Metamath Proof Explorer

Theorem ssdec

Description: Inclusion relation for a monotonic sequence of sets. (Contributed by Glauco Siliprandi, 8-Apr-2021)

Ref Expression
Hypotheses ssdec.1 ⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
ssdec.2 ⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐹 ‘ ( 𝑚 + 1 ) ) ⊆ ( 𝐹 ‘ 𝑚 ) )
Assertion ssdec ( 𝜑 → ( 𝐹 ‘ 𝑁 ) ⊆ ( 𝐹 ‘ 𝑀 ) )

Proof

Step Hyp Ref Expression
1 ssdec.1 ⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
2 ssdec.2 ⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐹 ‘ ( 𝑚 + 1 ) ) ⊆ ( 𝐹 ‘ 𝑚 ) )
3 eluzel2 ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℤ )
4 1 3 syl ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
5 eluzelz ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ℤ )
6 1 5 syl ⊢ ( 𝜑 → 𝑁 ∈ ℤ )
7 4 6 jca ⊢ ( 𝜑 → ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) )
8 eluzle ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ≤ 𝑁 )
9 1 8 syl ⊢ ( 𝜑 → 𝑀 ≤ 𝑁 )
10 6 zred ⊢ ( 𝜑 → 𝑁 ∈ ℝ )
11 10 leidd ⊢ ( 𝜑 → 𝑁 ≤ 𝑁 )
12 6 9 11 3jca ⊢ ( 𝜑 → ( 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ∧ 𝑁 ≤ 𝑁 ) )
13 7 12 jca ⊢ ( 𝜑 → ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ∧ 𝑁 ≤ 𝑁 ) ) )
14 fveq2 ⊢ ( 𝑛 = 𝑀 → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑀 ) )
15 14 sseq1d ⊢ ( 𝑛 = 𝑀 → ( ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ 𝑀 ) ↔ ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) )
16 15 imbi2d ⊢ ( 𝑛 = 𝑀 → ( ( 𝜑 → ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) ↔ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) ) )
17 fveq2 ⊢ ( 𝑛 = 𝑚 → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑚 ) )
18 17 sseq1d ⊢ ( 𝑛 = 𝑚 → ( ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ 𝑀 ) ↔ ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) )
19 18 imbi2d ⊢ ( 𝑛 = 𝑚 → ( ( 𝜑 → ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) ↔ ( 𝜑 → ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) ) )
20 fveq2 ⊢ ( 𝑛 = ( 𝑚 + 1 ) → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ ( 𝑚 + 1 ) ) )
21 20 sseq1d ⊢ ( 𝑛 = ( 𝑚 + 1 ) → ( ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ 𝑀 ) ↔ ( 𝐹 ‘ ( 𝑚 + 1 ) ) ⊆ ( 𝐹 ‘ 𝑀 ) ) )
22 21 imbi2d ⊢ ( 𝑛 = ( 𝑚 + 1 ) → ( ( 𝜑 → ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) ↔ ( 𝜑 → ( 𝐹 ‘ ( 𝑚 + 1 ) ) ⊆ ( 𝐹 ‘ 𝑀 ) ) ) )
23 fveq2 ⊢ ( 𝑛 = 𝑁 → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑁 ) )
24 23 sseq1d ⊢ ( 𝑛 = 𝑁 → ( ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ 𝑀 ) ↔ ( 𝐹 ‘ 𝑁 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) )
25 24 imbi2d ⊢ ( 𝑛 = 𝑁 → ( ( 𝜑 → ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) ↔ ( 𝜑 → ( 𝐹 ‘ 𝑁 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) ) )
26 ssidd ⊢ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑀 ) )
27 26 a1i ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) → ( 𝜑 → ( 𝐹 ‘ 𝑀 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) )
28 simpr ⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → 𝜑 )
29 simplll ⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → 𝑀 ∈ ℤ )
30 simplr1 ⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → 𝑚 ∈ ℤ )
31 simplr2 ⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → 𝑀 ≤ 𝑚 )
32 29 30 31 3jca ⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → ( 𝑀 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ) )
33 eluz2 ⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ ( 𝑀 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ) )
34 32 33 sylibr ⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → 𝑚 ∈ ( ℤ≥ ‘ 𝑀 ) )
35 simpllr ⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → 𝑁 ∈ ℤ )
36 simplr3 ⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → 𝑚 < 𝑁 )
37 34 35 36 3jca ⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → ( 𝑚 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑁 ∈ ℤ ∧ 𝑚 < 𝑁 ) )
38 elfzo2 ⊢ ( 𝑚 ∈ ( 𝑀 ..^ 𝑁 ) ↔ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑁 ∈ ℤ ∧ 𝑚 < 𝑁 ) )
39 37 38 sylibr ⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → 𝑚 ∈ ( 𝑀 ..^ 𝑁 ) )
40 28 39 2 syl2anc ⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ 𝜑 ) → ( 𝐹 ‘ ( 𝑚 + 1 ) ) ⊆ ( 𝐹 ‘ 𝑚 ) )
41 40 3adant2 ⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ ( 𝜑 → ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) ∧ 𝜑 ) → ( 𝐹 ‘ ( 𝑚 + 1 ) ) ⊆ ( 𝐹 ‘ 𝑚 ) )
42 simpr ⊢ ( ( ( 𝜑 → ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) ∧ 𝜑 ) → 𝜑 )
43 simpl ⊢ ( ( ( 𝜑 → ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) ∧ 𝜑 ) → ( 𝜑 → ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) )
44 pm3.35 ⊢ ( ( 𝜑 ∧ ( 𝜑 → ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) ) → ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ 𝑀 ) )
45 42 43 44 syl2anc ⊢ ( ( ( 𝜑 → ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) ∧ 𝜑 ) → ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ 𝑀 ) )
46 45 3adant1 ⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ ( 𝜑 → ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) ∧ 𝜑 ) → ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ 𝑀 ) )
47 41 46 sstrd ⊢ ( ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) ∧ ( 𝜑 → ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) ∧ 𝜑 ) → ( 𝐹 ‘ ( 𝑚 + 1 ) ) ⊆ ( 𝐹 ‘ 𝑀 ) )
48 47 3exp ⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ∧ 𝑚 < 𝑁 ) ) → ( ( 𝜑 → ( 𝐹 ‘ 𝑚 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) → ( 𝜑 → ( 𝐹 ‘ ( 𝑚 + 1 ) ) ⊆ ( 𝐹 ‘ 𝑀 ) ) ) )
49 16 19 22 25 27 48 fzind ⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ∧ 𝑁 ≤ 𝑁 ) ) → ( 𝜑 → ( 𝐹 ‘ 𝑁 ) ⊆ ( 𝐹 ‘ 𝑀 ) ) )
50 13 49 mpcom ⊢ ( 𝜑 → ( 𝐹 ‘ 𝑁 ) ⊆ ( 𝐹 ‘ 𝑀 ) )