Theorem smonoord 46611

Description: Ordering relation for a strictly monotonic sequence, increasing case. Analogous to monoord 14003 (except that the case 𝑀 = 𝑁 must be excluded). Duplicate of monoords 44579? (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 13515	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) → 𝑁 ∈ ((𝑀 + 1)...𝑁))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑁 ∈ ((𝑀 + 1)...𝑁))
4		eleq1 2815	. . . . . 6 ⊢ (𝑥 = (𝑀 + 1) → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ (𝑀 + 1) ∈ ((𝑀 + 1)...𝑁)))
5		fveq2 6885	. . . . . . 7 ⊢ (𝑥 = (𝑀 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑀 + 1)))
6	5	breq2d 5153	. . . . . 6 ⊢ (𝑥 = (𝑀 + 1) → ((𝐹‘𝑀) < (𝐹‘𝑥) ↔ (𝐹‘𝑀) < (𝐹‘(𝑀 + 1))))
7	4, 6	imbi12d 344	. . . . 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 340	. . . 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 2815	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ 𝑛 ∈ ((𝑀 + 1)...𝑁)))
10		fveq2 6885	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝐹‘𝑥) = (𝐹‘𝑛))
11	10	breq2d 5153	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((𝐹‘𝑀) < (𝐹‘𝑥) ↔ (𝐹‘𝑀) < (𝐹‘𝑛)))
12	9, 11	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑛 → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑥)) ↔ (𝑛 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑛))))
13	12	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑥))) ↔ (𝜑 → (𝑛 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑛)))))
14		eleq1 2815	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)))
15		fveq2 6885	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
16	15	breq2d 5153	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑀) < (𝐹‘𝑥) ↔ (𝐹‘𝑀) < (𝐹‘(𝑛 + 1))))
17	14, 16	imbi12d 344	. . . . 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 340	. . . 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 2815	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ 𝑁 ∈ ((𝑀 + 1)...𝑁)))
20		fveq2 6885	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐹‘𝑥) = (𝐹‘𝑁))
21	20	breq2d 5153	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝐹‘𝑀) < (𝐹‘𝑥) ↔ (𝐹‘𝑀) < (𝐹‘𝑁)))
22	19, 21	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑥)) ↔ (𝑁 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑁))))
23	22	imbi2d 340	. . . 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 12852	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑁 − 1) ∈ (ℤ≥‘𝑀))
26	24, 1, 25	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘𝑀))
27		eluzfz1 13514	. . . . . . . 8 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...(𝑁 − 1)))
28	26, 27	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀...(𝑁 − 1)))
29		smonoord.3	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) < (𝐹‘(𝑘 + 1)))
30	29	ralrimiva 3140	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘𝑘) < (𝐹‘(𝑘 + 1)))
31		fveq2 6885	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
32		fvoveq1 7428	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑀 + 1)))
33	31, 32	breq12d 5154	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) < (𝐹‘(𝑘 + 1)) ↔ (𝐹‘𝑀) < (𝐹‘(𝑀 + 1))))
34	33	rspcv 3602	. . . . . . 7 ⊢ (𝑀 ∈ (𝑀...(𝑁 − 1)) → (∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘𝑘) < (𝐹‘(𝑘 + 1)) → (𝐹‘𝑀) < (𝐹‘(𝑀 + 1))))
35	28, 30, 34	sylc 65	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑀) < (𝐹‘(𝑀 + 1)))
36	35	a1d 25	. . . . 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 13520	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → 𝑛 ∈ ((𝑀 + 1)...𝑁))
39	38	adantll 711	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑀 + 1))) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → 𝑛 ∈ ((𝑀 + 1)...𝑁))
40	39	ex 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → 𝑛 ∈ ((𝑀 + 1)...𝑁)))
41	40	imim1d 82	. . . . 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 12891	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑛 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑛 ∈ (ℤ≥‘𝑀))
43	42	ex 412	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) → 𝑛 ∈ (ℤ≥‘𝑀)))
44	43, 24	syl11 33	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) → (𝜑 → 𝑛 ∈ (ℤ≥‘𝑀)))
45	44	adantr 480	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → (𝜑 → 𝑛 ∈ (ℤ≥‘𝑀)))
46	45	impcom 407	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ (ℤ≥‘𝑀))
47		eluzelz 12836	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) → 𝑛 ∈ ℤ)
48	47	adantr 480	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → 𝑛 ∈ ℤ)
49	48	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ ℤ)
50		elfzuz3 13504	. . . . . . . . . 10 ⊢ ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
