Step |
Hyp |
Ref |
Expression |
1 |
|
hoidmvlelem2.l |
⊢ 𝐿 = ( 𝑥 ∈ Fin ↦ ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) , 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) ) ) |
2 |
|
hoidmvlelem2.x |
⊢ ( 𝜑 → 𝑋 ∈ Fin ) |
3 |
|
hoidmvlelem2.y |
⊢ ( 𝜑 → 𝑌 ⊆ 𝑋 ) |
4 |
|
hoidmvlelem2.z |
⊢ ( 𝜑 → 𝑍 ∈ ( 𝑋 ∖ 𝑌 ) ) |
5 |
|
hoidmvlelem2.w |
⊢ 𝑊 = ( 𝑌 ∪ { 𝑍 } ) |
6 |
|
hoidmvlelem2.a |
⊢ ( 𝜑 → 𝐴 : 𝑊 ⟶ ℝ ) |
7 |
|
hoidmvlelem2.b |
⊢ ( 𝜑 → 𝐵 : 𝑊 ⟶ ℝ ) |
8 |
|
hoidmvlelem2.c |
⊢ ( 𝜑 → 𝐶 : ℕ ⟶ ( ℝ ↑m 𝑊 ) ) |
9 |
|
hoidmvlelem2.f |
⊢ 𝐹 = ( 𝑦 ∈ 𝑌 ↦ 0 ) |
10 |
|
hoidmvlelem2.j |
⊢ 𝐽 = ( 𝑗 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ) |
11 |
|
hoidmvlelem2.d |
⊢ ( 𝜑 → 𝐷 : ℕ ⟶ ( ℝ ↑m 𝑊 ) ) |
12 |
|
hoidmvlelem2.k |
⊢ 𝐾 = ( 𝑗 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ) |
13 |
|
hoidmvlelem2.r |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ∈ ℝ ) |
14 |
|
hoidmvlelem2.h |
⊢ 𝐻 = ( 𝑥 ∈ ℝ ↦ ( 𝑐 ∈ ( ℝ ↑m 𝑊 ) ↦ ( 𝑗 ∈ 𝑊 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑐 ‘ 𝑗 ) , if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) ) ) ) ) |
15 |
|
hoidmvlelem2.g |
⊢ 𝐺 = ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) |
16 |
|
hoidmvlelem2.e |
⊢ ( 𝜑 → 𝐸 ∈ ℝ+ ) |
17 |
|
hoidmvlelem2.u |
⊢ 𝑈 = { 𝑧 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) } |
18 |
|
hoidmvlelem2.su |
⊢ ( 𝜑 → 𝑆 ∈ 𝑈 ) |
19 |
|
hoidmvlelem2.sb |
⊢ ( 𝜑 → 𝑆 < ( 𝐵 ‘ 𝑍 ) ) |
20 |
|
hoidmvlelem2.p |
⊢ 𝑃 = ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) |
21 |
|
hoidmvlelem2.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
22 |
|
hoidmvlelem2.le |
⊢ ( 𝜑 → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) ) ) |
23 |
|
hoidmvlelem2.O |
⊢ 𝑂 = ran ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) |
24 |
|
hoidmvlelem2.v |
⊢ 𝑉 = ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) |
25 |
|
hoidmvlelem2.q |
⊢ 𝑄 = inf ( 𝑉 , ℝ , < ) |
26 |
|
snidg |
⊢ ( 𝑍 ∈ ( 𝑋 ∖ 𝑌 ) → 𝑍 ∈ { 𝑍 } ) |
27 |
4 26
|
syl |
⊢ ( 𝜑 → 𝑍 ∈ { 𝑍 } ) |
28 |
|
elun2 |
⊢ ( 𝑍 ∈ { 𝑍 } → 𝑍 ∈ ( 𝑌 ∪ { 𝑍 } ) ) |
29 |
27 28
|
syl |
⊢ ( 𝜑 → 𝑍 ∈ ( 𝑌 ∪ { 𝑍 } ) ) |
30 |
29 5
|
eleqtrrdi |
⊢ ( 𝜑 → 𝑍 ∈ 𝑊 ) |
31 |
6 30
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) ∈ ℝ ) |
32 |
7 30
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐵 ‘ 𝑍 ) ∈ ℝ ) |
33 |
32
|
snssd |
⊢ ( 𝜑 → { ( 𝐵 ‘ 𝑍 ) } ⊆ ℝ ) |
34 |
|
nfv |
⊢ Ⅎ 𝑖 𝜑 |
35 |
|
eqid |
⊢ ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) = ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) |
36 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ) → 𝜑 ) |
37 |
|
fz1ssnn |
⊢ ( 1 ... 𝑀 ) ⊆ ℕ |
38 |
|
elrabi |
⊢ ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } → 𝑖 ∈ ( 1 ... 𝑀 ) ) |
39 |
37 38
|
sselid |
⊢ ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } → 𝑖 ∈ ℕ ) |
40 |
39
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ) → 𝑖 ∈ ℕ ) |
41 |
|
eleq1w |
⊢ ( 𝑗 = 𝑖 → ( 𝑗 ∈ ℕ ↔ 𝑖 ∈ ℕ ) ) |
42 |
41
|
anbi2d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ↔ ( 𝜑 ∧ 𝑖 ∈ ℕ ) ) ) |
43 |
|
fveq2 |
⊢ ( 𝑗 = 𝑖 → ( 𝐷 ‘ 𝑗 ) = ( 𝐷 ‘ 𝑖 ) ) |
44 |
43
|
fveq1d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) |
45 |
44
|
eleq1d |
⊢ ( 𝑗 = 𝑖 → ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ↔ ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ ) ) |
46 |
42 45
|
imbi12d |
⊢ ( 𝑗 = 𝑖 → ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ) ↔ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ ) ) ) |
47 |
11
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐷 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑊 ) ) |
48 |
|
elmapi |
⊢ ( ( 𝐷 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑊 ) → ( 𝐷 ‘ 𝑗 ) : 𝑊 ⟶ ℝ ) |
49 |
47 48
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐷 ‘ 𝑗 ) : 𝑊 ⟶ ℝ ) |
50 |
30
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑍 ∈ 𝑊 ) |
51 |
49 50
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ) |
52 |
46 51
|
chvarvv |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ ) |
53 |
36 40 52
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ) → ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ ) |
54 |
34 35 53
|
rnmptssd |
⊢ ( 𝜑 → ran ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ⊆ ℝ ) |
55 |
23 54
|
eqsstrid |
⊢ ( 𝜑 → 𝑂 ⊆ ℝ ) |
56 |
33 55
|
unssd |
⊢ ( 𝜑 → ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) ⊆ ℝ ) |
57 |
24 56
|
eqsstrid |
⊢ ( 𝜑 → 𝑉 ⊆ ℝ ) |
58 |
|
ltso |
⊢ < Or ℝ |
59 |
58
|
a1i |
⊢ ( 𝜑 → < Or ℝ ) |
60 |
|
snfi |
⊢ { ( 𝐵 ‘ 𝑍 ) } ∈ Fin |
61 |
60
|
a1i |
⊢ ( 𝜑 → { ( 𝐵 ‘ 𝑍 ) } ∈ Fin ) |
62 |
|
fzfi |
⊢ ( 1 ... 𝑀 ) ∈ Fin |
63 |
|
rabfi |
⊢ ( ( 1 ... 𝑀 ) ∈ Fin → { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ∈ Fin ) |
64 |
62 63
|
ax-mp |
⊢ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ∈ Fin |
65 |
64
|
a1i |
⊢ ( 𝜑 → { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ∈ Fin ) |
66 |
35
|
rnmptfi |
⊢ ( { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ∈ Fin → ran ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ∈ Fin ) |
67 |
65 66
|
syl |
⊢ ( 𝜑 → ran ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ∈ Fin ) |
68 |
23 67
|
eqeltrid |
⊢ ( 𝜑 → 𝑂 ∈ Fin ) |
69 |
|
unfi |
⊢ ( ( { ( 𝐵 ‘ 𝑍 ) } ∈ Fin ∧ 𝑂 ∈ Fin ) → ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) ∈ Fin ) |
70 |
61 68 69
|
syl2anc |
⊢ ( 𝜑 → ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) ∈ Fin ) |
71 |
24 70
|
eqeltrid |
⊢ ( 𝜑 → 𝑉 ∈ Fin ) |
72 |
|
fvex |
⊢ ( 𝐵 ‘ 𝑍 ) ∈ V |
73 |
72
|
snid |
⊢ ( 𝐵 ‘ 𝑍 ) ∈ { ( 𝐵 ‘ 𝑍 ) } |
74 |
|
elun1 |
⊢ ( ( 𝐵 ‘ 𝑍 ) ∈ { ( 𝐵 ‘ 𝑍 ) } → ( 𝐵 ‘ 𝑍 ) ∈ ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) ) |
75 |
73 74
|
ax-mp |
⊢ ( 𝐵 ‘ 𝑍 ) ∈ ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) |
76 |
24
|
eqcomi |
⊢ ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) = 𝑉 |
77 |
75 76
|
eleqtri |
⊢ ( 𝐵 ‘ 𝑍 ) ∈ 𝑉 |
78 |
77
|
a1i |
⊢ ( 𝜑 → ( 𝐵 ‘ 𝑍 ) ∈ 𝑉 ) |
79 |
|
ne0i |
⊢ ( ( 𝐵 ‘ 𝑍 ) ∈ 𝑉 → 𝑉 ≠ ∅ ) |
80 |
78 79
|
syl |
⊢ ( 𝜑 → 𝑉 ≠ ∅ ) |
81 |
|
fiinfcl |
⊢ ( ( < Or ℝ ∧ ( 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅ ∧ 𝑉 ⊆ ℝ ) ) → inf ( 𝑉 , ℝ , < ) ∈ 𝑉 ) |
82 |
59 71 80 57 81
|
syl13anc |
⊢ ( 𝜑 → inf ( 𝑉 , ℝ , < ) ∈ 𝑉 ) |
83 |
25 82
|
eqeltrid |
⊢ ( 𝜑 → 𝑄 ∈ 𝑉 ) |
84 |
57 83
|
sseldd |
⊢ ( 𝜑 → 𝑄 ∈ ℝ ) |
85 |
|
ssrab2 |
⊢ { 𝑧 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) } ⊆ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) |
86 |
17 85
|
eqsstri |
⊢ 𝑈 ⊆ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) |
87 |
86
|
a1i |
⊢ ( 𝜑 → 𝑈 ⊆ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ) |
88 |
31 32
|
iccssred |
⊢ ( 𝜑 → ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ⊆ ℝ ) |
89 |
87 88
|
sstrd |
⊢ ( 𝜑 → 𝑈 ⊆ ℝ ) |
90 |
89 18
|
sseldd |
⊢ ( 𝜑 → 𝑆 ∈ ℝ ) |
91 |
31
|
rexrd |
⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) ∈ ℝ* ) |
92 |
32
|
rexrd |
⊢ ( 𝜑 → ( 𝐵 ‘ 𝑍 ) ∈ ℝ* ) |
93 |
86 18
|
sselid |
⊢ ( 𝜑 → 𝑆 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ) |
94 |
|
iccgelb |
⊢ ( ( ( 𝐴 ‘ 𝑍 ) ∈ ℝ* ∧ ( 𝐵 ‘ 𝑍 ) ∈ ℝ* ∧ 𝑆 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ) → ( 𝐴 ‘ 𝑍 ) ≤ 𝑆 ) |
95 |
91 92 93 94
|
syl3anc |
⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) ≤ 𝑆 ) |
96 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 = ( 𝐵 ‘ 𝑍 ) ) → 𝑆 < ( 𝐵 ‘ 𝑍 ) ) |
97 |
|
id |
⊢ ( 𝑥 = ( 𝐵 ‘ 𝑍 ) → 𝑥 = ( 𝐵 ‘ 𝑍 ) ) |
98 |
97
|
eqcomd |
⊢ ( 𝑥 = ( 𝐵 ‘ 𝑍 ) → ( 𝐵 ‘ 𝑍 ) = 𝑥 ) |
99 |
98
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 = ( 𝐵 ‘ 𝑍 ) ) → ( 𝐵 ‘ 𝑍 ) = 𝑥 ) |
100 |
96 99
|
breqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 = ( 𝐵 ‘ 𝑍 ) ) → 𝑆 < 𝑥 ) |
101 |
100
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) ∧ 𝑥 = ( 𝐵 ‘ 𝑍 ) ) → 𝑆 < 𝑥 ) |
102 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) ∧ ¬ 𝑥 = ( 𝐵 ‘ 𝑍 ) ) → 𝜑 ) |
103 |
|
id |
⊢ ( 𝑥 ∈ 𝑉 → 𝑥 ∈ 𝑉 ) |
104 |
103 24
|
eleqtrdi |
⊢ ( 𝑥 ∈ 𝑉 → 𝑥 ∈ ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) ) |
105 |
104
|
adantr |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ¬ 𝑥 = ( 𝐵 ‘ 𝑍 ) ) → 𝑥 ∈ ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) ) |
106 |
|
elsni |
⊢ ( 𝑥 ∈ { ( 𝐵 ‘ 𝑍 ) } → 𝑥 = ( 𝐵 ‘ 𝑍 ) ) |
107 |
106
|
con3i |
⊢ ( ¬ 𝑥 = ( 𝐵 ‘ 𝑍 ) → ¬ 𝑥 ∈ { ( 𝐵 ‘ 𝑍 ) } ) |
108 |
107
|
adantl |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ¬ 𝑥 = ( 𝐵 ‘ 𝑍 ) ) → ¬ 𝑥 ∈ { ( 𝐵 ‘ 𝑍 ) } ) |
109 |
|
elunnel1 |
⊢ ( ( 𝑥 ∈ ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) ∧ ¬ 𝑥 ∈ { ( 𝐵 ‘ 𝑍 ) } ) → 𝑥 ∈ 𝑂 ) |
110 |
105 108 109
|
syl2anc |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ¬ 𝑥 = ( 𝐵 ‘ 𝑍 ) ) → 𝑥 ∈ 𝑂 ) |
111 |
110
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) ∧ ¬ 𝑥 = ( 𝐵 ‘ 𝑍 ) ) → 𝑥 ∈ 𝑂 ) |
112 |
|
id |
⊢ ( 𝑥 ∈ 𝑂 → 𝑥 ∈ 𝑂 ) |
113 |
112 23
|
eleqtrdi |
⊢ ( 𝑥 ∈ 𝑂 → 𝑥 ∈ ran ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) |
114 |
|
vex |
⊢ 𝑥 ∈ V |
115 |
35
|
elrnmpt |
⊢ ( 𝑥 ∈ V → ( 𝑥 ∈ ran ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ↔ ∃ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) |
116 |
114 115
|
ax-mp |
⊢ ( 𝑥 ∈ ran ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ↔ ∃ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) |
117 |
113 116
|
sylib |
⊢ ( 𝑥 ∈ 𝑂 → ∃ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) |
118 |
117
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑂 ) → ∃ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) |
119 |
|
fveq2 |
⊢ ( 𝑗 = 𝑖 → ( 𝐶 ‘ 𝑗 ) = ( 𝐶 ‘ 𝑖 ) ) |
120 |
119
|
fveq1d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ) |
121 |
120
|
eleq1d |
⊢ ( 𝑗 = 𝑖 → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ↔ ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ ) ) |
