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Theorem summo 15665

Description: A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

**Hypotheses**
Ref	Expression
summo.1	⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
summo.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
summo.3	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)

**Assertion**
Ref	Expression
summo	⊢ (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))))

Distinct variable groups: 𝑓,𝑘,𝑚,𝑛,𝑥,𝐴 𝑓,𝐹,𝑘,𝑚,𝑛,𝑥 𝑘,𝐺,𝑚,𝑛,𝑥 𝜑,𝑘,𝑚,𝑛 𝐵,𝑓,𝑚,𝑛,𝑥 𝜑,𝑥,𝑓 Allowed substitution hints: 𝐵(𝑘) 𝐺(𝑓)

Proof of Theorem summo Dummy variables 𝑔 𝑗 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (ℤ_≥‘𝑚) = (ℤ_≥‘𝑛))
2	1	sseq2d 4007	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝐴 ⊆ (ℤ_≥‘𝑚) ↔ 𝐴 ⊆ (ℤ_≥‘𝑛)))
3		seqeq1 13970	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → seq𝑚( + , 𝐹) = seq𝑛( + , 𝐹))
4	3	breq1d 5149	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (seq𝑚( + , 𝐹) ⇝ 𝑦 ↔ seq𝑛( + , 𝐹) ⇝ 𝑦))
5	2, 4	anbi12d 630	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ↔ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)))
6	5	cbvrexvw 3227	. . . . . . 7 ⊢ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ↔ ∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))
7		reeanv 3218	. . . . . . . . 9 ⊢ (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ ∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)))
8		simprlr 777	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → seq𝑚( + , 𝐹) ⇝ 𝑥)
9		summo.1	. . . . . . . . . . . . . 14 ⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
10		summo.2	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
11	10	ad4ant14 749	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
12		simplrl 774	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝑚 ∈ ℤ)
13		simplrr 775	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝑛 ∈ ℤ)
14		simprll 776	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝐴 ⊆ (ℤ_≥‘𝑚))
15		simprrl 778	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝐴 ⊆ (ℤ_≥‘𝑛))
16	9, 11, 12, 13, 14, 15	sumrb 15661	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → (seq𝑚( + , 𝐹) ⇝ 𝑥 ↔ seq𝑛( + , 𝐹) ⇝ 𝑥))
17	8, 16	mpbid 231	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → seq𝑛( + , 𝐹) ⇝ 𝑥)
18		simprrr 779	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → seq𝑛( + , 𝐹) ⇝ 𝑦)
19		climuni 15498	. . . . . . . . . . . 12 ⊢ ((seq𝑛( + , 𝐹) ⇝ 𝑥 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦)
20	17, 18, 19	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝑥 = 𝑦)
21	20	exp31 419	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) → 𝑥 = 𝑦)))
22	21	rexlimdvv 3202	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) → 𝑥 = 𝑦))
23	7, 22	biimtrrid 242	. . . . . . . 8 ⊢ (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ ∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) → 𝑥 = 𝑦))
24	23	expdimp 452	. . . . . . 7 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦))
25	6, 24	biimtrid 241	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦))
26		summo.3	. . . . . . 7 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
27	9, 10, 26	summolem2 15664	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
28	25, 27	jaod 856	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚))) → 𝑥 = 𝑦))
29	9, 10, 26	summolem2 15664	. . . . . . . 8 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) → 𝑦 = 𝑥))
30		equcom 2013	. . . . . . . 8 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
31	29, 30	imbitrdi 250	. . . . . . 7 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
32	31	impancom 451	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦))
33		oveq2 7410	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
34	33	f1oeq2d 6820	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑓:(1...𝑚)–1-1-onto→𝐴 ↔ 𝑓:(1...𝑛)–1-1-onto→𝐴))
35		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (seq1( + , 𝐺)‘𝑚) = (seq1( + , 𝐺)‘𝑛))
36	35	eqeq2d 2735	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑦 = (seq1( + , 𝐺)‘𝑚) ↔ 𝑦 = (seq1( + , 𝐺)‘𝑛)))
37	34, 36	anbi12d 630	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) ↔ (𝑓:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑛))))
38	37	exbidv 1916	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑛))))
39		f1oeq1 6812	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (𝑓:(1...𝑛)–1-1-onto→𝐴 ↔ 𝑔:(1...𝑛)–1-1-onto→𝐴))
40		fveq1 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑛) = (𝑔‘𝑛))
41	40	csbeq1d 3890	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑔 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑔‘𝑛) / 𝑘⦌𝐵)
42	41	mpteq2dv 5241	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑔 → (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) = (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))
43	26, 42	eqtrid 2776	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → 𝐺 = (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))
44	43	seqeq3d 13975	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵)))
45	44	fveq1d 6884	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (seq1( + , 𝐺)‘𝑛) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛))
46	45	eqeq2d 2735	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (𝑦 = (seq1( + , 𝐺)‘𝑛) ↔ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛)))
47	39, 46	anbi12d 630	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ((𝑓:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑛)) ↔ (𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛))))
48	47	cbvexvw 2032	. . . . . . . . 9 ⊢ (∃𝑓(𝑓:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑛)) ↔ ∃𝑔(𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛)))
