Metamath Proof Explorer


Theorem smfinfmpt

Description: The infimum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (c) of Fremlin1 p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses smfinfmpt.n 𝑛 𝜑
smfinfmpt.x 𝑥 𝜑
smfinfmpt.y 𝑦 𝜑
smfinfmpt.m ( 𝜑𝑀 ∈ ℤ )
smfinfmpt.z 𝑍 = ( ℤ𝑀 )
smfinfmpt.s ( 𝜑𝑆 ∈ SAlg )
smfinfmpt.b ( ( 𝜑𝑛𝑍𝑥𝐴 ) → 𝐵𝑉 )
smfinfmpt.f ( ( 𝜑𝑛𝑍 ) → ( 𝑥𝐴𝐵 ) ∈ ( SMblFn ‘ 𝑆 ) )
smfinfmpt.d 𝐷 = { 𝑥 𝑛𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 }
smfinfmpt.g 𝐺 = ( 𝑥𝐷 ↦ inf ( ran ( 𝑛𝑍𝐵 ) , ℝ , < ) )
Assertion smfinfmpt ( 𝜑𝐺 ∈ ( SMblFn ‘ 𝑆 ) )

Proof

Step Hyp Ref Expression
1 smfinfmpt.n 𝑛 𝜑
2 smfinfmpt.x 𝑥 𝜑
3 smfinfmpt.y 𝑦 𝜑
4 smfinfmpt.m ( 𝜑𝑀 ∈ ℤ )
5 smfinfmpt.z 𝑍 = ( ℤ𝑀 )
6 smfinfmpt.s ( 𝜑𝑆 ∈ SAlg )
7 smfinfmpt.b ( ( 𝜑𝑛𝑍𝑥𝐴 ) → 𝐵𝑉 )
8 smfinfmpt.f ( ( 𝜑𝑛𝑍 ) → ( 𝑥𝐴𝐵 ) ∈ ( SMblFn ‘ 𝑆 ) )
9 smfinfmpt.d 𝐷 = { 𝑥 𝑛𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 }
10 smfinfmpt.g 𝐺 = ( 𝑥𝐷 ↦ inf ( ran ( 𝑛𝑍𝐵 ) , ℝ , < ) )
11 10 a1i ( 𝜑𝐺 = ( 𝑥𝐷 ↦ inf ( ran ( 𝑛𝑍𝐵 ) , ℝ , < ) ) )
12 9 a1i ( 𝜑𝐷 = { 𝑥 𝑛𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 } )
13 eqidd ( 𝜑 → ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) = ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) )
14 13 8 fvmpt2d ( ( 𝜑𝑛𝑍 ) → ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) = ( 𝑥𝐴𝐵 ) )
15 14 dmeqd ( ( 𝜑𝑛𝑍 ) → dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) = dom ( 𝑥𝐴𝐵 ) )
16 nfcv 𝑥 𝑛
17 nfcv 𝑥 𝑍
18 16 17 nfel 𝑥 𝑛𝑍
19 2 18 nfan 𝑥 ( 𝜑𝑛𝑍 )
20 eqid ( 𝑥𝐴𝐵 ) = ( 𝑥𝐴𝐵 )
21 6 adantr ( ( 𝜑𝑛𝑍 ) → 𝑆 ∈ SAlg )
22 7 3expa ( ( ( 𝜑𝑛𝑍 ) ∧ 𝑥𝐴 ) → 𝐵𝑉 )
23 19 21 22 8 smffmpt ( ( 𝜑𝑛𝑍 ) → ( 𝑥𝐴𝐵 ) : 𝐴 ⟶ ℝ )
24 23 fvmptelrn ( ( ( 𝜑𝑛𝑍 ) ∧ 𝑥𝐴 ) → 𝐵 ∈ ℝ )
25 19 20 24 dmmptdf ( ( 𝜑𝑛𝑍 ) → dom ( 𝑥𝐴𝐵 ) = 𝐴 )
26 eqidd ( ( 𝜑𝑛𝑍 ) → 𝐴 = 𝐴 )
27 15 25 26 3eqtrrd ( ( 𝜑𝑛𝑍 ) → 𝐴 = dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) )
28 1 27 iineq2d ( 𝜑 𝑛𝑍 𝐴 = 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) )
29 nfcv 𝑥 𝑛𝑍 𝐴
30 nfmpt1 𝑥 ( 𝑥𝐴𝐵 )
31 17 30 nfmpt 𝑥 ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) )
32 31 16 nffv 𝑥 ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 )
33 32 nfdm 𝑥 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 )
34 17 33 nfiin 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 )
35 29 34 rabeqf ( 𝑛𝑍 𝐴 = 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) → { 𝑥 𝑛𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 } = { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 } )
36 28 35 syl ( 𝜑 → { 𝑥 𝑛𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 } = { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 } )
37 nfv 𝑦 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 )
38 3 37 nfan 𝑦 ( 𝜑𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) )
39 nfcv 𝑛 𝑥
