seqpo - Mathbox for Jeff Madsen

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Theorem seqpo 37771

Description: Two ways to say that a sequence respects a partial order. (Contributed by Jeff Madsen, 2-Sep-2009.)

**Assertion**
Ref	Expression
seqpo	⊢ ((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) → (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ↔ ∀𝑚 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘(𝑚 + 1))(𝐹‘𝑚)𝑅(𝐹‘𝑛)))

Distinct variable groups: 𝑚,𝐹,𝑛,𝑠 𝐴,𝑚,𝑛,𝑠 𝑅,𝑚,𝑛,𝑠

Proof of Theorem seqpo Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6876	. . . . . . . . . 10 ⊢ (𝑝 = (𝑚 + 1) → (𝐹‘𝑝) = (𝐹‘(𝑚 + 1)))
2	1	breq2d 5131	. . . . . . . . 9 ⊢ (𝑝 = (𝑚 + 1) → ((𝐹‘𝑚)𝑅(𝐹‘𝑝) ↔ (𝐹‘𝑚)𝑅(𝐹‘(𝑚 + 1))))
3	2	imbi2d 340	. . . . . . . 8 ⊢ (𝑝 = (𝑚 + 1) → ((((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ∧ 𝑚 ∈ ℕ)) → (𝐹‘𝑚)𝑅(𝐹‘𝑝)) ↔ (((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ∧ 𝑚 ∈ ℕ)) → (𝐹‘𝑚)𝑅(𝐹‘(𝑚 + 1)))))
4		fveq2 6876	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → (𝐹‘𝑝) = (𝐹‘𝑞))
5	4	breq2d 5131	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → ((𝐹‘𝑚)𝑅(𝐹‘𝑝) ↔ (𝐹‘𝑚)𝑅(𝐹‘𝑞)))
6	5	imbi2d 340	. . . . . . . 8 ⊢ (𝑝 = 𝑞 → ((((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ∧ 𝑚 ∈ ℕ)) → (𝐹‘𝑚)𝑅(𝐹‘𝑝)) ↔ (((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ∧ 𝑚 ∈ ℕ)) → (𝐹‘𝑚)𝑅(𝐹‘𝑞))))
7		fveq2 6876	. . . . . . . . . 10 ⊢ (𝑝 = (𝑞 + 1) → (𝐹‘𝑝) = (𝐹‘(𝑞 + 1)))
8	7	breq2d 5131	. . . . . . . . 9 ⊢ (𝑝 = (𝑞 + 1) → ((𝐹‘𝑚)𝑅(𝐹‘𝑝) ↔ (𝐹‘𝑚)𝑅(𝐹‘(𝑞 + 1))))
9	8	imbi2d 340	. . . . . . . 8 ⊢ (𝑝 = (𝑞 + 1) → ((((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ∧ 𝑚 ∈ ℕ)) → (𝐹‘𝑚)𝑅(𝐹‘𝑝)) ↔ (((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ∧ 𝑚 ∈ ℕ)) → (𝐹‘𝑚)𝑅(𝐹‘(𝑞 + 1)))))
10		fveq2 6876	. . . . . . . . . 10 ⊢ (𝑝 = 𝑛 → (𝐹‘𝑝) = (𝐹‘𝑛))
11	10	breq2d 5131	. . . . . . . . 9 ⊢ (𝑝 = 𝑛 → ((𝐹‘𝑚)𝑅(𝐹‘𝑝) ↔ (𝐹‘𝑚)𝑅(𝐹‘𝑛)))
12	11	imbi2d 340	. . . . . . . 8 ⊢ (𝑝 = 𝑛 → ((((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ∧ 𝑚 ∈ ℕ)) → (𝐹‘𝑚)𝑅(𝐹‘𝑝)) ↔ (((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ∧ 𝑚 ∈ ℕ)) → (𝐹‘𝑚)𝑅(𝐹‘𝑛))))
13		fveq2 6876	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑚 → (𝐹‘𝑠) = (𝐹‘𝑚))
14		fvoveq1 7428	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑚 → (𝐹‘(𝑠 + 1)) = (𝐹‘(𝑚 + 1)))