51	50	ad2antll 726	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
52		eluzp1m1 12852	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑁 − 1) ∈ (ℤ≥‘𝑛))
53	49, 51, 52	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝑁 − 1) ∈ (ℤ≥‘𝑛))
54		elfzuzb 13501	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑀...(𝑁 − 1)) ↔ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑁 − 1) ∈ (ℤ≥‘𝑛)))
55	46, 53, 54	sylanbrc 582	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ (𝑀...(𝑁 − 1)))
56	30	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘𝑘) < (𝐹‘(𝑘 + 1)))
57		fveq2 6885	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
58		fvoveq1 7428	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
59	57, 58	breq12d 5154	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) < (𝐹‘(𝑘 + 1)) ↔ (𝐹‘𝑛) < (𝐹‘(𝑛 + 1))))
60	59	rspcv 3602	. . . . . . 7 ⊢ (𝑛 ∈ (𝑀...(𝑁 − 1)) → (∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘𝑘) < (𝐹‘(𝑘 + 1)) → (𝐹‘𝑛) < (𝐹‘(𝑛 + 1))))
61	55, 56, 60	sylc 65	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝐹‘𝑛) < (𝐹‘(𝑛 + 1)))
62		zre 12566	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
63	62	lep1d 12149	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → 𝑀 ≤ (𝑀 + 1))
64	24, 63	jccir 521	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑀 ≤ (𝑀 + 1)))
65		eluzuzle 12835	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≤ (𝑀 + 1)) → (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) → 𝑁 ∈ (ℤ≥‘𝑀)))
66	64, 1, 65	sylc 65	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
67		eluzfz1 13514	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
68	66, 67	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
69		smonoord.2	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)
70	69	ralrimiva 3140	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ)
71	31	eleq1d 2812	. . . . . . . . . 10 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑀) ∈ ℝ))
72	71	rspcv 3602	. . . . . . . . 9 ⊢ (𝑀 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ → (𝐹‘𝑀) ∈ ℝ))
73	68, 70, 72	sylc 65	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℝ)
74	73	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝐹‘𝑀) ∈ ℝ)
75		fzp1ss 13558	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℤ → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁))
76	24, 75	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁))
77	76	sseld 3976	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁)))
78	77	com12 32	. . . . . . . . . . 11 ⊢ ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (𝜑 → (𝑛 + 1) ∈ (𝑀...𝑁)))
79	78	adantl 481	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → (𝜑 → (𝑛 + 1) ∈ (𝑀...𝑁)))
80	79	impcom 407	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁))
81		peano2fzr 13520	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
82	46, 80, 81	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
83	70	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ)
84	57	eleq1d 2812	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑛) ∈ ℝ))
85	84	rspcv 3602	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ → (𝐹‘𝑛) ∈ ℝ))
86	82, 83, 85	sylc 65	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝐹‘𝑛) ∈ ℝ)
87		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
88	87	eleq1d 2812	. . . . . . . . 9 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(𝑛 + 1)) ∈ ℝ))
89	88	rspcv 3602	. . . . . . . 8 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ → (𝐹‘(𝑛 + 1)) ∈ ℝ))
90	80, 83, 89	sylc 65	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
91		lttr 11294	. . . . . . 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 1368	. . . . . 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 691	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((𝐹‘𝑀) < (𝐹‘𝑛) → (𝐹‘𝑀) < (𝐹‘(𝑛 + 1))))
94	41, 93	animpimp2impd 843	. . . 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 12894	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) → (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑁))))
96	1, 95	mpcom 38	. 2 ⊢ (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (𝐹‘𝑀) < (𝐹‘𝑁)))
97	3, 96	mpd 15	1 ⊢ (𝜑 → (𝐹‘𝑀) < (𝐹‘𝑁))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3055   ⊆ wss 3943   class class class wbr 5141  ‘cfv 6537  (class class class)co 7405  ℝcr 11111  1c1 11113   + caddc 11115   < clt 11252   ≤ cle 11253   − cmin 11448  ℤcz 12562  ℤ_≥cuz 12826  ...cfz 13490

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 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491

This theorem is referenced by:  iccpartiltu  46662  iccpartigtl  46663  iccpartgt  46667