122 |
42 121
|
imbi12d |
⊢ ( 𝑗 = 𝑖 → ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ) ↔ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ ) ) ) |
123 |
8
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐶 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑊 ) ) |
124 |
|
elmapi |
⊢ ( ( 𝐶 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑊 ) → ( 𝐶 ‘ 𝑗 ) : 𝑊 ⟶ ℝ ) |
125 |
123 124
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐶 ‘ 𝑗 ) : 𝑊 ⟶ ℝ ) |
126 |
125 50
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ) |
127 |
122 126
|
chvarvv |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ ) |
128 |
127
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ* ) |
129 |
36 40 128
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ) → ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ* ) |
130 |
52
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ* ) |
131 |
36 40 130
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ) → ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ* ) |
132 |
120 44
|
oveq12d |
⊢ ( 𝑗 = 𝑖 → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) |
133 |
132
|
eleq2d |
⊢ ( 𝑗 = 𝑖 → ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ↔ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) ) |
134 |
133
|
elrab |
⊢ ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ↔ ( 𝑖 ∈ ( 1 ... 𝑀 ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) ) |
135 |
134
|
biimpi |
⊢ ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } → ( 𝑖 ∈ ( 1 ... 𝑀 ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) ) |
136 |
135
|
simprd |
⊢ ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } → 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) |
137 |
136
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ) → 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) |
138 |
|
icoltub |
⊢ ( ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ* ∧ ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ* ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) → 𝑆 < ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) |
139 |
129 131 137 138
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ) → 𝑆 < ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) |
140 |
139
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ∧ 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) → 𝑆 < ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) |
141 |
|
id |
⊢ ( 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) → 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) |
142 |
141
|
eqcomd |
⊢ ( 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) → ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) = 𝑥 ) |
143 |
142
|
3ad2ant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ∧ 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) → ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) = 𝑥 ) |
144 |
140 143
|
breqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ∧ 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) → 𝑆 < 𝑥 ) |
145 |
144
|
3exp |
⊢ ( 𝜑 → ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } → ( 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) → 𝑆 < 𝑥 ) ) ) |
146 |
145
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑂 ) → ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } → ( 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) → 𝑆 < 𝑥 ) ) ) |
147 |
146
|
rexlimdv |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑂 ) → ( ∃ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) → 𝑆 < 𝑥 ) ) |
148 |
118 147
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑂 ) → 𝑆 < 𝑥 ) |
149 |
102 111 148
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) ∧ ¬ 𝑥 = ( 𝐵 ‘ 𝑍 ) ) → 𝑆 < 𝑥 ) |
150 |
101 149
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → 𝑆 < 𝑥 ) |
151 |
150
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑉 𝑆 < 𝑥 ) |
152 |
|
breq2 |
⊢ ( 𝑥 = inf ( 𝑉 , ℝ , < ) → ( 𝑆 < 𝑥 ↔ 𝑆 < inf ( 𝑉 , ℝ , < ) ) ) |
153 |
152
|
rspcva |
⊢ ( ( inf ( 𝑉 , ℝ , < ) ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝑉 𝑆 < 𝑥 ) → 𝑆 < inf ( 𝑉 , ℝ , < ) ) |
154 |
82 151 153
|
syl2anc |
⊢ ( 𝜑 → 𝑆 < inf ( 𝑉 , ℝ , < ) ) |
155 |
25
|
eqcomi |
⊢ inf ( 𝑉 , ℝ , < ) = 𝑄 |
156 |
155
|
a1i |
⊢ ( 𝜑 → inf ( 𝑉 , ℝ , < ) = 𝑄 ) |
157 |
154 156
|
breqtrd |
⊢ ( 𝜑 → 𝑆 < 𝑄 ) |
158 |
31 90 84 95 157
|
lelttrd |
⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) < 𝑄 ) |
159 |
31 84 158
|
ltled |
⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) ≤ 𝑄 ) |
160 |
|
fiminre |
⊢ ( ( 𝑉 ⊆ ℝ ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅ ) → ∃ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 𝑥 ≤ 𝑦 ) |
161 |
57 71 80 160
|
syl3anc |
⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 𝑥 ≤ 𝑦 ) |
162 |
|
lbinfle |
⊢ ( ( 𝑉 ⊆ ℝ ∧ ∃ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 𝑥 ≤ 𝑦 ∧ ( 𝐵 ‘ 𝑍 ) ∈ 𝑉 ) → inf ( 𝑉 , ℝ , < ) ≤ ( 𝐵 ‘ 𝑍 ) ) |
163 |
57 161 78 162
|
syl3anc |
⊢ ( 𝜑 → inf ( 𝑉 , ℝ , < ) ≤ ( 𝐵 ‘ 𝑍 ) ) |
164 |
25 163
|
eqbrtrid |
⊢ ( 𝜑 → 𝑄 ≤ ( 𝐵 ‘ 𝑍 ) ) |
165 |
31 32 84 159 164
|
eliccd |
⊢ ( 𝜑 → 𝑄 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ) |
166 |
84
|
recnd |
⊢ ( 𝜑 → 𝑄 ∈ ℂ ) |
167 |
90
|
recnd |
⊢ ( 𝜑 → 𝑆 ∈ ℂ ) |
168 |
31
|
recnd |
⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) ∈ ℂ ) |
169 |
166 167 168
|
npncand |
⊢ ( 𝜑 → ( ( 𝑄 − 𝑆 ) + ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) = ( 𝑄 − ( 𝐴 ‘ 𝑍 ) ) ) |
170 |
169
|
eqcomd |
⊢ ( 𝜑 → ( 𝑄 − ( 𝐴 ‘ 𝑍 ) ) = ( ( 𝑄 − 𝑆 ) + ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ) |
171 |
170
|
oveq2d |
⊢ ( 𝜑 → ( 𝐺 · ( 𝑄 − ( 𝐴 ‘ 𝑍 ) ) ) = ( 𝐺 · ( ( 𝑄 − 𝑆 ) + ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ) ) |
172 |
|
rge0ssre |
⊢ ( 0 [,) +∞ ) ⊆ ℝ |
173 |
2 3
|
ssfid |
⊢ ( 𝜑 → 𝑌 ∈ Fin ) |
174 |
|
ssun1 |
⊢ 𝑌 ⊆ ( 𝑌 ∪ { 𝑍 } ) |
175 |
174 5
|
sseqtrri |
⊢ 𝑌 ⊆ 𝑊 |
176 |
175
|
a1i |
⊢ ( 𝜑 → 𝑌 ⊆ 𝑊 ) |
177 |
6 176
|
fssresd |
⊢ ( 𝜑 → ( 𝐴 ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) |
178 |
7 176
|
fssresd |
⊢ ( 𝜑 → ( 𝐵 ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) |
179 |
1 173 177 178
|
hoidmvcl |
⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ∈ ( 0 [,) +∞ ) ) |
180 |
15 179
|
eqeltrid |
⊢ ( 𝜑 → 𝐺 ∈ ( 0 [,) +∞ ) ) |
181 |
172 180
|
sselid |
⊢ ( 𝜑 → 𝐺 ∈ ℝ ) |
182 |
181
|
recnd |
⊢ ( 𝜑 → 𝐺 ∈ ℂ ) |
183 |
166 167
|
subcld |
⊢ ( 𝜑 → ( 𝑄 − 𝑆 ) ∈ ℂ ) |
184 |
167 168
|
subcld |
⊢ ( 𝜑 → ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ∈ ℂ ) |
185 |
182 183 184
|
adddid |
⊢ ( 𝜑 → ( 𝐺 · ( ( 𝑄 − 𝑆 ) + ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ) = ( ( 𝐺 · ( 𝑄 − 𝑆 ) ) + ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ) ) |
186 |
182 183
|
mulcld |
⊢ ( 𝜑 → ( 𝐺 · ( 𝑄 − 𝑆 ) ) ∈ ℂ ) |
187 |
182 184
|
mulcld |
⊢ ( 𝜑 → ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ∈ ℂ ) |
188 |
186 187
|
addcomd |
⊢ ( 𝜑 → ( ( 𝐺 · ( 𝑄 − 𝑆 ) ) + ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ) = ( ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) + ( 𝐺 · ( 𝑄 − 𝑆 ) ) ) ) |
189 |
171 185 188
|
3eqtrd |
⊢ ( 𝜑 → ( 𝐺 · ( 𝑄 − ( 𝐴 ‘ 𝑍 ) ) ) = ( ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) + ( 𝐺 · ( 𝑄 − 𝑆 ) ) ) ) |
190 |
84 90
|
jca |
⊢ ( 𝜑 → ( 𝑄 ∈ ℝ ∧ 𝑆 ∈ ℝ ) ) |
191 |
|
resubcl |
⊢ ( ( 𝑄 ∈ ℝ ∧ 𝑆 ∈ ℝ ) → ( 𝑄 − 𝑆 ) ∈ ℝ ) |
192 |
190 191
|
syl |
⊢ ( 𝜑 → ( 𝑄 − 𝑆 ) ∈ ℝ ) |
193 |
181 192
|
jca |
⊢ ( 𝜑 → ( 𝐺 ∈ ℝ ∧ ( 𝑄 − 𝑆 ) ∈ ℝ ) ) |
194 |
|
remulcl |
⊢ ( ( 𝐺 ∈ ℝ ∧ ( 𝑄 − 𝑆 ) ∈ ℝ ) → ( 𝐺 · ( 𝑄 − 𝑆 ) ) ∈ ℝ ) |
195 |
193 194
|
syl |
⊢ ( 𝜑 → ( 𝐺 · ( 𝑄 − 𝑆 ) ) ∈ ℝ ) |
196 |
90 31
|
jca |
⊢ ( 𝜑 → ( 𝑆 ∈ ℝ ∧ ( 𝐴 ‘ 𝑍 ) ∈ ℝ ) ) |
197 |
|
resubcl |
⊢ ( ( 𝑆 ∈ ℝ ∧ ( 𝐴 ‘ 𝑍 ) ∈ ℝ ) → ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ∈ ℝ ) |
198 |
196 197
|
syl |
⊢ ( 𝜑 → ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ∈ ℝ ) |
199 |
181 198
|
jca |
⊢ ( 𝜑 → ( 𝐺 ∈ ℝ ∧ ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ∈ ℝ ) ) |
200 |
|
remulcl |
⊢ ( ( 𝐺 ∈ ℝ ∧ ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ∈ ℝ ) → ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ∈ ℝ ) |
201 |
199 200
|
syl |
⊢ ( 𝜑 → ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ∈ ℝ ) |
202 |
195 201
|
jca |
⊢ ( 𝜑 → ( ( 𝐺 · ( 𝑄 − 𝑆 ) ) ∈ ℝ ∧ ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ∈ ℝ ) ) |
203 |
|
readdcl |
⊢ ( ( ( 𝐺 · ( 𝑄 − 𝑆 ) ) ∈ ℝ ∧ ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ∈ ℝ ) → ( ( 𝐺 · ( 𝑄 − 𝑆 ) ) + ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ) ∈ ℝ ) |
204 |
202 203
|
syl |
⊢ ( 𝜑 → ( ( 𝐺 · ( 𝑄 − 𝑆 ) ) + ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ) ∈ ℝ ) |
205 |
188 204
|
eqeltrrd |
⊢ ( 𝜑 → ( ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) + ( 𝐺 · ( 𝑄 − 𝑆 ) ) ) ∈ ℝ ) |
206 |
|
1red |
⊢ ( 𝜑 → 1 ∈ ℝ ) |
207 |
16
|
rpred |
⊢ ( 𝜑 → 𝐸 ∈ ℝ ) |
208 |
206 207
|
readdcld |
⊢ ( 𝜑 → ( 1 + 𝐸 ) ∈ ℝ ) |
209 |
4
|
eldifbd |
⊢ ( 𝜑 → ¬ 𝑍 ∈ 𝑌 ) |
210 |
30 209
|
eldifd |
⊢ ( 𝜑 → 𝑍 ∈ ( 𝑊 ∖ 𝑌 ) ) |
211 |
1 173 210 5 8 11 13 14 90
|
sge0hsphoire |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ∈ ℝ ) |
212 |
208 211
|
remulcld |
⊢ ( 𝜑 → ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ∈ ℝ ) |
213 |
|
fzfid |
⊢ ( 𝜑 → ( 1 ... 𝑀 ) ∈ Fin ) |
214 |
192
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 𝑄 − 𝑆 ) ∈ ℝ ) |
215 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 𝜑 ) |
216 |
|
elfznn |
⊢ ( 𝑗 ∈ ( 1 ... 𝑀 ) → 𝑗 ∈ ℕ ) |
217 |
216
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 𝑗 ∈ ℕ ) |
218 |
|
id |
⊢ ( 𝑗 ∈ ℕ → 𝑗 ∈ ℕ ) |
219 |
|
ovexd |
⊢ ( 𝑗 ∈ ℕ → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ∈ V ) |
