Metamath Proof Explorer


Theorem smflimsuplem5

Description: H converges to the superior limit of F . (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses smflimsuplem5.a 𝑛 𝜑
smflimsuplem5.b 𝑚 𝜑
smflimsuplem5.m ( 𝜑𝑀 ∈ ℤ )
smflimsuplem5.z 𝑍 = ( ℤ𝑀 )
smflimsuplem5.s ( 𝜑𝑆 ∈ SAlg )
smflimsuplem5.f ( 𝜑𝐹 : 𝑍 ⟶ ( SMblFn ‘ 𝑆 ) )
smflimsuplem5.e 𝐸 = ( 𝑛𝑍 ↦ { 𝑥 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ } )
smflimsuplem5.h 𝐻 = ( 𝑛𝑍 ↦ ( 𝑥 ∈ ( 𝐸𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ) )
smflimsuplem5.r ( 𝜑 → ( lim sup ‘ ( 𝑚𝑍 ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) ) ∈ ℝ )
smflimsuplem5.n ( 𝜑𝑁𝑍 )
smflimsuplem5.x ( 𝜑𝑋 𝑚 ∈ ( ℤ𝑁 ) dom ( 𝐹𝑚 ) )
Assertion smflimsuplem5 ( 𝜑 → ( 𝑛 ∈ ( ℤ𝑁 ) ↦ ( ( 𝐻𝑛 ) ‘ 𝑋 ) ) ⇝ ( lim sup ‘ ( 𝑚 ∈ ( ℤ𝑁 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) ) )

Proof

Step Hyp Ref Expression
1 smflimsuplem5.a 𝑛 𝜑
2 smflimsuplem5.b 𝑚 𝜑
3 smflimsuplem5.m ( 𝜑𝑀 ∈ ℤ )
4 smflimsuplem5.z 𝑍 = ( ℤ𝑀 )
5 smflimsuplem5.s ( 𝜑𝑆 ∈ SAlg )
6 smflimsuplem5.f ( 𝜑𝐹 : 𝑍 ⟶ ( SMblFn ‘ 𝑆 ) )
7 smflimsuplem5.e 𝐸 = ( 𝑛𝑍 ↦ { 𝑥 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ } )
8 smflimsuplem5.h 𝐻 = ( 𝑛𝑍 ↦ ( 𝑥 ∈ ( 𝐸𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ) )
9 smflimsuplem5.r ( 𝜑 → ( lim sup ‘ ( 𝑚𝑍 ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) ) ∈ ℝ )
10 smflimsuplem5.n ( 𝜑𝑁𝑍 )
11 smflimsuplem5.x ( 𝜑𝑋 𝑚 ∈ ( ℤ𝑁 ) dom ( 𝐹𝑚 ) )
12 4 eleq2i ( 𝑁𝑍𝑁 ∈ ( ℤ𝑀 ) )
13 12 biimpi ( 𝑁𝑍𝑁 ∈ ( ℤ𝑀 ) )
14 uzss ( 𝑁 ∈ ( ℤ𝑀 ) → ( ℤ𝑁 ) ⊆ ( ℤ𝑀 ) )
15 13 14 syl ( 𝑁𝑍 → ( ℤ𝑁 ) ⊆ ( ℤ𝑀 ) )
16 15 4 sseqtrrdi ( 𝑁𝑍 → ( ℤ𝑁 ) ⊆ 𝑍 )
17 10 16 syl ( 𝜑 → ( ℤ𝑁 ) ⊆ 𝑍 )
18 17 sselda ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → 𝑛𝑍 )
19 nfcv 𝑥 𝑍
20 nfrab1 𝑥 { 𝑥 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ }
21 19 20 nfmpt 𝑥 ( 𝑛𝑍 ↦ { 𝑥 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ } )
22 7 21 nfcxfr 𝑥 𝐸
23 nfcv 𝑥 𝑛
24 22 23 nffv 𝑥 ( 𝐸𝑛 )
25 fvex ( 𝐸𝑛 ) ∈ V
26 24 25 mptexf ( 𝑥 ∈ ( 𝐸𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ) ∈ V
27 26 a1i ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → ( 𝑥 ∈ ( 𝐸𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ) ∈ V )
28 8 fvmpt2 ( ( 𝑛𝑍 ∧ ( 𝑥 ∈ ( 𝐸𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ) ∈ V ) → ( 𝐻𝑛 ) = ( 𝑥 ∈ ( 𝐸𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ) )
29 18 27 28 syl2anc ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → ( 𝐻𝑛 ) = ( 𝑥 ∈ ( 𝐸𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ) )
30 29 fveq1d ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → ( ( 𝐻𝑛 ) ‘ 𝑋 ) = ( ( 𝑥 ∈ ( 𝐸𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ) ‘ 𝑋 ) )
31 nfcv 𝑦 ( 𝐸𝑛 )
32 nfcv 𝑦 sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < )
33 nfcv 𝑥 sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑦 ) ) , ℝ* , < )