40 nfii1 𝑛 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 )
41 39 40 nfel 𝑛 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 )
42 1 41 nfan 𝑛 ( 𝜑𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) )
43 simpll ( ( ( 𝜑𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛𝑍 ) → 𝜑 )
44 simpr ( ( ( 𝜑𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛𝑍 ) → 𝑛𝑍 )
45 eliinid ( ( 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∧ 𝑛𝑍 ) → 𝑥 ∈ dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) )
46 45 adantll ( ( ( 𝜑𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛𝑍 ) → 𝑥 ∈ dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) )
47 27 eqcomd ( ( 𝜑𝑛𝑍 ) → dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) = 𝐴 )
48 47 adantlr ( ( ( 𝜑𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛𝑍 ) → dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) = 𝐴 )
49 46 48 eleqtrd ( ( ( 𝜑𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛𝑍 ) → 𝑥𝐴 )
50 14 fveq1d ( ( 𝜑𝑛𝑍 ) → ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) = ( ( 𝑥𝐴𝐵 ) ‘ 𝑥 ) )
51 50 3adant3 ( ( 𝜑𝑛𝑍𝑥𝐴 ) → ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) = ( ( 𝑥𝐴𝐵 ) ‘ 𝑥 ) )
52 simp3 ( ( 𝜑𝑛𝑍𝑥𝐴 ) → 𝑥𝐴 )
53 20 fvmpt2 ( ( 𝑥𝐴𝐵𝑉 ) → ( ( 𝑥𝐴𝐵 ) ‘ 𝑥 ) = 𝐵 )
54 52 7 53 syl2anc ( ( 𝜑𝑛𝑍𝑥𝐴 ) → ( ( 𝑥𝐴𝐵 ) ‘ 𝑥 ) = 𝐵 )
55 51 54 eqtr2d ( ( 𝜑𝑛𝑍𝑥𝐴 ) → 𝐵 = ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) )
56 55 breq2d ( ( 𝜑𝑛𝑍𝑥𝐴 ) → ( 𝑦𝐵𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
57 43 44 49 56 syl3anc ( ( ( 𝜑𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛𝑍 ) → ( 𝑦𝐵𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
58 42 57 ralbida ( ( 𝜑𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ) → ( ∀ 𝑛𝑍 𝑦𝐵 ↔ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
59 38 58 rexbid ( ( 𝜑𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ) → ( ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 ↔ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
60 2 59 rabbida ( 𝜑 → { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 } = { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } )
61 36 60 eqtrd ( 𝜑 → { 𝑥 𝑛𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 } = { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } )
62 12 61 eqtrd ( 𝜑𝐷 = { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } )
63 2 62 alrimi ( 𝜑 → ∀ 𝑥 𝐷 = { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } )
64 nfcv 𝑛
65 nfra1 𝑛𝑛𝑍 𝑦𝐵
66 64 65 nfrex 𝑛𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵
67 nfii1 𝑛 𝑛𝑍 𝐴
68 66 67 nfrabw 𝑛 { 𝑥 𝑛𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 }
69 9 68 nfcxfr 𝑛 𝐷
70 39 69 nfel 𝑛 𝑥𝐷
71 1 70 nfan 𝑛 ( 𝜑𝑥𝐷 )
72 simpll ( ( ( 𝜑𝑥𝐷 ) ∧ 𝑛𝑍 ) → 𝜑 )
73 simpr ( ( ( 𝜑𝑥𝐷 ) ∧ 𝑛𝑍 ) → 𝑛𝑍 )