15	13, 14	breq12d 5132	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑚 → ((𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ↔ (𝐹‘𝑚)𝑅(𝐹‘(𝑚 + 1))))
16	15	rspccva 3600	. . . . . . . . . 10 ⊢ ((∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚)𝑅(𝐹‘(𝑚 + 1)))
17	16	adantl 481	. . . . . . . . 9 ⊢ (((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ∧ 𝑚 ∈ ℕ)) → (𝐹‘𝑚)𝑅(𝐹‘(𝑚 + 1)))
18	17	a1i 11	. . . . . . . 8 ⊢ ((𝑚 + 1) ∈ ℤ → (((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ∧ 𝑚 ∈ ℕ)) → (𝐹‘𝑚)𝑅(𝐹‘(𝑚 + 1))))
19		peano2nn 12252	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
20		elnnuz 12896	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 + 1) ∈ ℕ ↔ (𝑚 + 1) ∈ (ℤ_≥‘1))
21	19, 20	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ (ℤ_≥‘1))
22		uztrn 12870	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑞 ∈ (ℤ_≥‘(𝑚 + 1)) ∧ (𝑚 + 1) ∈ (ℤ_≥‘1)) → 𝑞 ∈ (ℤ_≥‘1))
23		elnnuz 12896	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 ∈ ℕ ↔ 𝑞 ∈ (ℤ_≥‘1))
24	22, 23	sylibr 234	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ (ℤ_≥‘(𝑚 + 1)) ∧ (𝑚 + 1) ∈ (ℤ_≥‘1)) → 𝑞 ∈ ℕ)
25	24	expcom 413	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 + 1) ∈ (ℤ_≥‘1) → (𝑞 ∈ (ℤ_≥‘(𝑚 + 1)) → 𝑞 ∈ ℕ))
26	21, 25	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → (𝑞 ∈ (ℤ_≥‘(𝑚 + 1)) → 𝑞 ∈ ℕ))
27	26	imdistani 568	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ ∧ 𝑞 ∈ (ℤ_≥‘(𝑚 + 1))) → (𝑚 ∈ ℕ ∧ 𝑞 ∈ ℕ))
28		fveq2 6876	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = 𝑞 → (𝐹‘𝑠) = (𝐹‘𝑞))
29		fvoveq1 7428	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = 𝑞 → (𝐹‘(𝑠 + 1)) = (𝐹‘(𝑞 + 1)))
30	28, 29	breq12d 5132	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = 𝑞 → ((𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ↔ (𝐹‘𝑞)𝑅(𝐹‘(𝑞 + 1))))
31	30	rspccva 3600	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ∧ 𝑞 ∈ ℕ) → (𝐹‘𝑞)𝑅(𝐹‘(𝑞 + 1)))
32	31	ad2ant2l 746	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ ∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1))) ∧ (𝑚 ∈ ℕ ∧ 𝑞 ∈ ℕ)) → (𝐹‘𝑞)𝑅(𝐹‘(𝑞 + 1)))
33	32	ex 412	. . . . . . . . . . . . . 14 ⊢ (((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ ∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1))) → ((𝑚 ∈ ℕ ∧ 𝑞 ∈ ℕ) → (𝐹‘𝑞)𝑅(𝐹‘(𝑞 + 1))))
