Metamath Proof Explorer


Theorem isercolllem1

Description: Lemma for isercoll . (Contributed by Mario Carneiro, 6-Apr-2015)

Ref Expression
Hypotheses isercoll.z 𝑍 = ( ℤ𝑀 )
isercoll.m ( 𝜑𝑀 ∈ ℤ )
isercoll.g ( 𝜑𝐺 : ℕ ⟶ 𝑍 )
isercoll.i ( ( 𝜑𝑘 ∈ ℕ ) → ( 𝐺𝑘 ) < ( 𝐺 ‘ ( 𝑘 + 1 ) ) )
Assertion isercolllem1 ( ( 𝜑𝑆 ⊆ ℕ ) → ( 𝐺𝑆 ) Isom < , < ( 𝑆 , ( 𝐺𝑆 ) ) )

Proof

Step Hyp Ref Expression
1 isercoll.z 𝑍 = ( ℤ𝑀 )
2 isercoll.m ( 𝜑𝑀 ∈ ℤ )
3 isercoll.g ( 𝜑𝐺 : ℕ ⟶ 𝑍 )
4 isercoll.i ( ( 𝜑𝑘 ∈ ℕ ) → ( 𝐺𝑘 ) < ( 𝐺 ‘ ( 𝑘 + 1 ) ) )
5 uzssz ( ℤ𝑀 ) ⊆ ℤ
6 1 5 eqsstri 𝑍 ⊆ ℤ
7 zssre ℤ ⊆ ℝ
8 6 7 sstri 𝑍 ⊆ ℝ
9 3 ad2antrr ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → 𝐺 : ℕ ⟶ 𝑍 )
10 simplrl ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → 𝑥 ∈ ℕ )
11 9 10 ffvelrnd ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( 𝐺𝑥 ) ∈ 𝑍 )
12 8 11 sselid ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( 𝐺𝑥 ) ∈ ℝ )
13 simplrr ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → 𝑦 ∈ ℕ )
14 13 nnred ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → 𝑦 ∈ ℝ )
15 12 14 resubcld ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝐺𝑥 ) − 𝑦 ) ∈ ℝ )
16 10 nnred ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → 𝑥 ∈ ℝ )
17 12 16 resubcld ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝐺𝑥 ) − 𝑥 ) ∈ ℝ )
18 9 13 ffvelrnd ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( 𝐺𝑦 ) ∈ 𝑍 )
19 8 18 sselid ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( 𝐺𝑦 ) ∈ ℝ )
20 19 14 resubcld ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝐺𝑦 ) − 𝑦 ) ∈ ℝ )
21 simpr ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → 𝑥 < 𝑦 )
22 16 14 12 21 ltsub2dd ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝐺𝑥 ) − 𝑦 ) < ( ( 𝐺𝑥 ) − 𝑥 ) )
23 10 nnzd ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → 𝑥 ∈ ℤ )
24 13 nnzd ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → 𝑦 ∈ ℤ )
25 16 14 21 ltled ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → 𝑥𝑦 )
26 eluz2 ( 𝑦 ∈ ( ℤ𝑥 ) ↔ ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑥𝑦 ) )
27 23 24 25 26 syl3anbrc ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → 𝑦 ∈ ( ℤ𝑥 ) )
28 elfzuz ( 𝑘 ∈ ( 𝑥 ... 𝑦 ) → 𝑘 ∈ ( ℤ𝑥 ) )
29 eluznn ( ( 𝑥 ∈ ℕ ∧ 𝑘 ∈ ( ℤ𝑥 ) ) → 𝑘 ∈ ℕ )
30 10 29 sylan ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ( ℤ𝑥 ) ) → 𝑘 ∈ ℕ )
31 fveq2 ( 𝑛 = 𝑘 → ( 𝐺𝑛 ) = ( 𝐺𝑘 ) )
32 id ( 𝑛 = 𝑘𝑛 = 𝑘 )
33 31 32 oveq12d ( 𝑛 = 𝑘 → ( ( 𝐺𝑛 ) − 𝑛 ) = ( ( 𝐺𝑘 ) − 𝑘 ) )
34 eqid ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) ) = ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) )
35 ovex ( ( 𝐺𝑘 ) − 𝑘 ) ∈ V
36 33 34 35 fvmpt ( 𝑘 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) ) ‘ 𝑘 ) = ( ( 𝐺𝑘 ) − 𝑘 ) )
37 36 adantl ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) ) ‘ 𝑘 ) = ( ( 𝐺𝑘 ) − 𝑘 ) )
38 9 ffvelrnda ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( 𝐺𝑘 ) ∈ 𝑍 )
39 8 38 sselid ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( 𝐺𝑘 ) ∈ ℝ )
40 nnre ( 𝑘 ∈ ℕ → 𝑘 ∈ ℝ )
41 40 adantl ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℝ )
42 39 41 resubcld ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝐺𝑘 ) − 𝑘 ) ∈ ℝ )