220 |
20
|
fvmpt2 |
⊢ ( ( 𝑗 ∈ ℕ ∧ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ∈ V ) → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) |
221 |
218 219 220
|
syl2anc |
⊢ ( 𝑗 ∈ ℕ → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) |
222 |
221
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) |
223 |
173
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑌 ∈ Fin ) |
224 |
175
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑌 ⊆ 𝑊 ) |
225 |
125 224
|
fssresd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) |
226 |
225
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) |
227 |
|
iftrue |
⊢ ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ) |
228 |
227
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ) |
229 |
228
|
feq1d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ ↔ ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) ) |
230 |
226 229
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ ) |
231 |
|
0red |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑌 ) → 0 ∈ ℝ ) |
232 |
231 9
|
fmptd |
⊢ ( 𝜑 → 𝐹 : 𝑌 ⟶ ℝ ) |
233 |
232
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → 𝐹 : 𝑌 ⟶ ℝ ) |
234 |
|
iffalse |
⊢ ( ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = 𝐹 ) |
235 |
234
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = 𝐹 ) |
236 |
235
|
feq1d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ ↔ 𝐹 : 𝑌 ⟶ ℝ ) ) |
237 |
233 236
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ ) |
238 |
230 237
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ ) |
239 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑗 ∈ ℕ ) |
240 |
|
fvex |
⊢ ( 𝐶 ‘ 𝑗 ) ∈ V |
241 |
240
|
resex |
⊢ ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ∈ V |
242 |
241
|
a1i |
⊢ ( 𝜑 → ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ∈ V ) |
243 |
2 3
|
ssexd |
⊢ ( 𝜑 → 𝑌 ∈ V ) |
244 |
|
mptexg |
⊢ ( 𝑌 ∈ V → ( 𝑦 ∈ 𝑌 ↦ 0 ) ∈ V ) |
245 |
243 244
|
syl |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝑌 ↦ 0 ) ∈ V ) |
246 |
9 245
|
eqeltrid |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
247 |
242 246
|
ifcld |
⊢ ( 𝜑 → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V ) |
248 |
247
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V ) |
249 |
10
|
fvmpt2 |
⊢ ( ( 𝑗 ∈ ℕ ∧ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V ) → ( 𝐽 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ) |
250 |
239 248 249
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐽 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ) |
251 |
250
|
feq1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ↔ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ ) ) |
252 |
238 251
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ) |
253 |
49 224
|
fssresd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) |
254 |
253
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) |
255 |
|
iftrue |
⊢ ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ) |
256 |
255
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ) |
257 |
256
|
feq1d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ ↔ ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) ) |
258 |
254 257
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ ) |
259 |
|
iffalse |
⊢ ( ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = 𝐹 ) |
260 |
259
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = 𝐹 ) |
261 |
260
|
feq1d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ ↔ 𝐹 : 𝑌 ⟶ ℝ ) ) |
262 |
233 261
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ ) |
263 |
258 262
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ ) |
264 |
|
fvex |
⊢ ( 𝐷 ‘ 𝑗 ) ∈ V |
265 |
264
|
resex |
⊢ ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ∈ V |
266 |
265
|
a1i |
⊢ ( 𝜑 → ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ∈ V ) |
267 |
266 246
|
ifcld |
⊢ ( 𝜑 → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V ) |
268 |
267
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V ) |
269 |
12
|
fvmpt2 |
⊢ ( ( 𝑗 ∈ ℕ ∧ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V ) → ( 𝐾 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ) |
270 |
239 268 269
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐾 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ) |
271 |
270
|
feq1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐾 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ↔ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ ) ) |
272 |
263 271
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐾 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ) |
273 |
1 223 252 272
|
hoidmvcl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ∈ ( 0 [,) +∞ ) ) |
274 |
222 273
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) ∈ ( 0 [,) +∞ ) ) |
275 |
172 274
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) ∈ ℝ ) |
276 |
215 217 275
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 𝑃 ‘ 𝑗 ) ∈ ℝ ) |
277 |
214 276
|
remulcld |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ∈ ℝ ) |
278 |
213 277
|
fsumrecl |
⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ∈ ℝ ) |
279 |
208 278
|
remulcld |
⊢ ( 𝜑 → ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) |
280 |
212 279
|
readdcld |
⊢ ( 𝜑 → ( ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) + ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ∈ ℝ ) |
281 |
1 173 210 5 8 11 13 14 84
|
sge0hsphoire |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ∈ ℝ ) |
282 |
208 281
|
remulcld |
⊢ ( 𝜑 → ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ∈ ℝ ) |
283 |
18 17
|
eleqtrdi |
⊢ ( 𝜑 → 𝑆 ∈ { 𝑧 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) } ) |
284 |
|
oveq1 |
⊢ ( 𝑧 = 𝑆 → ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) = ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) |
285 |
284
|
oveq2d |
⊢ ( 𝑧 = 𝑆 → ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) = ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ) |
286 |
|
fveq2 |
⊢ ( 𝑧 = 𝑆 → ( 𝐻 ‘ 𝑧 ) = ( 𝐻 ‘ 𝑆 ) ) |
287 |
286
|
fveq1d |
⊢ ( 𝑧 = 𝑆 → ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) = ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) |
288 |
287
|
oveq2d |
⊢ ( 𝑧 = 𝑆 → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) = ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) |
289 |
288
|
mpteq2dv |
⊢ ( 𝑧 = 𝑆 → ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) |
290 |
289
|
fveq2d |
⊢ ( 𝑧 = 𝑆 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) |
291 |
290
|
oveq2d |
⊢ ( 𝑧 = 𝑆 → ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) = ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) |
292 |
285 291
|
breq12d |
⊢ ( 𝑧 = 𝑆 → ( ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ↔ ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) ) |
293 |
292
|
elrab |
⊢ ( 𝑆 ∈ { 𝑧 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) } ↔ ( 𝑆 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∧ ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) ) |
294 |
283 293
|
sylib |
⊢ ( 𝜑 → ( 𝑆 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∧ ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) ) |
295 |
294
|
simprd |
⊢ ( 𝜑 → ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) |
296 |
213 276
|
fsumrecl |
⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) ∈ ℝ ) |
297 |
208 296
|
remulcld |
⊢ ( 𝜑 → ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) ) ∈ ℝ ) |
298 |
|
0red |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
299 |
90 84
|
posdifd |
⊢ ( 𝜑 → ( 𝑆 < 𝑄 ↔ 0 < ( 𝑄 − 𝑆 ) ) ) |
300 |
157 299
|
mpbid |
⊢ ( 𝜑 → 0 < ( 𝑄 − 𝑆 ) ) |
301 |
298 192 300
|
ltled |
⊢ ( 𝜑 → 0 ≤ ( 𝑄 − 𝑆 ) ) |
302 |
181 297 192 301 22
|
lemul1ad |
⊢ ( 𝜑 → ( 𝐺 · ( 𝑄 − 𝑆 ) ) ≤ ( ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) ) · ( 𝑄 − 𝑆 ) ) ) |
303 |
208
|
recnd |
⊢ ( 𝜑 → ( 1 + 𝐸 ) ∈ ℂ ) |
304 |
296
|
recnd |
⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) ∈ ℂ ) |
305 |
303 304 183
|
mulassd |
⊢ ( 𝜑 → ( ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) ) · ( 𝑄 − 𝑆 ) ) = ( ( 1 + 𝐸 ) · ( Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) · ( 𝑄 − 𝑆 ) ) ) ) |
306 |
276
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 𝑃 ‘ 𝑗 ) ∈ ℂ ) |
307 |
213 183 306
|
fsummulc1 |
⊢ ( 𝜑 → ( Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) · ( 𝑄 − 𝑆 ) ) = Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑃 ‘ 𝑗 ) · ( 𝑄 − 𝑆 ) ) ) |
308 |
183
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 𝑄 − 𝑆 ) ∈ ℂ ) |
309 |
306 308
|
mulcomd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝑃 ‘ 𝑗 ) · ( 𝑄 − 𝑆 ) ) = ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) |
310 |
309
|
sumeq2dv |
⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑃 ‘ 𝑗 ) · ( 𝑄 − 𝑆 ) ) = Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) |
311 |
307 310
|
eqtrd |
⊢ ( 𝜑 → ( Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) · ( 𝑄 − 𝑆 ) ) = Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) |
312 |
311
|
oveq2d |
⊢ ( 𝜑 → ( ( 1 + 𝐸 ) · ( Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) · ( 𝑄 − 𝑆 ) ) ) = ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) |
313 |
305 312
|
eqtrd |
⊢ ( 𝜑 → ( ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) ) · ( 𝑄 − 𝑆 ) ) = ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) |
314 |
302 313
|
breqtrd |
⊢ ( 𝜑 → ( 𝐺 · ( 𝑄 − 𝑆 ) ) ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) |
315 |
201 195 212 279 295 314
|
leadd12dd |