34		ffvelcdm 7071	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:ℕ⟶𝐴 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∈ 𝐴)
35	34	adantrr 717	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:ℕ⟶𝐴 ∧ (𝑚 ∈ ℕ ∧ 𝑞 ∈ ℕ)) → (𝐹‘𝑚) ∈ 𝐴)
36		ffvelcdm 7071	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:ℕ⟶𝐴 ∧ 𝑞 ∈ ℕ) → (𝐹‘𝑞) ∈ 𝐴)
37	36	adantrl 716	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:ℕ⟶𝐴 ∧ (𝑚 ∈ ℕ ∧ 𝑞 ∈ ℕ)) → (𝐹‘𝑞) ∈ 𝐴)
38		peano2nn 12252	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 ∈ ℕ → (𝑞 + 1) ∈ ℕ)
39		ffvelcdm 7071	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:ℕ⟶𝐴 ∧ (𝑞 + 1) ∈ ℕ) → (𝐹‘(𝑞 + 1)) ∈ 𝐴)
40	38, 39	sylan2 593	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:ℕ⟶𝐴 ∧ 𝑞 ∈ ℕ) → (𝐹‘(𝑞 + 1)) ∈ 𝐴)
41	40	adantrl 716	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:ℕ⟶𝐴 ∧ (𝑚 ∈ ℕ ∧ 𝑞 ∈ ℕ)) → (𝐹‘(𝑞 + 1)) ∈ 𝐴)
42	35, 37, 41	3jca 1128	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:ℕ⟶𝐴 ∧ (𝑚 ∈ ℕ ∧ 𝑞 ∈ ℕ)) → ((𝐹‘𝑚) ∈ 𝐴 ∧ (𝐹‘𝑞) ∈ 𝐴 ∧ (𝐹‘(𝑞 + 1)) ∈ 𝐴))
43		potr 5574	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 Po 𝐴 ∧ ((𝐹‘𝑚) ∈ 𝐴 ∧ (𝐹‘𝑞) ∈ 𝐴 ∧ (𝐹‘(𝑞 + 1)) ∈ 𝐴)) → (((𝐹‘𝑚)𝑅(𝐹‘𝑞) ∧ (𝐹‘𝑞)𝑅(𝐹‘(𝑞 + 1))) → (𝐹‘𝑚)𝑅(𝐹‘(𝑞 + 1))))
44	43	expcomd 416	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 Po 𝐴 ∧ ((𝐹‘𝑚) ∈ 𝐴 ∧ (𝐹‘𝑞) ∈ 𝐴 ∧ (𝐹‘(𝑞 + 1)) ∈ 𝐴)) → ((𝐹‘𝑞)𝑅(𝐹‘(𝑞 + 1)) → ((𝐹‘𝑚)𝑅(𝐹‘𝑞) → (𝐹‘𝑚)𝑅(𝐹‘(𝑞 + 1)))))
45	44	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 Po 𝐴 → (((𝐹‘𝑚) ∈ 𝐴 ∧ (𝐹‘𝑞) ∈ 𝐴 ∧ (𝐹‘(𝑞 + 1)) ∈ 𝐴) → ((𝐹‘𝑞)𝑅(𝐹‘(𝑞 + 1)) → ((𝐹‘𝑚)𝑅(𝐹‘𝑞) → (𝐹‘𝑚)𝑅(𝐹‘(𝑞 + 1))))))
46	42, 45	syl5 34	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 Po 𝐴 → ((𝐹:ℕ⟶𝐴 ∧ (𝑚 ∈ ℕ ∧ 𝑞 ∈ ℕ)) → ((𝐹‘𝑞)𝑅(𝐹‘(𝑞 + 1)) → ((𝐹‘𝑚)𝑅(𝐹‘𝑞) → (𝐹‘𝑚)𝑅(𝐹‘(𝑞 + 1))))))
47	46	expdimp 452	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) → ((𝑚 ∈ ℕ ∧ 𝑞 ∈ ℕ) → ((𝐹‘𝑞)𝑅(𝐹‘(𝑞 + 1)) → ((𝐹‘𝑚)𝑅(𝐹‘𝑞) → (𝐹‘𝑚)𝑅(𝐹‘(𝑞 + 1))))))
48	47	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ ∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1))) → ((𝑚 ∈ ℕ ∧ 𝑞 ∈ ℕ) → ((𝐹‘𝑞)𝑅(𝐹‘(𝑞 + 1)) → ((𝐹‘𝑚)𝑅(𝐹‘𝑞) → (𝐹‘𝑚)𝑅(𝐹‘(𝑞 + 1))))))
49	33, 48	mpdd 43	. . . . . . . . . . . . 13 ⊢ (((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ ∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1))) → ((𝑚 ∈ ℕ ∧ 𝑞 ∈ ℕ) → ((𝐹‘𝑚)𝑅(𝐹‘𝑞) → (𝐹‘𝑚)𝑅(𝐹‘(𝑞 + 1)))))
