Metamath Proof Explorer

Theorem sge0seq

Description: A series of nonnegative reals agrees with the generalized sum of nonnegative reals. (Contributed by Glauco Siliprandi, 3-Mar-2021)

Ref Expression
Hypotheses sge0seq.m ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
sge0seq.z ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
sge0seq.f ⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ( 0 [,) +∞ ) )
sge0seq.g ⊢ 𝐺 = seq 𝑀 ( + , 𝐹 )
Assertion sge0seq ( 𝜑 → ( Σ^ ‘ 𝐹 ) = sup ( ran 𝐺 , ℝ* , < ) )

Proof

Step Hyp Ref Expression
1 sge0seq.m ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
2 sge0seq.z ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
3 sge0seq.f ⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ( 0 [,) +∞ ) )
4 sge0seq.g ⊢ 𝐺 = seq 𝑀 ( + , 𝐹 )
5 rge0ssre ⊢ ( 0 [,) +∞ ) ⊆ ℝ
6 3 ffvelcdmda ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ( 0 [,) +∞ ) )
7 5 6 sselid ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ )
8 readdcl ⊢ ( ( 𝑘 ∈ ℝ ∧ 𝑖 ∈ ℝ ) → ( 𝑘 + 𝑖 ) ∈ ℝ )
9 8 adantl ⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℝ ∧ 𝑖 ∈ ℝ ) ) → ( 𝑘 + 𝑖 ) ∈ ℝ )
10 2 1 7 9 seqf ⊢ ( 𝜑 → seq 𝑀 ( + , 𝐹 ) : 𝑍 ⟶ ℝ )
11 4 a1i ⊢ ( 𝜑 → 𝐺 = seq 𝑀 ( + , 𝐹 ) )
12 11 feq1d ⊢ ( 𝜑 → ( 𝐺 : 𝑍 ⟶ ℝ ↔ seq 𝑀 ( + , 𝐹 ) : 𝑍 ⟶ ℝ ) )
13 10 12 mpbird ⊢ ( 𝜑 → 𝐺 : 𝑍 ⟶ ℝ )
14 13 frnd ⊢ ( 𝜑 → ran 𝐺 ⊆ ℝ )
15 ressxr ⊢ ℝ ⊆ ℝ*
16 15 a1i ⊢ ( 𝜑 → ℝ ⊆ ℝ* )
17 14 16 sstrd ⊢ ( 𝜑 → ran 𝐺 ⊆ ℝ* )
18 2 fvexi ⊢ 𝑍 ∈ V
19 18 a1i ⊢ ( 𝜑 → 𝑍 ∈ V )
20 icossicc ⊢ ( 0 [,) +∞ ) ⊆ ( 0 [,] +∞ )
21 20 a1i ⊢ ( 𝜑 → ( 0 [,) +∞ ) ⊆ ( 0 [,] +∞ ) )
22 3 21 fssd ⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ( 0 [,] +∞ ) )
23 19 22 sge0xrcl ⊢ ( 𝜑 → ( Σ^ ‘ 𝐹 ) ∈ ℝ* )
24 simpr ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ran 𝐺 ) → 𝑧 ∈ ran 𝐺 )
25 13 ffnd ⊢ ( 𝜑 → 𝐺 Fn 𝑍 )
26 fvelrnb ⊢ ( 𝐺 Fn 𝑍 → ( 𝑧 ∈ ran 𝐺 ↔ ∃ 𝑗 ∈ 𝑍 ( 𝐺 ‘ 𝑗 ) = 𝑧 ) )
27 25 26 syl ⊢ ( 𝜑 → ( 𝑧 ∈ ran 𝐺 ↔ ∃ 𝑗 ∈ 𝑍 ( 𝐺 ‘ 𝑗 ) = 𝑧 ) )
28 27 adantr ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ran 𝐺 ) → ( 𝑧 ∈ ran 𝐺 ↔ ∃ 𝑗 ∈ 𝑍 ( 𝐺 ‘ 𝑗 ) = 𝑧 ) )
29 24 28 mpbid ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ran 𝐺 ) → ∃ 𝑗 ∈ 𝑍 ( 𝐺 ‘ 𝑗 ) = 𝑧 )
30 20 6 sselid ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ( 0 [,] +∞ ) )
31 elfzuz ⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) )
32 31 2 eleqtrrdi ⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) → 𝑘 ∈ 𝑍 )
33 32 ssriv ⊢ ( 𝑀 ... 𝑗 ) ⊆ 𝑍
34 33 a1i ⊢ ( 𝜑 → ( 𝑀 ... 𝑗 ) ⊆ 𝑍 )
35 19 30 34 sge0lessmpt ⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ↦ ( 𝐹 ‘ 𝑘 ) ) ) ≤ ( Σ^ ‘ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) )
36 35 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝐺 ‘ 𝑗 ) = 𝑧 ) → ( Σ^ ‘ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ↦ ( 𝐹 ‘ 𝑘 ) ) ) ≤ ( Σ^ ‘ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) )
37 fzfid ⊢ ( 𝜑 → ( 𝑀 ... 𝑗 ) ∈ Fin )
38 32 6 sylan2 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ( 0 [,) +∞ ) )
39 37 38 sge0fsummpt ⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ↦ ( 𝐹 ‘ 𝑘 ) ) ) = Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) )
40 39 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝐺 ‘ 𝑗 ) = 𝑧 ) → ( Σ^ ‘ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ↦ ( 𝐹 ‘ 𝑘 ) ) ) = Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) )
41 simpll ⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → 𝜑 )
42 32 adantl ⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → 𝑘 ∈ 𝑍 )
43 eqidd ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) )
44 41 42 43 syl2anc ⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) )
45 2 eleq2i ⊢ ( 𝑗 ∈ 𝑍 ↔ 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) )