⊢ ( 𝜑 → ( ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) + ( 𝐺 · ( 𝑄 − 𝑆 ) ) ) ≤ ( ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) + ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ) |
316 |
|
nnsplit |
⊢ ( 𝑀 ∈ ℕ → ℕ = ( ( 1 ... 𝑀 ) ∪ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ) |
317 |
21 316
|
syl |
⊢ ( 𝜑 → ℕ = ( ( 1 ... 𝑀 ) ∪ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ) |
318 |
|
uncom |
⊢ ( ( 1 ... 𝑀 ) ∪ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) = ( ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∪ ( 1 ... 𝑀 ) ) |
319 |
318
|
a1i |
⊢ ( 𝜑 → ( ( 1 ... 𝑀 ) ∪ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) = ( ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∪ ( 1 ... 𝑀 ) ) ) |
320 |
317 319
|
eqtr2d |
⊢ ( 𝜑 → ( ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∪ ( 1 ... 𝑀 ) ) = ℕ ) |
321 |
320
|
eqcomd |
⊢ ( 𝜑 → ℕ = ( ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∪ ( 1 ... 𝑀 ) ) ) |
322 |
321
|
mpteq1d |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) = ( 𝑗 ∈ ( ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∪ ( 1 ... 𝑀 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) |
323 |
322
|
fveq2d |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ( ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∪ ( 1 ... 𝑀 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) |
324 |
|
nfv |
⊢ Ⅎ 𝑗 𝜑 |
325 |
|
fvexd |
⊢ ( 𝜑 → ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∈ V ) |
326 |
|
ovexd |
⊢ ( 𝜑 → ( 1 ... 𝑀 ) ∈ V ) |
327 |
|
incom |
⊢ ( ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∩ ( 1 ... 𝑀 ) ) = ( ( 1 ... 𝑀 ) ∩ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) |
328 |
|
nnuzdisj |
⊢ ( ( 1 ... 𝑀 ) ∩ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) = ∅ |
329 |
327 328
|
eqtri |
⊢ ( ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∩ ( 1 ... 𝑀 ) ) = ∅ |
330 |
329
|
a1i |
⊢ ( 𝜑 → ( ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∩ ( 1 ... 𝑀 ) ) = ∅ ) |
331 |
|
icossicc |
⊢ ( 0 [,) +∞ ) ⊆ ( 0 [,] +∞ ) |
332 |
|
ssid |
⊢ ( 0 [,) +∞ ) ⊆ ( 0 [,) +∞ ) |
333 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → 𝜑 ) |
334 |
21
|
peano2nnd |
⊢ ( 𝜑 → ( 𝑀 + 1 ) ∈ ℕ ) |
335 |
|
uznnssnn |
⊢ ( ( 𝑀 + 1 ) ∈ ℕ → ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ⊆ ℕ ) |
336 |
334 335
|
syl |
⊢ ( 𝜑 → ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ⊆ ℕ ) |
337 |
336
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ⊆ ℕ ) |
338 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) |
339 |
337 338
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → 𝑗 ∈ ℕ ) |
340 |
|
snfi |
⊢ { 𝑍 } ∈ Fin |
341 |
340
|
a1i |
⊢ ( 𝜑 → { 𝑍 } ∈ Fin ) |
342 |
|
unfi |
⊢ ( ( 𝑌 ∈ Fin ∧ { 𝑍 } ∈ Fin ) → ( 𝑌 ∪ { 𝑍 } ) ∈ Fin ) |
343 |
173 341 342
|
syl2anc |
⊢ ( 𝜑 → ( 𝑌 ∪ { 𝑍 } ) ∈ Fin ) |
344 |
5 343
|
eqeltrid |
⊢ ( 𝜑 → 𝑊 ∈ Fin ) |
345 |
344
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑊 ∈ Fin ) |
346 |
|
eleq1w |
⊢ ( 𝑗 = 𝑙 → ( 𝑗 ∈ 𝑌 ↔ 𝑙 ∈ 𝑌 ) ) |
347 |
|
fveq2 |
⊢ ( 𝑗 = 𝑙 → ( 𝑐 ‘ 𝑗 ) = ( 𝑐 ‘ 𝑙 ) ) |
348 |
347
|
breq1d |
⊢ ( 𝑗 = 𝑙 → ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 ↔ ( 𝑐 ‘ 𝑙 ) ≤ 𝑥 ) ) |
349 |
348 347
|
ifbieq1d |
⊢ ( 𝑗 = 𝑙 → if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) = if ( ( 𝑐 ‘ 𝑙 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑙 ) , 𝑥 ) ) |
350 |
346 347 349
|
ifbieq12d |
⊢ ( 𝑗 = 𝑙 → if ( 𝑗 ∈ 𝑌 , ( 𝑐 ‘ 𝑗 ) , if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) ) = if ( 𝑙 ∈ 𝑌 , ( 𝑐 ‘ 𝑙 ) , if ( ( 𝑐 ‘ 𝑙 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑙 ) , 𝑥 ) ) ) |
351 |
350
|
cbvmptv |
⊢ ( 𝑗 ∈ 𝑊 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑐 ‘ 𝑗 ) , if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) ) ) = ( 𝑙 ∈ 𝑊 ↦ if ( 𝑙 ∈ 𝑌 , ( 𝑐 ‘ 𝑙 ) , if ( ( 𝑐 ‘ 𝑙 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑙 ) , 𝑥 ) ) ) |
352 |
351
|
mpteq2i |
⊢ ( 𝑐 ∈ ( ℝ ↑m 𝑊 ) ↦ ( 𝑗 ∈ 𝑊 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑐 ‘ 𝑗 ) , if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) ) ) ) = ( 𝑐 ∈ ( ℝ ↑m 𝑊 ) ↦ ( 𝑙 ∈ 𝑊 ↦ if ( 𝑙 ∈ 𝑌 , ( 𝑐 ‘ 𝑙 ) , if ( ( 𝑐 ‘ 𝑙 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑙 ) , 𝑥 ) ) ) ) |
353 |
352
|
mpteq2i |
⊢ ( 𝑥 ∈ ℝ ↦ ( 𝑐 ∈ ( ℝ ↑m 𝑊 ) ↦ ( 𝑗 ∈ 𝑊 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑐 ‘ 𝑗 ) , if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) ) ) ) ) = ( 𝑥 ∈ ℝ ↦ ( 𝑐 ∈ ( ℝ ↑m 𝑊 ) ↦ ( 𝑙 ∈ 𝑊 ↦ if ( 𝑙 ∈ 𝑌 , ( 𝑐 ‘ 𝑙 ) , if ( ( 𝑐 ‘ 𝑙 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑙 ) , 𝑥 ) ) ) ) ) |
354 |
14 353
|
eqtri |
⊢ 𝐻 = ( 𝑥 ∈ ℝ ↦ ( 𝑐 ∈ ( ℝ ↑m 𝑊 ) ↦ ( 𝑙 ∈ 𝑊 ↦ if ( 𝑙 ∈ 𝑌 , ( 𝑐 ‘ 𝑙 ) , if ( ( 𝑐 ‘ 𝑙 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑙 ) , 𝑥 ) ) ) ) ) |
355 |
90
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑆 ∈ ℝ ) |
356 |
354 355 345 49
|
hsphoif |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) : 𝑊 ⟶ ℝ ) |
357 |
1 345 125 356
|
hoidmvcl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,) +∞ ) ) |
358 |
333 339 357
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,) +∞ ) ) |
359 |
332 358
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,) +∞ ) ) |
360 |
331 359
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,] +∞ ) ) |
361 |
215 217 357
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,) +∞ ) ) |
362 |
331 361
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,] +∞ ) ) |
363 |
324 325 326 330 360 362
|
sge0splitmpt |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ( ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∪ ( 1 ... 𝑀 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) = ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) +𝑒 ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) |
364 |
|
nnex |
⊢ ℕ ∈ V |
365 |
364
|
a1i |
⊢ ( 𝜑 → ℕ ∈ V ) |
366 |
331 357
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,] +∞ ) ) |
367 |
324 365 366 211 336
|
sge0ssrempt |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ∈ ℝ ) |
368 |
37
|
a1i |
⊢ ( 𝜑 → ( 1 ... 𝑀 ) ⊆ ℕ ) |
369 |
324 365 366 211 368
|
sge0ssrempt |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ∈ ℝ ) |
370 |
|
rexadd |
⊢ ( ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ∈ ℝ ∧ ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ∈ ℝ ) → ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) +𝑒 ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) = ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) |
371 |
367 369 370
|
syl2anc |
⊢ ( 𝜑 → ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) +𝑒 ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) = ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) |
372 |
323 363 371
|
3eqtrd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) = ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) |
373 |
372
|
oveq2d |
⊢ ( 𝜑 → ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) = ( ( 1 + 𝐸 ) · ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) ) |
374 |
373
|
oveq1d |
⊢ ( 𝜑 → ( ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) + ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) = ( ( ( 1 + 𝐸 ) · ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) + ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ) |
375 |
372 211
|
eqeltrrd |
⊢ ( 𝜑 → ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ∈ ℝ ) |
376 |
375
|
recnd |
⊢ ( 𝜑 → ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ∈ ℂ ) |
377 |
278
|
recnd |
⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ∈ ℂ ) |
378 |
303 376 377
|
adddid |
⊢ ( 𝜑 → ( ( 1 + 𝐸 ) · ( ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) = ( ( ( 1 + 𝐸 ) · ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) + ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ) |
379 |
378
|
eqcomd |
⊢ ( 𝜑 → ( ( ( 1 + 𝐸 ) · ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) + ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) = ( ( 1 + 𝐸 ) · ( ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ) |
380 |
367
|
recnd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ∈ ℂ ) |
381 |
369
|
recnd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ∈ ℂ ) |
382 |
380 381 377
|
addassd |
⊢ ( 𝜑 → ( ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) = ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + ( ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ) |
383 |
213 361
|
sge0fsummpt |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) = Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) |
384 |
383
|
oveq1d |
⊢ ( 𝜑 → ( ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) = ( Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) |
385 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
386 |
172 385
|
sstri |
⊢ ( 0 [,) +∞ ) ⊆ ℂ |
387 |
386 357
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ℂ ) |
388 |
215 217 387
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ℂ ) |
389 |
192
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑄 − 𝑆 ) ∈ ℝ ) |
390 |
389 275
|
remulcld |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ∈ ℝ ) |
391 |
390
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ∈ ℂ ) |
392 |
217 391
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ∈ ℂ ) |
393 |
213 388 392
|
fsumadd |
⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) = ( Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) |
394 |
393
|
eqcomd |
⊢ ( 𝜑 → ( Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) = Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) |
395 |
384 394
|
eqtrd |