50	27, 49	syl5 34	. . . . . . . . . . . 12 ⊢ (((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ ∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1))) → ((𝑚 ∈ ℕ ∧ 𝑞 ∈ (ℤ_≥‘(𝑚 + 1))) → ((𝐹‘𝑚)𝑅(𝐹‘𝑞) → (𝐹‘𝑚)𝑅(𝐹‘(𝑞 + 1)))))
51	50	expdimp 452	. . . . . . . . . . 11 ⊢ ((((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ ∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1))) ∧ 𝑚 ∈ ℕ) → (𝑞 ∈ (ℤ_≥‘(𝑚 + 1)) → ((𝐹‘𝑚)𝑅(𝐹‘𝑞) → (𝐹‘𝑚)𝑅(𝐹‘(𝑞 + 1)))))
52	51	anasss 466	. . . . . . . . . 10 ⊢ (((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ∧ 𝑚 ∈ ℕ)) → (𝑞 ∈ (ℤ_≥‘(𝑚 + 1)) → ((𝐹‘𝑚)𝑅(𝐹‘𝑞) → (𝐹‘𝑚)𝑅(𝐹‘(𝑞 + 1)))))
53	52	com12 32	. . . . . . . . 9 ⊢ (𝑞 ∈ (ℤ_≥‘(𝑚 + 1)) → (((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ∧ 𝑚 ∈ ℕ)) → ((𝐹‘𝑚)𝑅(𝐹‘𝑞) → (𝐹‘𝑚)𝑅(𝐹‘(𝑞 + 1)))))
54	53	a2d 29	. . . . . . . 8 ⊢ (𝑞 ∈ (ℤ_≥‘(𝑚 + 1)) → ((((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ∧ 𝑚 ∈ ℕ)) → (𝐹‘𝑚)𝑅(𝐹‘𝑞)) → (((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ∧ 𝑚 ∈ ℕ)) → (𝐹‘𝑚)𝑅(𝐹‘(𝑞 + 1)))))
55	3, 6, 9, 12, 18, 54	uzind4 12922	. . . . . . 7 ⊢ (𝑛 ∈ (ℤ_≥‘(𝑚 + 1)) → (((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ∧ 𝑚 ∈ ℕ)) → (𝐹‘𝑚)𝑅(𝐹‘𝑛)))
56	55	com12 32	. . . . . 6 ⊢ (((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ∧ 𝑚 ∈ ℕ)) → (𝑛 ∈ (ℤ_≥‘(𝑚 + 1)) → (𝐹‘𝑚)𝑅(𝐹‘𝑛)))
57	56	ralrimiv 3131	. . . . 5 ⊢ (((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ∧ 𝑚 ∈ ℕ)) → ∀𝑛 ∈ (ℤ_≥‘(𝑚 + 1))(𝐹‘𝑚)𝑅(𝐹‘𝑛))
58	57	anassrs 467	. . . 4 ⊢ ((((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ ∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1))) ∧ 𝑚 ∈ ℕ) → ∀𝑛 ∈ (ℤ_≥‘(𝑚 + 1))(𝐹‘𝑚)𝑅(𝐹‘𝑛))
59	58	ralrimiva 3132	. . 3 ⊢ (((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) ∧ ∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1))) → ∀𝑚 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘(𝑚 + 1))(𝐹‘𝑚)𝑅(𝐹‘𝑛))
60	59	ex 412	. 2 ⊢ ((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) → (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) → ∀𝑚 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘(𝑚 + 1))(𝐹‘𝑚)𝑅(𝐹‘𝑛)))
61		fvoveq1 7428	. . . . . . 7 ⊢ (𝑚 = 𝑠 → (ℤ_≥‘(𝑚 + 1)) = (ℤ_≥‘(𝑠 + 1)))
62		fveq2 6876	. . . . . . . 8 ⊢ (𝑚 = 𝑠 → (𝐹‘𝑚) = (𝐹‘𝑠))