43 37 42 eqeltrd ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) ) ‘ 𝑘 ) ∈ ℝ )
44 30 43 syldan ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ( ℤ𝑥 ) ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) ) ‘ 𝑘 ) ∈ ℝ )
45 28 44 sylan2 ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ( 𝑥 ... 𝑦 ) ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) ) ‘ 𝑘 ) ∈ ℝ )
46 elfzuz ( 𝑘 ∈ ( 𝑥 ... ( 𝑦 − 1 ) ) → 𝑘 ∈ ( ℤ𝑥 ) )
47 peano2nn ( 𝑘 ∈ ℕ → ( 𝑘 + 1 ) ∈ ℕ )
48 ffvelrn ( ( 𝐺 : ℕ ⟶ 𝑍 ∧ ( 𝑘 + 1 ) ∈ ℕ ) → ( 𝐺 ‘ ( 𝑘 + 1 ) ) ∈ 𝑍 )
49 9 47 48 syl2an ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ ( 𝑘 + 1 ) ) ∈ 𝑍 )
50 8 49 sselid ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ ( 𝑘 + 1 ) ) ∈ ℝ )
51 peano2rem ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) ∈ ℝ → ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − 1 ) ∈ ℝ )
52 50 51 syl ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − 1 ) ∈ ℝ )
53 4 ad4ant14 ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( 𝐺𝑘 ) < ( 𝐺 ‘ ( 𝑘 + 1 ) ) )
54 6 38 sselid ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( 𝐺𝑘 ) ∈ ℤ )
55 6 49 sselid ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ ( 𝑘 + 1 ) ) ∈ ℤ )
56 zltlem1 ( ( ( 𝐺𝑘 ) ∈ ℤ ∧ ( 𝐺 ‘ ( 𝑘 + 1 ) ) ∈ ℤ ) → ( ( 𝐺𝑘 ) < ( 𝐺 ‘ ( 𝑘 + 1 ) ) ↔ ( 𝐺𝑘 ) ≤ ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − 1 ) ) )
57 54 55 56 syl2anc ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝐺𝑘 ) < ( 𝐺 ‘ ( 𝑘 + 1 ) ) ↔ ( 𝐺𝑘 ) ≤ ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − 1 ) ) )
58 53 57 mpbid ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( 𝐺𝑘 ) ≤ ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − 1 ) )
59 39 52 41 58 lesub1dd ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝐺𝑘 ) − 𝑘 ) ≤ ( ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − 1 ) − 𝑘 ) )
60 50 recnd ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ ( 𝑘 + 1 ) ) ∈ ℂ )
61 1cnd ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → 1 ∈ ℂ )
62 41 recnd ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℂ )
63 60 61 62 sub32d ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − 1 ) − 𝑘 ) = ( ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − 𝑘 ) − 1 ) )
64 60 62 61 subsub4d ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − 𝑘 ) − 1 ) = ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − ( 𝑘 + 1 ) ) )
65 63 64 eqtrd ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − 1 ) − 𝑘 ) = ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − ( 𝑘 + 1 ) ) )
66 59 65 breqtrd ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝐺𝑘 ) − 𝑘 ) ≤ ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − ( 𝑘 + 1 ) ) )
67 47 adantl ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( 𝑘 + 1 ) ∈ ℕ )
68 fveq2 ( 𝑛 = ( 𝑘 + 1 ) → ( 𝐺𝑛 ) = ( 𝐺 ‘ ( 𝑘 + 1 ) ) )
69 id ( 𝑛 = ( 𝑘 + 1 ) → 𝑛 = ( 𝑘 + 1 ) )
70 68 69 oveq12d ( 𝑛 = ( 𝑘 + 1 ) → ( ( 𝐺𝑛 ) − 𝑛 ) = ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − ( 𝑘 + 1 ) ) )
71 ovex ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − ( 𝑘 + 1 ) ) ∈ V
72 70 34 71 fvmpt ( ( 𝑘 + 1 ) ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) ) ‘ ( 𝑘 + 1 ) ) = ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − ( 𝑘 + 1 ) ) )
