Theorem smonoord 48583

Description: Ordering relation for a strictly monotonic sequence, increasing case. Analogous to monoord 14182 (except that the case 𝑀 = 𝑁 must be excluded). Duplicate of monoords 46449? (Contributed by AV, 12-Jul-2020.)

Hypotheses

Ref	Expression
smonoord.0	⊢ (𝜑 → 𝑀 ∈ ℤ)
smonoord.1	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))
smonoord.2	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)
smonoord.3	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) < (𝐹‘(𝑘 + 1)))

Assertion

Ref	Expression
smonoord	⊢ (𝜑 → (𝐹‘𝑀) < (𝐹‘𝑁))

Distinct variable groups:   𝑘,𝐹   𝑘,𝑀   𝑘,𝑁   𝜑,𝑘

Proof of Theorem smonoord

Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		smonoord.1	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))
2		eluzfz2 13672	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) → 𝑁 ∈ ((𝑀 + 1)...𝑁))
3	1, 2	syl 18	. 2 ⊢ (𝜑 → 𝑁 ∈ ((𝑀 + 1)...𝑁))
4		eleq1 2848	. . . . . 6 ⊢ (𝑥 = (𝑀 + 1) → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ (𝑀 + 1) ∈ ((𝑀 + 1)...𝑁)))
5		fveq2 6890	. . . . . . 7 ⊢ (𝑥 = (𝑀 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑀 + 1)))
6	5	breq2d 5114	. . . . . 6 ⊢ (𝑥 = (𝑀 + 1) → ((𝐹‘𝑀) < (𝐹‘𝑥) ↔ (𝐹‘𝑀) < (𝐹‘(𝑀 + 1))))
7	4, 6	imbi12d 347	. . . . 5 ⊢ (𝑥 = (𝑀 + 1) → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑥)) ↔ ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘(𝑀 + 1)))))
8	7	imbi2d 343	. . . 4 ⊢ (𝑥 = (𝑀 + 1) → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑥))) ↔ (𝜑 → ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘(𝑀 + 1))))))
9		eleq1 2848	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ 𝑛 ∈ ((𝑀 + 1)...𝑁)))
10		fveq2 6890	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝐹‘𝑥) = (𝐹‘𝑛))
11	10	breq2d 5114	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((𝐹‘𝑀) < (𝐹‘𝑥) ↔ (𝐹‘𝑀) < (𝐹‘𝑛)))
12	9, 11	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑛 → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑥)) ↔ (𝑛 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑛))))
13	12	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑥))) ↔ (𝜑 → (𝑛 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑛)))))
14		eleq1 2848	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)))
15		fveq2 6890	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
16	15	breq2d 5114	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑀) < (𝐹‘𝑥) ↔ (𝐹‘𝑀) < (𝐹‘(𝑛 + 1))))
17	14, 16	imbi12d 347	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑥)) ↔ ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘(𝑛 + 1)))))
18	17	imbi2d 343	. . . 4 ⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑥))) ↔ (𝜑 → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘(𝑛 + 1))))))
19		eleq1 2848	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ 𝑁 ∈ ((𝑀 + 1)...𝑁)))
20		fveq2 6890	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐹‘𝑥) = (𝐹‘𝑁))
21	20	breq2d 5114	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝐹‘𝑀) < (𝐹‘𝑥) ↔ (𝐹‘𝑀) < (𝐹‘𝑁)))
22	19, 21	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑥)) ↔ (𝑁 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑁))))
23	22	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑥))) ↔ (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑁)))))
24		smonoord.0	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
25		eluzp1m1 12998	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑁 − 1) ∈ (ℤ≥‘𝑀))
26	24, 1, 25	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘𝑀))
27		eluzfz1 13671	. . . . . . . 8 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...(𝑁 − 1)))
28	26, 27	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀...(𝑁 − 1)))
29		smonoord.3	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) < (𝐹‘(𝑘 + 1)))
30	29	ralrimiva 3154	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘𝑘) < (𝐹‘(𝑘 + 1)))
31		fveq2 6890	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
32		fvoveq1 7447	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑀 + 1)))
33	31, 32	breq12d 5115	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) < (𝐹‘(𝑘 + 1)) ↔ (𝐹‘𝑀) < (𝐹‘(𝑀 + 1))))
34	33	rspcv 3572	. . . . . . 7 ⊢ (𝑀 ∈ (𝑀...(𝑁 − 1)) → (∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘𝑘) < (𝐹‘(𝑘 + 1)) → (𝐹‘𝑀) < (𝐹‘(𝑀 + 1))))
35	28, 30, 34	sylc 66	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑀) < (𝐹‘(𝑀 + 1)))
36	35	a1d 26	. . . . 5 ⊢ (𝜑 → ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘(𝑀 + 1))))
37	36	a1i 11	. . . 4 ⊢ ((𝑀 + 1) ∈ ℤ → (𝜑 → ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘(𝑀 + 1)))))
38		peano2fzr 13677	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → 𝑛 ∈ ((𝑀 + 1)...𝑁))
39	38	adantll 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑀 + 1))) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → 𝑛 ∈ ((𝑀 + 1)...𝑁))
40	39	ex 418	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → 𝑛 ∈ ((𝑀 + 1)...𝑁)))
41	40	imim1d 83	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝑛 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑛)) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑛))))
42		peano2uzr 13037	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑛 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑛 ∈ (ℤ≥‘𝑀))
43	42	ex 418	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) → 𝑛 ∈ (ℤ≥‘𝑀)))
44	43, 24	syl11 34	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) → (𝜑 → 𝑛 ∈ (ℤ≥‘𝑀)))
45	44	adantr 486	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → (𝜑 → 𝑛 ∈ (ℤ≥‘𝑀)))