46 45 biimpi ⊢ ( 𝑗 ∈ 𝑍 → 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) )
47 46 adantl ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) )
48 7 recnd ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ )
49 41 42 48 syl2anc ⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ )
50 44 47 49 fsumser ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) = ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) )
51 50 3adant3 ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝐺 ‘ 𝑗 ) = 𝑧 ) → Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) = ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) )
52 40 51 eqtrd ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝐺 ‘ 𝑗 ) = 𝑧 ) → ( Σ^ ‘ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ↦ ( 𝐹 ‘ 𝑘 ) ) ) = ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) )
53 4 eqcomi ⊢ seq 𝑀 ( + , 𝐹 ) = 𝐺
54 53 fveq1i ⊢ ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) = ( 𝐺 ‘ 𝑗 )
55 54 a1i ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝐺 ‘ 𝑗 ) = 𝑧 ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) = ( 𝐺 ‘ 𝑗 ) )
56 simp3 ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝐺 ‘ 𝑗 ) = 𝑧 ) → ( 𝐺 ‘ 𝑗 ) = 𝑧 )
57 52 55 56 3eqtrrd ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝐺 ‘ 𝑗 ) = 𝑧 ) → 𝑧 = ( Σ^ ‘ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ↦ ( 𝐹 ‘ 𝑘 ) ) ) )
58 3 feqmptd ⊢ ( 𝜑 → 𝐹 = ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) )
59 58 fveq2d ⊢ ( 𝜑 → ( Σ^ ‘ 𝐹 ) = ( Σ^ ‘ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) )
60 59 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝐺 ‘ 𝑗 ) = 𝑧 ) → ( Σ^ ‘ 𝐹 ) = ( Σ^ ‘ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) )
61 57 60 breq12d ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝐺 ‘ 𝑗 ) = 𝑧 ) → ( 𝑧 ≤ ( Σ^ ‘ 𝐹 ) ↔ ( Σ^ ‘ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ↦ ( 𝐹 ‘ 𝑘 ) ) ) ≤ ( Σ^ ‘ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) ) )
62 36 61 mpbird ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝐺 ‘ 𝑗 ) = 𝑧 ) → 𝑧 ≤ ( Σ^ ‘ 𝐹 ) )
63 62 3exp ⊢ ( 𝜑 → ( 𝑗 ∈ 𝑍 → ( ( 𝐺 ‘ 𝑗 ) = 𝑧 → 𝑧 ≤ ( Σ^ ‘ 𝐹 ) ) ) )
64 63 adantr ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ran 𝐺 ) → ( 𝑗 ∈ 𝑍 → ( ( 𝐺 ‘ 𝑗 ) = 𝑧 → 𝑧 ≤ ( Σ^ ‘ 𝐹 ) ) ) )
65 64 rexlimdv ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ran 𝐺 ) → ( ∃ 𝑗 ∈ 𝑍 ( 𝐺 ‘ 𝑗 ) = 𝑧 → 𝑧 ≤ ( Σ^ ‘ 𝐹 ) ) )
66 29 65 mpd ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ran 𝐺 ) → 𝑧 ≤ ( Σ^ ‘ 𝐹 ) )
67 66 ralrimiva ⊢ ( 𝜑 → ∀ 𝑧 ∈ ran 𝐺 𝑧 ≤ ( Σ^ ‘ 𝐹 ) )
68 nfv ⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) )
69 18 a1i ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → 𝑍 ∈ V )
70 6 ad4ant14 ⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ( 0 [,) +∞ ) )
71 simplr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → 𝑧 ∈ ℝ )
72 simpr ⊢ ( ( 𝜑 ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → 𝑧 < ( Σ^ ‘ 𝐹 ) )
73 59 adantr ⊢ ( ( 𝜑 ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → ( Σ^ ‘ 𝐹 ) = ( Σ^ ‘ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) )
74 72 73 breqtrd ⊢ ( ( 𝜑 ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → 𝑧 < ( Σ^ ‘ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) )
75 74 adantlr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → 𝑧 < ( Σ^ ‘ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) )
76 68 69 70 71 75 sge0gtfsumgt ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → ∃ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) )
77 1 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) → 𝑀 ∈ ℤ )
78 elpwinss ⊢ ( 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) → 𝑤 ⊆ 𝑍 )