⊢ ( 𝜑 → ( ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) = Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) |
396 |
395
|
oveq2d |
⊢ ( 𝜑 → ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + ( ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) = ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ) |
397 |
382 396
|
eqtrd |
⊢ ( 𝜑 → ( ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) = ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ) |
398 |
397
|
oveq2d |
⊢ ( 𝜑 → ( ( 1 + 𝐸 ) · ( ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) = ( ( 1 + 𝐸 ) · ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ) ) |
399 |
374 379 398
|
3eqtrd |
⊢ ( 𝜑 → ( ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) + ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) = ( ( 1 + 𝐸 ) · ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ) ) |
400 |
172 357
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ℝ ) |
401 |
400 390
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) |
402 |
215 217 401
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) |
403 |
213 402
|
fsumrecl |
⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) |
404 |
367 403
|
readdcld |
⊢ ( 𝜑 → ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ∈ ℝ ) |
405 |
|
0le1 |
⊢ 0 ≤ 1 |
406 |
405
|
a1i |
⊢ ( 𝜑 → 0 ≤ 1 ) |
407 |
16
|
rpge0d |
⊢ ( 𝜑 → 0 ≤ 𝐸 ) |
408 |
206 207 406 407
|
addge0d |
⊢ ( 𝜑 → 0 ≤ ( 1 + 𝐸 ) ) |
409 |
84
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑄 ∈ ℝ ) |
410 |
354 409 345 49
|
hsphoif |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) : 𝑊 ⟶ ℝ ) |
411 |
1 345 125 410
|
hoidmvcl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,) +∞ ) ) |
412 |
331 411
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,] +∞ ) ) |
413 |
324 365 412 281 336
|
sge0ssrempt |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ∈ ℝ ) |
414 |
172 411
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ℝ ) |
415 |
215 217 414
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ℝ ) |
416 |
213 415
|
fsumrecl |
⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ℝ ) |
417 |
333 339 412
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,] +∞ ) ) |
418 |
210
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑍 ∈ ( 𝑊 ∖ 𝑌 ) ) |
419 |
90 84 157
|
ltled |
⊢ ( 𝜑 → 𝑆 ≤ 𝑄 ) |
420 |
419
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑆 ≤ 𝑄 ) |
421 |
1 345 418 5 355 409 420 354 125 49
|
hsphoidmvle2 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ≤ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) |
422 |
333 339 421
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ≤ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) |
423 |
324 325 360 417 422
|
sge0lempt |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) |
424 |
215
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) = 0 ) → 𝜑 ) |
425 |
217
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) = 0 ) → 𝑗 ∈ ℕ ) |
426 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) = 0 ) → ( 𝑃 ‘ 𝑗 ) = 0 ) |
427 |
|
oveq2 |
⊢ ( ( 𝑃 ‘ 𝑗 ) = 0 → ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) = ( ( 𝑄 − 𝑆 ) · 0 ) ) |
428 |
427
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) = 0 ) → ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) = ( ( 𝑄 − 𝑆 ) · 0 ) ) |
429 |
183
|
mul01d |
⊢ ( 𝜑 → ( ( 𝑄 − 𝑆 ) · 0 ) = 0 ) |
430 |
429
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) = 0 ) → ( ( 𝑄 − 𝑆 ) · 0 ) = 0 ) |
431 |
428 430
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) = 0 ) → ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) = 0 ) |
432 |
431
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) = 0 ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + 0 ) ) |
433 |
387
|
addid1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + 0 ) = ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) |
434 |
433
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) = 0 ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + 0 ) = ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) |
435 |
432 434
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) = 0 ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) = ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) |
436 |
421
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) = 0 ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ≤ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) |
437 |
435 436
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) = 0 ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ≤ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) |
438 |
424 425 426 437
|
syl21anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) = 0 ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ≤ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) |
439 |
|
simpl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ¬ ( 𝑃 ‘ 𝑗 ) = 0 ) → ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ) |
440 |
|
neqne |
⊢ ( ¬ ( 𝑃 ‘ 𝑗 ) = 0 → ( 𝑃 ‘ 𝑗 ) ≠ 0 ) |
441 |
440
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ¬ ( 𝑃 ‘ 𝑗 ) = 0 ) → ( 𝑃 ‘ 𝑗 ) ≠ 0 ) |
442 |
402
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) |
443 |
215
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝜑 ) |
444 |
217
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑗 ∈ ℕ ) |
445 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( 𝑃 ‘ 𝑗 ) ≠ 0 ) |
446 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑍 ∈ ( 𝑋 ∖ 𝑌 ) ) |
447 |
209
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ¬ 𝑍 ∈ 𝑌 ) |
448 |
|
eqid |
⊢ ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
449 |
1 223 446 447 5 125 356 448
|
hoiprodp1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) = ( ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) · ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) ) ) ) |
450 |
449
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) = ( ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) · ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) ) ) ) |
451 |
222
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) |
452 |
223
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑌 ∈ Fin ) |
453 |
222
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) |
454 |
|
fveq2 |
⊢ ( 𝑌 = ∅ → ( 𝐿 ‘ 𝑌 ) = ( 𝐿 ‘ ∅ ) ) |
455 |
454
|
oveqd |
⊢ ( 𝑌 = ∅ → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝐾 ‘ 𝑗 ) ) ) |
456 |
455
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝐾 ‘ 𝑗 ) ) ) |
457 |
252
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ) |
458 |
|
id |
⊢ ( 𝑌 = ∅ → 𝑌 = ∅ ) |
459 |
458
|
eqcomd |
⊢ ( 𝑌 = ∅ → ∅ = 𝑌 ) |
460 |
459
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ∅ = 𝑌 ) |
461 |
460
|
feq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ( ( 𝐽 ‘ 𝑗 ) : ∅ ⟶ ℝ ↔ ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ) ) |
462 |
457 461
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ( 𝐽 ‘ 𝑗 ) : ∅ ⟶ ℝ ) |
463 |
272
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ( 𝐾 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ) |
464 |
460
|
feq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ( ( 𝐾 ‘ 𝑗 ) : ∅ ⟶ ℝ ↔ ( 𝐾 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ) ) |
465 |
463 464
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ( 𝐾 ‘ 𝑗 ) : ∅ ⟶ ℝ ) |
466 |
1 462 465
|
hoidmv0val |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝐾 ‘ 𝑗 ) ) = 0 ) |
467 |
453 456 466
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ( 𝑃 ‘ 𝑗 ) = 0 ) |
468 |
467
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ 𝑌 = ∅ ) → ( 𝑃 ‘ 𝑗 ) = 0 ) |
469 |
|
neneq |
⊢ ( ( 𝑃 ‘ 𝑗 ) ≠ 0 → ¬ ( 𝑃 ‘ 𝑗 ) = 0 ) |
470 |
469
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ 𝑌 = ∅ ) → ¬ ( 𝑃 ‘ 𝑗 ) = 0 ) |
471 |
468 470
|
pm2.65da |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ¬ 𝑌 = ∅ ) |
472 |
471
|
neqned |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑌 ≠ ∅ ) |
473 |
252
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ) |
474 |
272
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( 𝐾 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ) |
475 |
1 452 472 473 474
|
hoidmvn0val |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) = ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
476 |
250
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( 𝐽 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ) |
477 |
222
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) |
478 |
250
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝐽 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ) |
479 |
478 235
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝐽 ‘ 𝑗 ) = 𝐹 ) |
480 |
270
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝐾 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ) |
481 |
480 260
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝐾 ‘ 𝑗 ) = 𝐹 ) |
482 |
479 481
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) = ( 𝐹 ( 𝐿 ‘ 𝑌 ) 𝐹 ) ) |
483 |
1 173 232
|
hoidmvval0b |
⊢ ( 𝜑 → ( 𝐹 ( 𝐿 ‘ 𝑌 ) 𝐹 ) = 0 ) |
484 |
483
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝐹 ( 𝐿 ‘ 𝑌 ) 𝐹 ) = 0 ) |
485 |
477 482 484
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝑃 ‘ 𝑗 ) = 0 ) |
486 |
485
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝑃 ‘ 𝑗 ) = 0 ) |
487 |
469
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ¬ ( 𝑃 ‘ 𝑗 ) = 0 ) |
488 |
486 487
|
condan |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
489 |
488
|