34 fveq2 ( 𝑥 = 𝑦 → ( ( 𝐹𝑚 ) ‘ 𝑥 ) = ( ( 𝐹𝑚 ) ‘ 𝑦 ) )
35 34 mpteq2dv ( 𝑥 = 𝑦 → ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) = ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑦 ) ) )
36 35 rneqd ( 𝑥 = 𝑦 → ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) = ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑦 ) ) )
37 36 supeq1d ( 𝑥 = 𝑦 → sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) = sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑦 ) ) , ℝ* , < ) )
38 24 31 32 33 37 cbvmptf ( 𝑥 ∈ ( 𝐸𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ) = ( 𝑦 ∈ ( 𝐸𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑦 ) ) , ℝ* , < ) )
39 simpl ( ( 𝑦 = 𝑋𝑚 ∈ ( ℤ𝑛 ) ) → 𝑦 = 𝑋 )
40 39 fveq2d ( ( 𝑦 = 𝑋𝑚 ∈ ( ℤ𝑛 ) ) → ( ( 𝐹𝑚 ) ‘ 𝑦 ) = ( ( 𝐹𝑚 ) ‘ 𝑋 ) )
41 40 mpteq2dva ( 𝑦 = 𝑋 → ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑦 ) ) = ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) )
42 41 rneqd ( 𝑦 = 𝑋 → ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑦 ) ) = ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) )
43 42 supeq1d ( 𝑦 = 𝑋 → sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑦 ) ) , ℝ* , < ) = sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) , ℝ* , < ) )
44 43 eleq1d ( 𝑦 = 𝑋 → ( sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑦 ) ) , ℝ* , < ) ∈ ℝ ↔ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) , ℝ* , < ) ∈ ℝ ) )
45 uzss ( 𝑛 ∈ ( ℤ𝑁 ) → ( ℤ𝑛 ) ⊆ ( ℤ𝑁 ) )
46 iinss1 ( ( ℤ𝑛 ) ⊆ ( ℤ𝑁 ) → 𝑚 ∈ ( ℤ𝑁 ) dom ( 𝐹𝑚 ) ⊆ 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) )
47 45 46 syl ( 𝑛 ∈ ( ℤ𝑁 ) → 𝑚 ∈ ( ℤ𝑁 ) dom ( 𝐹𝑚 ) ⊆ 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) )
48 47 adantl ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → 𝑚 ∈ ( ℤ𝑁 ) dom ( 𝐹𝑚 ) ⊆ 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) )
49 11 adantr ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → 𝑋 𝑚 ∈ ( ℤ𝑁 ) dom ( 𝐹𝑚 ) )
50 48 49 sseldd ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → 𝑋 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) )
51 nfv 𝑚 𝑛 ∈ ( ℤ𝑁 )
52 2 51 nfan 𝑚 ( 𝜑𝑛 ∈ ( ℤ𝑁 ) )
53 eqid ( ℤ𝑛 ) = ( ℤ𝑛 )
54 simpll ( ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) ∧ 𝑚 ∈ ( ℤ𝑛 ) ) → 𝜑 )
55 45 sselda ( ( 𝑛 ∈ ( ℤ𝑁 ) ∧ 𝑚 ∈ ( ℤ𝑛 ) ) → 𝑚 ∈ ( ℤ𝑁 ) )
56 55 adantll ( ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) ∧ 𝑚 ∈ ( ℤ𝑛 ) ) → 𝑚 ∈ ( ℤ𝑁 ) )
57 5 adantr ( ( 𝜑𝑚 ∈ ( ℤ𝑁 ) ) → 𝑆 ∈ SAlg )
58 simpl ( ( 𝜑𝑚 ∈ ( ℤ𝑁 ) ) → 𝜑 )
59 17 sselda ( ( 𝜑𝑚 ∈ ( ℤ𝑁 ) ) → 𝑚𝑍 )
60 6 ffvelrnda ( ( 𝜑𝑚𝑍 ) → ( 𝐹𝑚 ) ∈ ( SMblFn ‘ 𝑆 ) )
61 58 59 60 syl2anc ( ( 𝜑𝑚 ∈ ( ℤ𝑁 ) ) → ( 𝐹𝑚 ) ∈ ( SMblFn ‘ 𝑆 ) )
62 eqid dom ( 𝐹𝑚 ) = dom ( 𝐹𝑚 )
63 57 61 62 smff ( ( 𝜑𝑚 ∈ ( ℤ𝑁 ) ) → ( 𝐹𝑚 ) : dom ( 𝐹𝑚 ) ⟶ ℝ )
64 eliin ( 𝑋 𝑚 ∈ ( ℤ𝑁 ) dom ( 𝐹𝑚 ) → ( 𝑋 𝑚 ∈ ( ℤ𝑁 ) dom ( 𝐹𝑚 ) ↔ ∀ 𝑚 ∈ ( ℤ𝑁 ) 𝑋 ∈ dom ( 𝐹𝑚 ) ) )
65 11 64 syl ( 𝜑 → ( 𝑋 𝑚 ∈ ( ℤ𝑁 ) dom ( 𝐹𝑚 ) ↔ ∀ 𝑚 ∈ ( ℤ𝑁 ) 𝑋 ∈ dom ( 𝐹𝑚 ) ) )
