summo - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > summo		Structured version Visualization version GIF version

Theorem summo 15076

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 6672	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (ℤ_≥‘𝑚) = (ℤ_≥‘𝑛))
2	1	sseq2d 4001	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝐴 ⊆ (ℤ_≥‘𝑚) ↔ 𝐴 ⊆ (ℤ_≥‘𝑛)))
3		seqeq1 13375	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → seq𝑚( + , 𝐹) = seq𝑛( + , 𝐹))
4	3	breq1d 5078	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (seq𝑚( + , 𝐹) ⇝ 𝑦 ↔ seq𝑛( + , 𝐹) ⇝ 𝑦))
5	2, 4	anbi12d 632	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ↔ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)))
6	5	cbvrexvw 3452	. . . . . . 7 ⊢ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ↔ ∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))
7		reeanv 3369	. . . . . . . . 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 778	. . . . . . . . . . . . 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 750	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
12		simplrl 775	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝑚 ∈ ℤ)
13		simplrr 776	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝑛 ∈ ℤ)
14		simprll 777	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝐴 ⊆ (ℤ_≥‘𝑚))
15		simprrl 779	. . . . . . . . . . . . . 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 15072	. . . . . . . . . . . . 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 234	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → seq𝑛( + , 𝐹) ⇝ 𝑥)
18		simprrr 780	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → seq𝑛( + , 𝐹) ⇝ 𝑦)
19		climuni 14911	. . . . . . . . . . . 12 ⊢ ((seq𝑛( + , 𝐹) ⇝ 𝑥 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦)
20	17, 18, 19	syl2anc 586	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝑥 = 𝑦)
21	20	exp31 422	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) → 𝑥 = 𝑦)))
22	21	rexlimdvv 3295	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) → 𝑥 = 𝑦))
23	7, 22	syl5bir 245	. . . . . . . 8 ⊢ (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ ∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) → 𝑥 = 𝑦))
24	23	expdimp 455	. . . . . . 7 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦))
25	6, 24	syl5bi 244	. . . . . 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 15075	. . . . . 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 855	. . . . 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 15075	. . . . . . . 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 2025	. . . . . . . 8 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
31	29, 30	syl6ib 253	. . . . . . 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 454	. . . . . 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 7166	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
34	33	f1oeq2d 6613	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑓:(1...𝑚)–1-1-onto→𝐴 ↔ 𝑓:(1...𝑛)–1-1-onto→𝐴))
35		fveq2 6672	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (seq1( + , 𝐺)‘𝑚) = (seq1( + , 𝐺)‘𝑛))
36	35	eqeq2d 2834	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑦 = (seq1( + , 𝐺)‘𝑚) ↔ 𝑦 = (seq1( + , 𝐺)‘𝑛)))
37	34, 36	anbi12d 632	. . . . . . . . . 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 1922	. . . . . . . . 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 6606	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (𝑓:(1...𝑛)–1-1-onto→𝐴 ↔ 𝑔:(1...𝑛)–1-1-onto→𝐴))
40		fveq1 6671	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑛) = (𝑔‘𝑛))
41	40	csbeq1d 3889	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑔 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑔‘𝑛) / 𝑘⦌𝐵)
42	41	mpteq2dv 5164	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑔 → (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) = (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))
43	26, 42	syl5eq 2870	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → 𝐺 = (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))
44	43	seqeq3d 13380	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵)))
45	44	fveq1d 6674	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (seq1( + , 𝐺)‘𝑛) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛))
46	45	eqeq2d 2834	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (𝑦 = (seq1( + , 𝐺)‘𝑛) ↔ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵))‘𝑛)))
47	39, 46	anbi12d 632	. . . . . . . . . 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 2044	. . . . . . . . 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	syl6bb 289	. . . . . . . 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 3452	. . . . . . 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 3369	. . . . . . . . 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 1956	. . . . . . . . . . 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 654	. . . . . . . . . . . . 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 750	. . . . . . . . . . . . . . . 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 6672	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑗 → (𝑓‘𝑛) = (𝑓‘𝑗))
56	55	csbeq1d 3889	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑗 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
57	56	cbvmptv 5171	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) = (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
58	26, 57	eqtri 2846	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
59		fveq2 6672	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑗 → (𝑔‘𝑛) = (𝑔‘𝑗))
60	59	csbeq1d 3889	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑗 → ⦋(𝑔‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑔‘𝑗) / 𝑘⦌𝐵)
61	60	cbvmptv 5171	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ ↦ ⦋(𝑔‘𝑛) / 𝑘⦌𝐵) = (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵)
62		simplr 767	. . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . 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 15073	. . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . 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 249	. . . . . . . . . . . . . 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 456	. . . . . . . . . . . . 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	syl5bi 244	. . . . . . . . . . . 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 1935	. . . . . . . . . . 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	syl5bir 245	. . . . . . . . . 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 3296	. . . . . . . . 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	syl5bir 245	. . . . . . . 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 455	. . . . . . 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	syl5bi 244	. . . . . 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 855	. . . . 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 456	. . 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 1929	. 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 5072	. . . . . 6 ⊢ (𝑥 = 𝑦 → (seq𝑚( + , 𝐹) ⇝ 𝑥 ↔ seq𝑚( + , 𝐹) ⇝ 𝑦))
81	80	anbi2d 630	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)))
82	81	rexbidv 3299	. . . 4 ⊢ (𝑥 = 𝑦 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)))
83		eqeq1 2827	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = (seq1( + , 𝐺)‘𝑚) ↔ 𝑦 = (seq1( + , 𝐺)‘𝑚)))
84	83	anbi2d 630	. . . . . 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 1922	. . . . 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 3299	. . . 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 2650	. 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 236	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 398 ∨ wo 843 ∀wal 1535 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∃*wmo 2620 ∃wrex 3141 ⦋csb 3885 ⊆ wss 3938 ifcif 4469 class class class wbr 5068 ↦ cmpt 5148 –1-1-onto→wf1o 6356 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 1c1 10540 + caddc 10542 ℕcn 11640 ℤcz 11984 ℤ_≥cuz 12246 ...cfz 12895 seqcseq 13372 ⇝ cli 14843

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847

This theorem is referenced by: fsum 15079

W3C validator