iftrued |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ) |
490 |
476 489
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( 𝐽 ‘ 𝑗 ) = ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ) |
491 |
490
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) ) |
492 |
491
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) ) |
493 |
|
fvres |
⊢ ( 𝑘 ∈ 𝑌 → ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) ) |
494 |
493
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) ) |
495 |
492 494
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) ) |
496 |
270
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( 𝐾 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ) |
497 |
488 255
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ) |
498 |
496 497
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( 𝐾 ‘ 𝑗 ) = ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ) |
499 |
498
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) ) |
500 |
499
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) ) |
501 |
|
fvres |
⊢ ( 𝑘 ∈ 𝑌 → ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) |
502 |
501
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) |
503 |
500 502
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) |
504 |
495 503
|
oveq12d |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
505 |
504
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ 𝑘 ∈ 𝑌 ) → ( vol ‘ ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) = ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
506 |
505
|
prodeq2dv |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
507 |
475 506
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) = ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
508 |
355
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → 𝑆 ∈ ℝ ) |
509 |
345
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → 𝑊 ∈ Fin ) |
510 |
49
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝐷 ‘ 𝑗 ) : 𝑊 ⟶ ℝ ) |
511 |
|
elun1 |
⊢ ( 𝑘 ∈ 𝑌 → 𝑘 ∈ ( 𝑌 ∪ { 𝑍 } ) ) |
512 |
511 5
|
eleqtrrdi |
⊢ ( 𝑘 ∈ 𝑌 → 𝑘 ∈ 𝑊 ) |
513 |
512
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → 𝑘 ∈ 𝑊 ) |
514 |
354 508 509 510 513
|
hsphoival |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) = if ( 𝑘 ∈ 𝑌 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) , if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ≤ 𝑆 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) , 𝑆 ) ) ) |
515 |
|
iftrue |
⊢ ( 𝑘 ∈ 𝑌 → if ( 𝑘 ∈ 𝑌 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) , if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ≤ 𝑆 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) , 𝑆 ) ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) |
516 |
515
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → if ( 𝑘 ∈ 𝑌 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) , if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ≤ 𝑆 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) , 𝑆 ) ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) |
517 |
514 516
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) |
518 |
517
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
519 |
518
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
520 |
519
|
prodeq2dv |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
521 |
520
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) |
522 |
521
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) |
523 |
451 507 522
|
3eqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ( 𝑃 ‘ 𝑗 ) ) |
524 |
354 355 345 49 50
|
hsphoival |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) = if ( 𝑍 ∈ 𝑌 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑆 ) ) ) |
525 |
209
|
iffalsed |
⊢ ( 𝜑 → if ( 𝑍 ∈ 𝑌 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑆 ) ) = if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑆 ) ) |
526 |
525
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → if ( 𝑍 ∈ 𝑌 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑆 ) ) = if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑆 ) ) |
527 |
524 526
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) = if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑆 ) ) |
528 |
527
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑆 ) ) ) |
529 |
528
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑆 ) ) ) |
530 |
126
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ* ) |
531 |
530
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ* ) |
532 |
51
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ* ) |
533 |
532
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ* ) |
534 |
|
icoltub |
⊢ ( ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ* ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ* ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → 𝑆 < ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) |
535 |
531 533 488 534
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑆 < ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) |
536 |
355
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑆 ∈ ℝ ) |
537 |
51
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ) |
538 |
536 537
|
ltnled |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( 𝑆 < ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ↔ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 ) ) |
539 |
535 538
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 ) |
540 |
539
|
iffalsed |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑆 ) = 𝑆 ) |
541 |
540
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑆 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) 𝑆 ) ) |
542 |
529 541
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) 𝑆 ) ) |
543 |
542
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) ) = ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) 𝑆 ) ) ) |
544 |
|
volico |
⊢ ( ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ∧ 𝑆 ∈ ℝ ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) 𝑆 ) ) = if ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 , ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) , 0 ) ) |
545 |
126 536 544
|
syl2an |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) 𝑆 ) ) = if ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 , ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) , 0 ) ) |
546 |
545
|
anabss5 |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) 𝑆 ) ) = if ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 , ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) , 0 ) ) |
547 |
|
iftrue |
⊢ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 → if ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 , ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) , 0 ) = ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
548 |
547
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → if ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 , ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) , 0 ) = ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
549 |
|
iffalse |
⊢ ( ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 → if ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 , ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) , 0 ) = 0 ) |
550 |
549
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → if ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 , ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) , 0 ) = 0 ) |
551 |
|
simpll |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → ( 𝜑 ∧ 𝑗 ∈ ℕ ) ) |
552 |
|
icogelb |
⊢ ( ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ* ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ* ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 ) |
553 |
531 533 488 552
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 ) |
554 |
553
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 ) |
555 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) |
556 |
554 555
|
jca |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) ) |
557 |
551 126
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ) |
558 |
551 355
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → 𝑆 ∈ ℝ ) |
559 |
557 558
|
eqleltd |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑆 ↔ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) ) ) |
560 |
556 559
|
mpbird |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑆 ) |
561 |
|
id |
⊢ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑆 → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑆 ) |
562 |
561
|
eqcomd |
⊢ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑆 → 𝑆 = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) |
563 |
562
|
oveq1d |
⊢ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑆 → ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
564 |
563
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑆 ) → ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
565 |
385 126
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℂ ) |
566 |
565
|
subidd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) = 0 ) |
567 |
566
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑆 ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) = 0 ) |
568 |
564 567
|
eqtr2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑆 ) → 0 = ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
569 |
551 560 568
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → 0 = ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
570 |
550 569