66 11 65 mpbid ( 𝜑 → ∀ 𝑚 ∈ ( ℤ𝑁 ) 𝑋 ∈ dom ( 𝐹𝑚 ) )
67 66 adantr ( ( 𝜑𝑚 ∈ ( ℤ𝑁 ) ) → ∀ 𝑚 ∈ ( ℤ𝑁 ) 𝑋 ∈ dom ( 𝐹𝑚 ) )
68 simpr ( ( 𝜑𝑚 ∈ ( ℤ𝑁 ) ) → 𝑚 ∈ ( ℤ𝑁 ) )
69 rspa ( ( ∀ 𝑚 ∈ ( ℤ𝑁 ) 𝑋 ∈ dom ( 𝐹𝑚 ) ∧ 𝑚 ∈ ( ℤ𝑁 ) ) → 𝑋 ∈ dom ( 𝐹𝑚 ) )
70 67 68 69 syl2anc ( ( 𝜑𝑚 ∈ ( ℤ𝑁 ) ) → 𝑋 ∈ dom ( 𝐹𝑚 ) )
71 63 70 ffvelrnd ( ( 𝜑𝑚 ∈ ( ℤ𝑁 ) ) → ( ( 𝐹𝑚 ) ‘ 𝑋 ) ∈ ℝ )
72 54 56 71 syl2anc ( ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) ∧ 𝑚 ∈ ( ℤ𝑛 ) ) → ( ( 𝐹𝑚 ) ‘ 𝑋 ) ∈ ℝ )
73 eluzelz ( 𝑛 ∈ ( ℤ𝑁 ) → 𝑛 ∈ ℤ )
74 73 adantl ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → 𝑛 ∈ ℤ )
75 3 adantr ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → 𝑀 ∈ ℤ )
76 fvex ( ( 𝐹𝑚 ) ‘ 𝑋 ) ∈ V
77 76 a1i ( ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) ∧ 𝑚𝑍 ) → ( ( 𝐹𝑚 ) ‘ 𝑋 ) ∈ V )
78 52 74 75 53 4 72 77 limsupequzmpt ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → ( lim sup ‘ ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) ) = ( lim sup ‘ ( 𝑚𝑍 ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) ) )
79 9 adantr ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → ( lim sup ‘ ( 𝑚𝑍 ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) ) ∈ ℝ )
80 78 79 eqeltrd ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → ( lim sup ‘ ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) ) ∈ ℝ )
81 80 renepnfd ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → ( lim sup ‘ ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) ) ≠ +∞ )
82 52 53 72 81 limsupubuzmpt ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → ∃ 𝑦 ∈ ℝ ∀ 𝑚 ∈ ( ℤ𝑛 ) ( ( 𝐹𝑚 ) ‘ 𝑋 ) ≤ 𝑦 )
83 uzid2 ( 𝑛 ∈ ( ℤ𝑁 ) → 𝑛 ∈ ( ℤ𝑛 ) )
84 83 ne0d ( 𝑛 ∈ ( ℤ𝑁 ) → ( ℤ𝑛 ) ≠ ∅ )
85 84 adantl ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → ( ℤ𝑛 ) ≠ ∅ )
86 52 85 72 supxrre3rnmpt ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → ( sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) , ℝ* , < ) ∈ ℝ ↔ ∃ 𝑦 ∈ ℝ ∀ 𝑚 ∈ ( ℤ𝑛 ) ( ( 𝐹𝑚 ) ‘ 𝑋 ) ≤ 𝑦 ) )
87 82 86 mpbird ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) , ℝ* , < ) ∈ ℝ )
88 44 50 87 elrabd ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → 𝑋 ∈ { 𝑦 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑦 ) ) , ℝ* , < ) ∈ ℝ } )
89 simpl ( ( 𝑦 = 𝑥𝑚 ∈ ( ℤ𝑛 ) ) → 𝑦 = 𝑥 )
90 89 fveq2d ( ( 𝑦 = 𝑥𝑚 ∈ ( ℤ𝑛 ) ) → ( ( 𝐹𝑚 ) ‘ 𝑦 ) = ( ( 𝐹𝑚 ) ‘ 𝑥 ) )
91 90 mpteq2dva ( 𝑦 = 𝑥 → ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑦 ) ) = ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) )
92 91 rneqd ( 𝑦 = 𝑥 → ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑦 ) ) = ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) )
93 92 supeq1d ( 𝑦 = 𝑥 → sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑦 ) ) , ℝ* , < ) = sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) )