46	45	impcom 413	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ (ℤ≥‘𝑀))
47		eluzelz 12982	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) → 𝑛 ∈ ℤ)
48	47	adantr 486	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → 𝑛 ∈ ℤ)
49	48	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ ℤ)
50		elfzuz3 13660	. . . . . . . . . 10 ⊢ ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
51	50	ad2antll 742	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
52		eluzp1m1 12998	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑁 − 1) ∈ (ℤ≥‘𝑛))
53	49, 51, 52	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝑁 − 1) ∈ (ℤ≥‘𝑛))
54		elfzuzb 13657	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑀...(𝑁 − 1)) ↔ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑁 − 1) ∈ (ℤ≥‘𝑛)))
55	46, 53, 54	sylanbrc 595	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ (𝑀...(𝑁 − 1)))
56	30	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘𝑘) < (𝐹‘(𝑘 + 1)))
57		fveq2 6890	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
58		fvoveq1 7447	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
59	57, 58	breq12d 5115	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) < (𝐹‘(𝑘 + 1)) ↔ (𝐹‘𝑛) < (𝐹‘(𝑛 + 1))))
60	59	rspcv 3572	. . . . . . 7 ⊢ (𝑛 ∈ (𝑀...(𝑁 − 1)) → (∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘𝑘) < (𝐹‘(𝑘 + 1)) → (𝐹‘𝑛) < (𝐹‘(𝑛 + 1))))
61	55, 56, 60	sylc 66	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝐹‘𝑛) < (𝐹‘(𝑛 + 1)))
62		zre 12704	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
63	62	lep1d 12255	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → 𝑀 ≤ (𝑀 + 1))
64	24, 63	jccir 531	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑀 ≤ (𝑀 + 1)))
65		eluzuzle 12981	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≤ (𝑀 + 1)) → (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) → 𝑁 ∈ (ℤ≥‘𝑀)))
66	64, 1, 65	sylc 66	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
67		eluzfz1 13671	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
68	66, 67	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
69		smonoord.2	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)
70	69	ralrimiva 3154	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ)
71	31	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑀) ∈ ℝ))
72	71	rspcv 3572	. . . . . . . . 9 ⊢ (𝑀 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ → (𝐹‘𝑀) ∈ ℝ))
73	68, 70, 72	sylc 66	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℝ)
74	73	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝐹‘𝑀) ∈ ℝ)
75		fzp1ss 13716	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℤ → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁))
76	24, 75	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁))
77	76	sseld 3908	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁)))
78	77	com12 33	. . . . . . . . . . 11 ⊢ ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (𝜑 → (𝑛 + 1) ∈ (𝑀...𝑁)))
79	78	adantl 487	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → (𝜑 → (𝑛 + 1) ∈ (𝑀...𝑁)))
80	79	impcom 413	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁))
81		peano2fzr 13677	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
82	46, 80, 81	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
83	70	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ)
84	57	eleq1d 2845	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑛) ∈ ℝ))
85	84	rspcv 3572	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ → (𝐹‘𝑛) ∈ ℝ))
86	82, 83, 85	sylc 66	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝐹‘𝑛) ∈ ℝ)
87		fveq2 6890	. . . . . . . . . 10 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
88	87	eleq1d 2845	. . . . . . . . 9 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(𝑛 + 1)) ∈ ℝ))
89	88	rspcv 3572	. . . . . . . 8 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ → (𝐹‘(𝑛 + 1)) ∈ ℝ))
90	80, 83, 89	sylc 66	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
91		lttr 11393	. . . . . . 7 ⊢ (((𝐹‘𝑀) ∈ ℝ ∧ (𝐹‘𝑛) ∈ ℝ ∧ (𝐹‘(𝑛 + 1)) ∈ ℝ) → (((𝐹‘𝑀) < (𝐹‘𝑛) ∧ (𝐹‘𝑛) < (𝐹‘(𝑛 + 1))) → (𝐹‘𝑀) < (𝐹‘(𝑛 + 1))))
92	74, 86, 90, 91	syl3anc 1398	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (((𝐹‘𝑀) < (𝐹‘𝑛) ∧ (𝐹‘𝑛) < (𝐹‘(𝑛 + 1))) → (𝐹‘𝑀) < (𝐹‘(𝑛 + 1))))
93	61, 92	mpan2d 707	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((𝐹‘𝑀) < (𝐹‘𝑛) → (𝐹‘𝑀) < (𝐹‘(𝑛 + 1))))
94	41, 93	animpimp2impd 860	. . . 4 ⊢ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) → ((𝜑 → (𝑛 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑛))) → (𝜑 → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘(𝑛 + 1))))))
95	8, 13, 18, 23, 37, 94	uzind4 13040	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) → (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑁))))
96	1, 95	mpcom 39	. 2 ⊢ (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑁)))
97	3, 96	mpd 16	1 ⊢ (𝜑 → (𝐹‘𝑀) < (𝐹‘𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3896   class class class wbr 5102  ‘cfv 6543  (class class class)co 7424  ℝcr 11206  1c1 11208   + caddc 11210   < clt 11350   ≤ cle 11351   − cmin 11548  ℤcz 12700  ℤ_≥cuz 12972  ...cfz 13646

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-er 8725  df-en 8982  df-dom 8983  df-sdom 8984  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-nn 12343  df-n0 12614  df-z 12701  df-uz 12973  df-fz 13647

This theorem is used by:  iccpartiltu  48640  iccpartigtl  48641  iccpartgt  48645