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → if ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 , ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) , 0 ) = ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
571 |
548 570
|
pm2.61dan |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → if ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 , ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) , 0 ) = ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
572 |
543 546 571
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) ) = ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
573 |
523 572
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) · ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) ) ) = ( ( 𝑃 ‘ 𝑗 ) · ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) |
574 |
386 274
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) ∈ ℂ ) |
575 |
355 126
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ∈ ℝ ) |
576 |
575
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ∈ ℂ ) |
577 |
574 576
|
mulcomd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑃 ‘ 𝑗 ) · ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) = ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) ) |
578 |
577
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝑃 ‘ 𝑗 ) · ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) = ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) ) |
579 |
450 573 578
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) = ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) ) |
580 |
579
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) = ( ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) |
581 |
183
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑄 − 𝑆 ) ∈ ℂ ) |
582 |
576 581 574
|
adddird |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) + ( 𝑄 − 𝑆 ) ) · ( 𝑃 ‘ 𝑗 ) ) = ( ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) |
583 |
582
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) = ( ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) + ( 𝑄 − 𝑆 ) ) · ( 𝑃 ‘ 𝑗 ) ) ) |
584 |
583
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) = ( ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) + ( 𝑄 − 𝑆 ) ) · ( 𝑃 ‘ 𝑗 ) ) ) |
585 |
576 581
|
addcomd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) + ( 𝑄 − 𝑆 ) ) = ( ( 𝑄 − 𝑆 ) + ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) |
586 |
166
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑄 ∈ ℂ ) |
587 |
167
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑆 ∈ ℂ ) |
588 |
586 587 565
|
npncand |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑄 − 𝑆 ) + ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) = ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
589 |
585 588
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) + ( 𝑄 − 𝑆 ) ) = ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
590 |
589
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) + ( 𝑄 − 𝑆 ) ) · ( 𝑃 ‘ 𝑗 ) ) = ( ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) ) |
591 |
590
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) + ( 𝑄 − 𝑆 ) ) · ( 𝑃 ‘ 𝑗 ) ) = ( ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) ) |
592 |
580 584 591
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) = ( ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) ) |
593 |
443 444 445 592
|
syl21anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) = ( ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) ) |
594 |
|
eqid |
⊢ ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
595 |
1 223 50 447 5 125 410 594
|
hoiprodp1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) = ( ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) · ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) ) ) ) |
596 |
215 217 595
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) = ( ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) · ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) ) ) ) |
597 |
596
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) = ( ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) · ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) ) ) ) |
598 |
507
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) |
599 |
409
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → 𝑄 ∈ ℝ ) |
600 |
354 599 509 510 513
|
hsphoival |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) = if ( 𝑘 ∈ 𝑌 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) , if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) , 𝑄 ) ) ) |
601 |
|
iftrue |
⊢ ( 𝑘 ∈ 𝑌 → if ( 𝑘 ∈ 𝑌 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) , if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) , 𝑄 ) ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) |
602 |
601
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → if ( 𝑘 ∈ 𝑌 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) , if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) , 𝑄 ) ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) |
603 |
600 602
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) |
604 |
603
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
605 |
604
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
606 |
605
|
prodeq2dv |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
607 |
606
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
608 |
598 607 451
|
3eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ( 𝑃 ‘ 𝑗 ) ) |
609 |
443 444 445 608
|
syl21anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ( 𝑃 ‘ 𝑗 ) ) |
610 |
354 409 345 49 50
|
hsphoival |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) = if ( 𝑍 ∈ 𝑌 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑄 ) ) ) |
611 |
217 610
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) = if ( 𝑍 ∈ 𝑌 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑄 ) ) ) |
612 |
611
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) = if ( 𝑍 ∈ 𝑌 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑄 ) ) ) |
613 |
209
|
iffalsed |
⊢ ( 𝜑 → if ( 𝑍 ∈ 𝑌 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑄 ) ) = if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑄 ) ) |
614 |
613
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → if ( 𝑍 ∈ 𝑌 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑄 ) ) = if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑄 ) ) |
615 |
217 51
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ) |
616 |
615
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ) |
617 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) |
618 |
616 617
|
eqled |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 ) |
619 |
618
|
iftrued |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑄 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) |
620 |
619 617
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑄 ) = 𝑄 ) |
621 |
620
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑄 ) = 𝑄 ) |
622 |
84
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 𝑄 ∈ ℝ ) |
623 |
622
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → 𝑄 ∈ ℝ ) |
624 |
623
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → 𝑄 ∈ ℝ ) |
625 |
615
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ) |
626 |
625
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ) |
627 |
25
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑄 = inf ( 𝑉 , ℝ , < ) ) |
628 |
443 57
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑉 ⊆ ℝ ) |
629 |
161
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ∃ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 𝑥 ≤ 𝑦 ) |
630 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑗 ∈ ( 1 ... 𝑀 ) ) |
631 |
216 488
|
sylanl2 |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
632 |
630 631
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( 𝑗 ∈ ( 1 ... 𝑀 ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) |
633 |
|
rabid |
⊢ ( 𝑗 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ↔ ( 𝑗 ∈ ( 1 ... 𝑀 ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) |
634 |
632 633
|
sylibr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑗 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ) |
635 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) |
636 |
|
fveq2 |
⊢ ( 𝑖 = 𝑗 → ( 𝐷 ‘ 𝑖 ) = ( 𝐷 ‘ 𝑗 ) ) |
637 |
636
|
fveq1d |
⊢ ( 𝑖 = 𝑗 → ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) |
638 |
637
|
eqeq2d |
⊢ ( 𝑖 = 𝑗 → ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ↔ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
639 |
638
|
rspcev |
⊢ ( ( 𝑗 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → ∃ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) |
640 |
634 635 639
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ∃ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) |
641 |
|
fvexd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ V ) |
642 |
35 640 641
|
elrnmptd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ran ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) |
643 |
642 23
|
eleqtrrdi |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ 𝑂 ) |
644 |
|
elun2 |
⊢ ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ 𝑂 → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) ) |
645 |
643 644
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) ) |
646 |
76
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) = 𝑉 ) |
647 |
645 646
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ 𝑉 ) |
648 |
|
lbinfle |