49	38, 48	bitrdi 287	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑔(𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛))))
50	49	cbvrexvw 3227	. . . . . . 7 ⊢ (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑛 ∈ ℕ ∃𝑔(𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛)))
51		reeanv 3218	. . . . . . . . 9 ⊢ (∃𝑚 ∈ ℕ ∃𝑛 ∈ ℕ (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛))) ↔ (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑛 ∈ ℕ ∃𝑔(𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛))))
52		exdistrv 1951	. . . . . . . . . . 11 ⊢ (∃𝑓∃𝑔((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛))) ↔ (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛))))
53		an4 653	. . . . . . . . . . . . 13 ⊢ (((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛))) ↔ ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴) ∧ (𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛))))
54	10	ad4ant14 749	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
55		fveq2 6882	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑗 → (𝑓‘𝑛) = (𝑓‘𝑗))
56	55	csbeq1d 3890	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑗 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
57	56	cbvmptv 5252	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) = (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
58	26, 57	eqtri 2752	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
59		fveq2 6882	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑗 → (𝑔‘𝑛) = (𝑔‘𝑗))
60	59	csbeq1d 3890	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑗 → ⦋(𝑔‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑔‘𝑗) / 𝑘⦌𝐵)
61	60	cbvmptv 5252	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵) = (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵)
62		simplr 766	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ))
63		simprl 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → 𝑓:(1...𝑚)–1-1-onto→𝐴)
64		simprr 770	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → 𝑔:(1...𝑛)–1-1-onto→𝐴)
65	9, 54, 58, 61, 62, 63, 64	summolem3 15662	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → (seq1( + , 𝐺)‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛))
66		eqeq12 2741	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛)) → (𝑥 = 𝑦 ↔ (seq1( + , 𝐺)‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛)))
67	65, 66	syl5ibrcom 246	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → ((𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛)) → 𝑥 = 𝑦))
68	67	expimpd 453	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) → (((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴) ∧ (𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛))) → 𝑥 = 𝑦))
69	53, 68	biimtrid 241	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) → (((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛))) → 𝑥 = 𝑦))
70	69	exlimdvv 1929	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) → (∃𝑓∃𝑔((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛))) → 𝑥 = 𝑦))
71	52, 70	biimtrrid 242	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) → ((∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛))) → 𝑥 = 𝑦))
72	71	rexlimdvva 3203	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑚 ∈ ℕ ∃𝑛 ∈ ℕ (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛))) → 𝑥 = 𝑦))
73	51, 72	biimtrrid 242	. . . . . . . 8 ⊢ (𝜑 → ((∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑛 ∈ ℕ ∃𝑔(𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛))) → 𝑥 = 𝑦))
74	73	expdimp 452	. . . . . . 7 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))) → (∃𝑛 ∈ ℕ ∃𝑔(𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛)) → 𝑥 = 𝑦))
75	50, 74	biimtrid 241	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
76	32, 75	jaod 856	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚))) → 𝑥 = 𝑦))
77	28, 76	jaodan 954	. . . 4 ⊢ ((𝜑 ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)))) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚))) → 𝑥 = 𝑦))
78	77	expimpd 453	. . 3 ⊢ (𝜑 → (((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)))) → 𝑥 = 𝑦))
79	78	alrimivv 1923	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦(((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)))) → 𝑥 = 𝑦))
80		breq2 5143	. . . . . 6 ⊢ (𝑥 = 𝑦 → (seq𝑚( + , 𝐹) ⇝ 𝑥 ↔ seq𝑚( + , 𝐹) ⇝ 𝑦))
81	80	anbi2d 628	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)))
82	81	rexbidv 3170	. . . 4 ⊢ (𝑥 = 𝑦 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)))
83		eqeq1 2728	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = (seq1( + , 𝐺)‘𝑚) ↔ 𝑦 = (seq1( + , 𝐺)‘𝑚)))
84	83	anbi2d 628	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚))))
85	84	exbidv 1916	. . . . 5 ⊢ (𝑥 = 𝑦 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚))))
86	85	rexbidv 3170	. . . 4 ⊢ (𝑥 = 𝑦 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚))))
87	82, 86	orbi12d 915	. . 3 ⊢ (𝑥 = 𝑦 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)))))
88	87	mo4 2552	. 2 ⊢ (∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))) ↔ ∀𝑥∀𝑦(((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)))) → 𝑥 = 𝑦))
89	79, 88	sylibr 233	1 ⊢ (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 844 ∀wal 1531 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∃*wmo 2524 ∃wrex 3062 ⦋csb 3886 ⊆ wss 3941 ifcif 4521 class class class wbr 5139 ↦ cmpt 5222 –1-1-onto→wf1o 6533 ‘cfv 6534 (class class class)co 7402 ℂcc 11105 0cc0 11107 1c1 11108 + caddc 11110 ℕcn 12211 ℤcz 12557 ℤ_≥cuz 12821 ...cfz 13485 seqcseq 13967 ⇝ cli 15430

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-oi 9502 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12976 df-fz 13486 df-fzo 13629 df-seq 13968 df-exp 14029 df-hash 14292 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434

This theorem is referenced by: fsum 15668

W3C validator