79 78 3ad2ant2 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) → 𝑤 ⊆ 𝑍 )
80 elinel2 ⊢ ( 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) → 𝑤 ∈ Fin )
81 80 3ad2ant2 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) → 𝑤 ∈ Fin )
82 77 2 79 81 uzfissfz ⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) → ∃ 𝑗 ∈ 𝑍 𝑤 ⊆ ( 𝑀 ... 𝑗 ) )
83 82 3adant1r ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) → ∃ 𝑗 ∈ 𝑍 𝑤 ⊆ ( 𝑀 ... 𝑗 ) )
84 simpl1r ⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → 𝑧 ∈ ℝ )
85 80 adantl ⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ) → 𝑤 ∈ Fin )
86 58 7 fmpt3d ⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℝ )
87 86 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑤 ) → 𝐹 : 𝑍 ⟶ ℝ )
88 78 sselda ⊢ ( ( 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑘 ∈ 𝑤 ) → 𝑘 ∈ 𝑍 )
89 88 adantll ⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑤 ) → 𝑘 ∈ 𝑍 )
90 87 89 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑤 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ )
91 85 90 fsumrecl ⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ) → Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ∈ ℝ )
92 91 ad4ant13 ⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ) ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ∈ ℝ )
93 92 3adantl3 ⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ∈ ℝ )
94 32 7 sylan2 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ )
95 37 94 fsumrecl ⊢ ( 𝜑 → Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ ℝ )
96 95 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ) ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ ℝ )
97 96 3adantl3 ⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ ℝ )
98 simpl3 ⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) )
99 37 adantr ⊢ ( ( 𝜑 ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → ( 𝑀 ... 𝑗 ) ∈ Fin )
100 94 adantlr ⊢ ( ( ( 𝜑 ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ )
101 0xr ⊢ 0 ∈ ℝ*
102 101 a1i ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 0 ∈ ℝ* )
103 pnfxr ⊢ +∞ ∈ ℝ*
104 103 a1i ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → +∞ ∈ ℝ* )
105 icogelb ⊢ ( ( 0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 0 [,) +∞ ) ) → 0 ≤ ( 𝐹 ‘ 𝑘 ) )
106 102 104 6 105 syl3anc ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 0 ≤ ( 𝐹 ‘ 𝑘 ) )
107 32 106 sylan2 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → 0 ≤ ( 𝐹 ‘ 𝑘 ) )
108 107 adantlr ⊢ ( ( ( 𝜑 ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → 0 ≤ ( 𝐹 ‘ 𝑘 ) )
109 simpr ⊢ ( ( 𝜑 ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → 𝑤 ⊆ ( 𝑀 ... 𝑗 ) )
110 99 100 108 109 fsumless ⊢ ( ( 𝜑 ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ≤ Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) )
111 110 adantlr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ≤ Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) )
112 111 3ad2antl1 ⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ≤ Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) )
113 84 93 97 98 112 ltletrd ⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) )
114 113 ex ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) → ( 𝑤 ⊆ ( 𝑀 ... 𝑗 ) → 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) )
115 114 reximdv ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) → ( ∃ 𝑗 ∈ 𝑍 𝑤 ⊆ ( 𝑀 ... 𝑗 ) → ∃ 𝑗 ∈ 𝑍 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) )
116 83 115 mpd ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) → ∃ 𝑗 ∈ 𝑍 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) )