94 93 eleq1d ( 𝑦 = 𝑥 → ( sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑦 ) ) , ℝ* , < ) ∈ ℝ ↔ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ ) )
95 94 cbvrabv { 𝑦 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑦 ) ) , ℝ* , < ) ∈ ℝ } = { 𝑥 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ }
96 88 95 eleqtrdi ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → 𝑋 ∈ { 𝑥 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ } )
97 eqid { 𝑥 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ } = { 𝑥 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ }
98 fvex ( 𝐹𝑚 ) ∈ V
99 98 dmex dom ( 𝐹𝑚 ) ∈ V
100 99 rgenw 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) ∈ V
101 100 a1i ( 𝑛 ∈ ( ℤ𝑁 ) → ∀ 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) ∈ V )
102 84 101 iinexd ( 𝑛 ∈ ( ℤ𝑁 ) → 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) ∈ V )
103 102 adantl ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) ∈ V )
104 97 103 rabexd ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → { 𝑥 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ } ∈ V )
105 7 fvmpt2 ( ( 𝑛𝑍 ∧ { 𝑥 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ } ∈ V ) → ( 𝐸𝑛 ) = { 𝑥 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ } )
106 18 104 105 syl2anc ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → ( 𝐸𝑛 ) = { 𝑥 𝑚 ∈ ( ℤ𝑛 ) dom ( 𝐹𝑚 ) ∣ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ∈ ℝ } )
107 96 106 eleqtrrd ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → 𝑋 ∈ ( 𝐸𝑛 ) )
108 38 43 107 87 fvmptd3 ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → ( ( 𝑥 ∈ ( 𝐸𝑛 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑥 ) ) , ℝ* , < ) ) ‘ 𝑋 ) = sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) , ℝ* , < ) )
109 30 108 eqtrd ( ( 𝜑𝑛 ∈ ( ℤ𝑁 ) ) → ( ( 𝐻𝑛 ) ‘ 𝑋 ) = sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) , ℝ* , < ) )
110 1 109 mpteq2da ( 𝜑 → ( 𝑛 ∈ ( ℤ𝑁 ) ↦ ( ( 𝐻𝑛 ) ‘ 𝑋 ) ) = ( 𝑛 ∈ ( ℤ𝑁 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) , ℝ* , < ) ) )
111 4 eluzelz2 ( 𝑁𝑍𝑁 ∈ ℤ )
112 10 111 syl ( 𝜑𝑁 ∈ ℤ )
113 eqid ( ℤ𝑁 ) = ( ℤ𝑁 )
114 76 a1i ( ( 𝜑𝑚 ∈ ( ℤ𝑁 ) ) → ( ( 𝐹𝑚 ) ‘ 𝑋 ) ∈ V )
115 76 a1i ( ( 𝜑𝑚𝑍 ) → ( ( 𝐹𝑚 ) ‘ 𝑋 ) ∈ V )
116 2 112 3 113 4 114 115 limsupequzmpt ( 𝜑 → ( lim sup ‘ ( 𝑚 ∈ ( ℤ𝑁 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) ) = ( lim sup ‘ ( 𝑚𝑍 ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) ) )
117 116 9 eqeltrd ( 𝜑 → ( lim sup ‘ ( 𝑚 ∈ ( ℤ𝑁 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) ) ∈ ℝ )
118 2 112 113 71 117 supcnvlimsupmpt ( 𝜑 → ( 𝑛 ∈ ( ℤ𝑁 ) ↦ sup ( ran ( 𝑚 ∈ ( ℤ𝑛 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) , ℝ* , < ) ) ⇝ ( lim sup ‘ ( 𝑚 ∈ ( ℤ𝑁 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) ) )
119 110 118 eqbrtrd ( 𝜑 → ( 𝑛 ∈ ( ℤ𝑁 ) ↦ ( ( 𝐻𝑛 ) ‘ 𝑋 ) ) ⇝ ( lim sup ‘ ( 𝑚 ∈ ( ℤ𝑁 ) ↦ ( ( 𝐹𝑚 ) ‘ 𝑋 ) ) ) )