74 9 eleq2i ( 𝑥𝐷𝑥 ∈ { 𝑥 𝑛𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 } )
75 74 biimpi ( 𝑥𝐷𝑥 ∈ { 𝑥 𝑛𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 } )
76 rabidim1 ( 𝑥 ∈ { 𝑥 𝑛𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 } → 𝑥 𝑛𝑍 𝐴 )
77 75 76 syl ( 𝑥𝐷𝑥 𝑛𝑍 𝐴 )
78 77 adantr ( ( 𝑥𝐷𝑛𝑍 ) → 𝑥 𝑛𝑍 𝐴 )
79 simpr ( ( 𝑥𝐷𝑛𝑍 ) → 𝑛𝑍 )
80 eliinid ( ( 𝑥 𝑛𝑍 𝐴𝑛𝑍 ) → 𝑥𝐴 )
81 78 79 80 syl2anc ( ( 𝑥𝐷𝑛𝑍 ) → 𝑥𝐴 )
82 81 adantll ( ( ( 𝜑𝑥𝐷 ) ∧ 𝑛𝑍 ) → 𝑥𝐴 )
83 55 idi ( ( 𝜑𝑛𝑍𝑥𝐴 ) → 𝐵 = ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) )
84 72 73 82 83 syl3anc ( ( ( 𝜑𝑥𝐷 ) ∧ 𝑛𝑍 ) → 𝐵 = ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) )
85 71 84 mpteq2da ( ( 𝜑𝑥𝐷 ) → ( 𝑛𝑍𝐵 ) = ( 𝑛𝑍 ↦ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
86 85 rneqd ( ( 𝜑𝑥𝐷 ) → ran ( 𝑛𝑍𝐵 ) = ran ( 𝑛𝑍 ↦ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
87 86 infeq1d ( ( 𝜑𝑥𝐷 ) → inf ( ran ( 𝑛𝑍𝐵 ) , ℝ , < ) = inf ( ran ( 𝑛𝑍 ↦ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) )
88 87 ex ( 𝜑 → ( 𝑥𝐷 → inf ( ran ( 𝑛𝑍𝐵 ) , ℝ , < ) = inf ( ran ( 𝑛𝑍 ↦ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) )
89 2 88 ralrimi ( 𝜑 → ∀ 𝑥𝐷 inf ( ran ( 𝑛𝑍𝐵 ) , ℝ , < ) = inf ( ran ( 𝑛𝑍 ↦ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) )
90 mpteq12f ( ( ∀ 𝑥 𝐷 = { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ∧ ∀ 𝑥𝐷 inf ( ran ( 𝑛𝑍𝐵 ) , ℝ , < ) = inf ( ran ( 𝑛𝑍 ↦ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) → ( 𝑥𝐷 ↦ inf ( ran ( 𝑛𝑍𝐵 ) , ℝ , < ) ) = ( 𝑥 ∈ { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛𝑍 ↦ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) )
91 63 89 90 syl2anc ( 𝜑 → ( 𝑥𝐷 ↦ inf ( ran ( 𝑛𝑍𝐵 ) , ℝ , < ) ) = ( 𝑥 ∈ { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛𝑍 ↦ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) )
92 11 91 eqtrd ( 𝜑𝐺 = ( 𝑥 ∈ { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛𝑍 ↦ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) )
93 nfmpt1 𝑛 ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) )
94 eqid ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) = ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) )
95 1 8 94 fmptdf ( 𝜑 → ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) : 𝑍 ⟶ ( SMblFn ‘ 𝑆 ) )
96 eqid { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } = { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) }
97 eqid ( 𝑥 ∈ { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛𝑍 ↦ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) = ( 𝑥 ∈ { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛𝑍 ↦ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) )
98 93 31 4 5 6 95 96 97 smfinf ( 𝜑 → ( 𝑥 ∈ { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛𝑍 ↦ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) ∈ ( SMblFn ‘ 𝑆 ) )
99 92 98 eqeltrd ( 𝜑𝐺 ∈ ( SMblFn ‘ 𝑆 ) )