝑦) ∧ (𝐺‘𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)))
81	67, 80	syl5bir 233	. . . . . . . 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 230	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → ((𝑧 ∈ (^◡𝐹 “ {(𝐴 − 𝑦)}) ∧ 𝑧 ∈ (^◡𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)))
83	59, 82	syl5bi 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝑧 ∈ ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)))
84	83	rexlimdvw 3034	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴)))
85	58, 84	impbid 202	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → ((𝑧 ∈ ℝ ∧ ((𝐹 ∘_𝑓 + 𝐺)‘𝑧) = 𝐴) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦}))))
86	17, 85	bitrd 268	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝑧 ∈ (^◡(𝐹 ∘_𝑓 + 𝐺) “ {𝐴}) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦}))))
87		eliun 4524	. . 3 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ ran 𝐺((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦})) ↔ ∃𝑦 ∈ ran 𝐺 𝑧 ∈ ((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦})))
88	86, 87	syl6bbr 278	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (𝑧 ∈ (^◡(𝐹 ∘_𝑓 + 𝐺) “ {𝐴}) ↔ 𝑧 ∈ ∪ 𝑦 ∈ ran 𝐺((^◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (^◡𝐺 “ {𝑦}))))
89	88	eqrdv 2620	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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 {csn 4177 ∪ ciun 4520 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 “ cima 5117 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘_𝑓 cof 6895 ℂcc 9934 ℝcr 9935 + caddc 9939 − cmin 10266 ∫₁citg1 23384

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-sum 14417 df-itg1 23389

This theorem is referenced by: i1fadd 23462 itg1addlem4 23466

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