⊢ ( ( 𝑉 ⊆ ℝ ∧ ∃ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 𝑥 ≤ 𝑦 ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ 𝑉 ) → inf ( 𝑉 , ℝ , < ) ≤ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) |
649 |
628 629 647 648
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → inf ( 𝑉 , ℝ , < ) ≤ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) |
650 |
627 649
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑄 ≤ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) |
651 |
650
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → 𝑄 ≤ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) |
652 |
|
neqne |
⊢ ( ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≠ 𝑄 ) |
653 |
652
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≠ 𝑄 ) |
654 |
624 626 651 653
|
leneltd |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → 𝑄 < ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) |
655 |
624 626
|
ltnled |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → ( 𝑄 < ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ↔ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 ) ) |
656 |
654 655
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 ) |
657 |
656
|
iffalsed |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑄 ) = 𝑄 ) |
658 |
621 657
|
pm2.61dan |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑄 ) = 𝑄 ) |
659 |
612 614 658
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) = 𝑄 ) |
660 |
659
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) 𝑄 ) ) |
661 |
660
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) ) = ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) 𝑄 ) ) ) |
662 |
215 217 126
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ) |
663 |
662
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ) |
664 |
443 84
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑄 ∈ ℝ ) |
665 |
|
volico |
⊢ ( ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ∧ 𝑄 ∈ ℝ ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) 𝑄 ) ) = if ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑄 , ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) , 0 ) ) |
666 |
663 664 665
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) 𝑄 ) ) = if ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑄 , ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) , 0 ) ) |
667 |
443 90
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑆 ∈ ℝ ) |
668 |
443 444 445 553
|
syl21anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 ) |
669 |
443 157
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑆 < 𝑄 ) |
670 |
663 667 664 668 669
|
lelttrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑄 ) |
671 |
670
|
iftrued |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → if ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑄 , ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) , 0 ) = ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
672 |
661 666 671
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) ) = ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
673 |
609 672
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) · ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) ) ) = ( ( 𝑃 ‘ 𝑗 ) · ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) |
674 |
215 166
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 𝑄 ∈ ℂ ) |
675 |
385 662
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℂ ) |
676 |
674 675
|
subcld |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ∈ ℂ ) |
677 |
306 676
|
mulcomd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝑃 ‘ 𝑗 ) · ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) = ( ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) ) |
678 |
677
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝑃 ‘ 𝑗 ) · ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) = ( ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) ) |
679 |
597 673 678
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) = ( ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) ) |
680 |
593 679
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) = ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) |
681 |
442 680
|
eqled |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ≤ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) |
682 |
439 441 681
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ¬ ( 𝑃 ‘ 𝑗 ) = 0 ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ≤ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) |
683 |
438 682
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ≤ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) |
684 |
213 402 415 683
|
fsumle |
⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ≤ Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) |
685 |
367 403 413 416 423 684
|
leadd12dd |
⊢ ( 𝜑 → ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ≤ ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) |
686 |
321
|
mpteq1d |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) = ( 𝑗 ∈ ( ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∪ ( 1 ... 𝑀 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) |
687 |
686
|
fveq2d |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ( ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∪ ( 1 ... 𝑀 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) |
688 |
217 412
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,] +∞ ) ) |
689 |
324 325 326 330 417 688
|
sge0splitmpt |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ( ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∪ ( 1 ... 𝑀 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) = ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) +𝑒 ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) |
690 |
687 689
|
eqtrd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) = ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) +𝑒 ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) |
691 |
215 217 411
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,) +∞ ) ) |
692 |
213 691
|
sge0fsummpt |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) = Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) |
693 |
692 416
|
eqeltrd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ∈ ℝ ) |
694 |
|
rexadd |
⊢ ( ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ∈ ℝ ∧ ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ∈ ℝ ) → ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) +𝑒 ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) = ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) |
695 |
413 693 694
|
syl2anc |
⊢ ( 𝜑 → ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) +𝑒 ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) = ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) |
696 |
692
|
oveq2d |
⊢ ( 𝜑 → ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) = ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) |
697 |
690 695 696
|
3eqtrrd |
⊢ ( 𝜑 → ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) |
698 |
685 697
|
breqtrd |
⊢ ( 𝜑 → ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) |
699 |
404 281 208 408 698
|
lemul2ad |
⊢ ( 𝜑 → ( ( 1 + 𝐸 ) · ( ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) |
700 |
399 699
|
eqbrtrd |
⊢ ( 𝜑 → ( ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) + ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) |
701 |
205 280 282 315 700
|
letrd |
⊢ ( 𝜑 → ( ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) + ( 𝐺 · ( 𝑄 − 𝑆 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) |
702 |
189 701
|
eqbrtrd |
⊢ ( 𝜑 → ( 𝐺 · ( 𝑄 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) |
703 |
165 702
|
jca |
⊢ ( 𝜑 → ( 𝑄 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∧ ( 𝐺 · ( 𝑄 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) ) |
704 |
|
oveq1 |
⊢ ( 𝑧 = 𝑄 → ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) = ( 𝑄 − ( 𝐴 ‘ 𝑍 ) ) ) |
705 |
704
|
oveq2d |
⊢ ( 𝑧 = 𝑄 → ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) = ( 𝐺 · ( 𝑄 − ( 𝐴 ‘ 𝑍 ) ) ) ) |
706 |
|
fveq2 |
⊢ ( 𝑧 = 𝑄 → ( 𝐻 ‘ 𝑧 ) = ( 𝐻 ‘ 𝑄 ) ) |
707 |
706
|
fveq1d |
⊢ ( 𝑧 = 𝑄 → ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) = ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) |
708 |
707
|
oveq2d |
⊢ ( 𝑧 = 𝑄 → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) = ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) |
709 |
708
|
mpteq2dv |
⊢ ( 𝑧 = 𝑄 → ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) |
710 |
709
|
fveq2d |
⊢ ( 𝑧 = 𝑄 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) |
711 |
710
|
oveq2d |
⊢ ( 𝑧 = 𝑄 → ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) = ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) |
712 |
705 711
|
breq12d |
⊢ ( 𝑧 = 𝑄 → ( ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ↔ ( 𝐺 · ( 𝑄 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) ) |
713 |
712
|
elrab |
⊢ ( 𝑄 ∈ { 𝑧 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) } ↔ ( 𝑄 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∧ ( 𝐺 · ( 𝑄 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) ) |
714 |
703 713
|
sylibr |
⊢ ( 𝜑 → 𝑄 ∈ { 𝑧 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) } ) |
715 |
714 17
|
eleqtrrdi |
⊢ ( 𝜑 → 𝑄 ∈ 𝑈 ) |
716 |
|
breq2 |
⊢ ( 𝑢 = 𝑄 → ( 𝑆 < 𝑢 ↔ 𝑆 < 𝑄 ) ) |
717 |
716
|
rspcev |
⊢ ( ( 𝑄 ∈ 𝑈 ∧ 𝑆 < 𝑄 ) → ∃ 𝑢 ∈ 𝑈 𝑆 < 𝑢 ) |
718 |
715 157 717
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑢 ∈ 𝑈 𝑆 < 𝑢 ) |