63	62	breq1d 5129	. . . . . . 7 ⊢ (𝑚 = 𝑠 → ((𝐹‘𝑚)𝑅(𝐹‘𝑛) ↔ (𝐹‘𝑠)𝑅(𝐹‘𝑛)))
64	61, 63	raleqbidv 3325	. . . . . 6 ⊢ (𝑚 = 𝑠 → (∀𝑛 ∈ (ℤ_≥‘(𝑚 + 1))(𝐹‘𝑚)𝑅(𝐹‘𝑛) ↔ ∀𝑛 ∈ (ℤ_≥‘(𝑠 + 1))(𝐹‘𝑠)𝑅(𝐹‘𝑛)))
65	64	rspcv 3597	. . . . 5 ⊢ (𝑠 ∈ ℕ → (∀𝑚 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘(𝑚 + 1))(𝐹‘𝑚)𝑅(𝐹‘𝑛) → ∀𝑛 ∈ (ℤ_≥‘(𝑠 + 1))(𝐹‘𝑠)𝑅(𝐹‘𝑛)))
66	65	imdistanri 569	. . . 4 ⊢ ((∀𝑚 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘(𝑚 + 1))(𝐹‘𝑚)𝑅(𝐹‘𝑛) ∧ 𝑠 ∈ ℕ) → (∀𝑛 ∈ (ℤ_≥‘(𝑠 + 1))(𝐹‘𝑠)𝑅(𝐹‘𝑛) ∧ 𝑠 ∈ ℕ))
67		peano2nn 12252	. . . . . . 7 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℕ)
68	67	nnzd 12615	. . . . . 6 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℤ)
69		uzid 12867	. . . . . 6 ⊢ ((𝑠 + 1) ∈ ℤ → (𝑠 + 1) ∈ (ℤ_≥‘(𝑠 + 1)))
70	68, 69	syl 17	. . . . 5 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ (ℤ_≥‘(𝑠 + 1)))
71		fveq2 6876	. . . . . . 7 ⊢ (𝑛 = (𝑠 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑠 + 1)))
72	71	breq2d 5131	. . . . . 6 ⊢ (𝑛 = (𝑠 + 1) → ((𝐹‘𝑠)𝑅(𝐹‘𝑛) ↔ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1))))
73	72	rspccva 3600	. . . . 5 ⊢ ((∀𝑛 ∈ (ℤ_≥‘(𝑠 + 1))(𝐹‘𝑠)𝑅(𝐹‘𝑛) ∧ (𝑠 + 1) ∈ (ℤ_≥‘(𝑠 + 1))) → (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)))
74	70, 73	sylan2 593	. . . 4 ⊢ ((∀𝑛 ∈ (ℤ_≥‘(𝑠 + 1))(𝐹‘𝑠)𝑅(𝐹‘𝑛) ∧ 𝑠 ∈ ℕ) → (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)))
75	66, 74	syl 17	. . 3 ⊢ ((∀𝑚 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘(𝑚 + 1))(𝐹‘𝑚)𝑅(𝐹‘𝑛) ∧ 𝑠 ∈ ℕ) → (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)))
76	75	ralrimiva 3132	. 2 ⊢ (∀𝑚 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘(𝑚 + 1))(𝐹‘𝑚)𝑅(𝐹‘𝑛) → ∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)))
77	60, 76	impbid1 225	1 ⊢ ((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) → (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ↔ ∀𝑚 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘(𝑚 + 1))(𝐹‘𝑚)𝑅(𝐹‘𝑛)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ∀wral 3051 class class class wbr 5119 Po wpo 5559 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 1c1 11130 + caddc 11132 ℕcn 12240 ℤcz 12588 ℤ_≥cuz 12852

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-n0 12502 df-z 12589 df-uz 12853

This theorem is referenced by: incsequz2 37773

W3C validator