73 67 72 syl ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) ) ‘ ( 𝑘 + 1 ) ) = ( ( 𝐺 ‘ ( 𝑘 + 1 ) ) − ( 𝑘 + 1 ) ) )
74 66 37 73 3brtr4d ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) ) ‘ 𝑘 ) ≤ ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) ) ‘ ( 𝑘 + 1 ) ) )
75 30 74 syldan ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ( ℤ𝑥 ) ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) ) ‘ 𝑘 ) ≤ ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) ) ‘ ( 𝑘 + 1 ) ) )
76 46 75 sylan2 ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) ∧ 𝑘 ∈ ( 𝑥 ... ( 𝑦 − 1 ) ) ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) ) ‘ 𝑘 ) ≤ ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) ) ‘ ( 𝑘 + 1 ) ) )
77 27 45 76 monoord ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) ) ‘ 𝑥 ) ≤ ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) ) ‘ 𝑦 ) )
78 fveq2 ( 𝑛 = 𝑥 → ( 𝐺𝑛 ) = ( 𝐺𝑥 ) )
79 id ( 𝑛 = 𝑥𝑛 = 𝑥 )
80 78 79 oveq12d ( 𝑛 = 𝑥 → ( ( 𝐺𝑛 ) − 𝑛 ) = ( ( 𝐺𝑥 ) − 𝑥 ) )
81 ovex ( ( 𝐺𝑥 ) − 𝑥 ) ∈ V
82 80 34 81 fvmpt ( 𝑥 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) ) ‘ 𝑥 ) = ( ( 𝐺𝑥 ) − 𝑥 ) )
83 10 82 syl ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) ) ‘ 𝑥 ) = ( ( 𝐺𝑥 ) − 𝑥 ) )
84 fveq2 ( 𝑛 = 𝑦 → ( 𝐺𝑛 ) = ( 𝐺𝑦 ) )
85 id ( 𝑛 = 𝑦𝑛 = 𝑦 )
86 84 85 oveq12d ( 𝑛 = 𝑦 → ( ( 𝐺𝑛 ) − 𝑛 ) = ( ( 𝐺𝑦 ) − 𝑦 ) )
87 ovex ( ( 𝐺𝑦 ) − 𝑦 ) ∈ V
88 86 34 87 fvmpt ( 𝑦 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) ) ‘ 𝑦 ) = ( ( 𝐺𝑦 ) − 𝑦 ) )
89 13 88 syl ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝑛 ∈ ℕ ↦ ( ( 𝐺𝑛 ) − 𝑛 ) ) ‘ 𝑦 ) = ( ( 𝐺𝑦 ) − 𝑦 ) )
90 77 83 89 3brtr3d ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝐺𝑥 ) − 𝑥 ) ≤ ( ( 𝐺𝑦 ) − 𝑦 ) )
91 15 17 20 22 90 ltletrd ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝐺𝑥 ) − 𝑦 ) < ( ( 𝐺𝑦 ) − 𝑦 ) )
92 12 19 14 ltsub1d ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( ( 𝐺𝑥 ) < ( 𝐺𝑦 ) ↔ ( ( 𝐺𝑥 ) − 𝑦 ) < ( ( 𝐺𝑦 ) − 𝑦 ) ) )
93 91 92 mpbird ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) ∧ 𝑥 < 𝑦 ) → ( 𝐺𝑥 ) < ( 𝐺𝑦 ) )
94 93 ex ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) → ( 𝑥 < 𝑦 → ( 𝐺𝑥 ) < ( 𝐺𝑦 ) ) )
95 94 ralrimivva ( 𝜑 → ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ( 𝑥 < 𝑦 → ( 𝐺𝑥 ) < ( 𝐺𝑦 ) ) )
96 ss2ralv ( 𝑆 ⊆ ℕ → ( ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ( 𝑥 < 𝑦 → ( 𝐺𝑥 ) < ( 𝐺𝑦 ) ) → ∀ 𝑥𝑆𝑦𝑆 ( 𝑥 < 𝑦 → ( 𝐺𝑥 ) < ( 𝐺𝑦 ) ) ) )
97 95 96 mpan9 ( ( 𝜑𝑆 ⊆ ℕ ) → ∀ 𝑥𝑆𝑦𝑆 ( 𝑥 < 𝑦 → ( 𝐺𝑥 ) < ( 𝐺𝑦 ) ) )
98 nnssre ℕ ⊆ ℝ
99 ltso < Or ℝ
100 soss ( ℕ ⊆ ℝ → ( < Or ℝ → < Or ℕ ) )
101 98 99 100 mp2 < Or ℕ
102 101 a1i ( ( 𝜑𝑆 ⊆ ℕ ) → < Or ℕ )
103 soss ( 𝑍 ⊆ ℝ → ( < Or ℝ → < Or 𝑍 ) )
104 8 99 103 mp2 < Or 𝑍
105 104 a1i ( ( 𝜑𝑆 ⊆ ℕ ) → < Or 𝑍 )
106 3 adantr ( ( 𝜑𝑆 ⊆ ℕ ) → 𝐺 : ℕ ⟶ 𝑍 )
107 simpr ( ( 𝜑𝑆 ⊆ ℕ ) → 𝑆 ⊆ ℕ )
108 soisores ( ( ( < Or ℕ ∧ < Or 𝑍 ) ∧ ( 𝐺 : ℕ ⟶ 𝑍𝑆 ⊆ ℕ ) ) → ( ( 𝐺𝑆 ) Isom < , < ( 𝑆 , ( 𝐺𝑆 ) ) ↔ ∀ 𝑥𝑆𝑦𝑆 ( 𝑥 < 𝑦 → ( 𝐺𝑥 ) < ( 𝐺𝑦 ) ) ) )
109 102 105 106 107 108 syl22anc ( ( 𝜑𝑆 ⊆ ℕ ) → ( ( 𝐺𝑆 ) Isom < , < ( 𝑆 , ( 𝐺𝑆 ) ) ↔ ∀ 𝑥𝑆𝑦𝑆 ( 𝑥 < 𝑦 → ( 𝐺𝑥 ) < ( 𝐺𝑦 ) ) ) )
110 97 109 mpbird ( ( 𝜑𝑆 ⊆ ℕ ) → ( 𝐺𝑆 ) Isom < , < ( 𝑆 , ( 𝐺𝑆 ) ) )