117 116 3exp ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) → ( 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) → ( 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) → ∃ 𝑗 ∈ 𝑍 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) ) )
118 117 adantr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → ( 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) → ( 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) → ∃ 𝑗 ∈ 𝑍 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) ) )
119 118 rexlimdv ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → ( ∃ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) → ∃ 𝑗 ∈ 𝑍 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) )
120 76 119 mpd ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → ∃ 𝑗 ∈ 𝑍 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) )
121 10 ffnd ⊢ ( 𝜑 → seq 𝑀 ( + , 𝐹 ) Fn 𝑍 )
122 121 adantr ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → seq 𝑀 ( + , 𝐹 ) Fn 𝑍 )
123 47 45 sylibr ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝑗 ∈ 𝑍 )
124 fnfvelrn ⊢ ( ( seq 𝑀 ( + , 𝐹 ) Fn 𝑍 ∧ 𝑗 ∈ 𝑍 ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) ∈ ran seq 𝑀 ( + , 𝐹 ) )
125 122 123 124 syl2anc ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) ∈ ran seq 𝑀 ( + , 𝐹 ) )
126 4 a1i ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝐺 = seq 𝑀 ( + , 𝐹 ) )
127 126 rneqd ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ran 𝐺 = ran seq 𝑀 ( + , 𝐹 ) )
128 50 127 eleq12d ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ ran 𝐺 ↔ ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) ∈ ran seq 𝑀 ( + , 𝐹 ) ) )
129 125 128 mpbird ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ ran 𝐺 )
130 129 adantlr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑗 ∈ 𝑍 ) → Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ ran 𝐺 )
131 130 3adant3 ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑗 ∈ 𝑍 ∧ 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) → Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ ran 𝐺 )
132 simp3 ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑗 ∈ 𝑍 ∧ 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) → 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) )
133 breq2 ⊢ ( 𝑦 = Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) → ( 𝑧 < 𝑦 ↔ 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) )
134 133 rspcev ⊢ ( ( Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ ran 𝐺 ∧ 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 < 𝑦 )
135 131 132 134 syl2anc ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑗 ∈ 𝑍 ∧ 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 < 𝑦 )
136 135 3exp ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) → ( 𝑗 ∈ 𝑍 → ( 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 < 𝑦 ) ) )
137 136 rexlimdv ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) → ( ∃ 𝑗 ∈ 𝑍 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 < 𝑦 ) )
138 137 adantr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → ( ∃ 𝑗 ∈ 𝑍 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 < 𝑦 ) )
139 120 138 mpd ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 < 𝑦 )
140 139 ex ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) → ( 𝑧 < ( Σ^ ‘ 𝐹 ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 < 𝑦 ) )
141 140 ralrimiva ⊢ ( 𝜑 → ∀ 𝑧 ∈ ℝ ( 𝑧 < ( Σ^ ‘ 𝐹 ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 < 𝑦 ) )
142 supxr2 ⊢ ( ( ( ran 𝐺 ⊆ ℝ* ∧ ( Σ^ ‘ 𝐹 ) ∈ ℝ* ) ∧ ( ∀ 𝑧 ∈ ran 𝐺 𝑧 ≤ ( Σ^ ‘ 𝐹 ) ∧ ∀ 𝑧 ∈ ℝ ( 𝑧 < ( Σ^ ‘ 𝐹 ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 < 𝑦 ) ) ) → sup ( ran 𝐺 , ℝ* , < ) = ( Σ^ ‘ 𝐹 ) )
143 17 23 67 141 142 syl22anc ⊢ ( 𝜑 → sup ( ran 𝐺 , ℝ* , < ) = ( Σ^ ‘ 𝐹 ) )
144 143 eqcomd ⊢ ( 𝜑 → ( Σ^ ‘ 𝐹 ) = sup ( ran 𝐺 , ℝ* , < ) )