Step |
Hyp |
Ref |
Expression |
1 |
|
hoidmvlelem3.l |
⊢ 𝐿 = ( 𝑥 ∈ Fin ↦ ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) , 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) ) ) |
2 |
|
hoidmvlelem3.x |
⊢ ( 𝜑 → 𝑋 ∈ Fin ) |
3 |
|
hoidmvlelem3.y |
⊢ ( 𝜑 → 𝑌 ⊆ 𝑋 ) |
4 |
|
hoidmvlelem3.z |
⊢ ( 𝜑 → 𝑍 ∈ ( 𝑋 ∖ 𝑌 ) ) |
5 |
|
hoidmvlelem3.w |
⊢ 𝑊 = ( 𝑌 ∪ { 𝑍 } ) |
6 |
|
hoidmvlelem3.a |
⊢ ( 𝜑 → 𝐴 : 𝑊 ⟶ ℝ ) |
7 |
|
hoidmvlelem3.b |
⊢ ( 𝜑 → 𝐵 : 𝑊 ⟶ ℝ ) |
8 |
|
hoidmvlelem3.lt |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( 𝐴 ‘ 𝑘 ) < ( 𝐵 ‘ 𝑘 ) ) |
9 |
|
hoidmvlelem3.f |
⊢ 𝐹 = ( 𝑦 ∈ 𝑌 ↦ 0 ) |
10 |
|
hoidmvlelem3.c |
⊢ ( 𝜑 → 𝐶 : ℕ ⟶ ( ℝ ↑m 𝑊 ) ) |
11 |
|
hoidmvlelem3.j |
⊢ 𝐽 = ( 𝑗 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ) |
12 |
|
hoidmvlelem3.d |
⊢ ( 𝜑 → 𝐷 : ℕ ⟶ ( ℝ ↑m 𝑊 ) ) |
13 |
|
hoidmvlelem3.k |
⊢ 𝐾 = ( 𝑗 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ) |
14 |
|
hoidmvlelem3.r |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ∈ ℝ ) |
15 |
|
hoidmvlelem3.h |
⊢ 𝐻 = ( 𝑥 ∈ ℝ ↦ ( 𝑐 ∈ ( ℝ ↑m 𝑊 ) ↦ ( 𝑗 ∈ 𝑊 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑐 ‘ 𝑗 ) , if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) ) ) ) ) |
16 |
|
hoidmvlelem3.g |
⊢ 𝐺 = ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) |
17 |
|
hoidmvlelem3.e |
⊢ ( 𝜑 → 𝐸 ∈ ℝ+ ) |
18 |
|
hoidmvlelem3.u |
⊢ 𝑈 = { 𝑧 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) } |
19 |
|
hoidmvlelem3.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝑈 ) |
20 |
|
hoidmvlelem3.sb |
⊢ ( 𝜑 → 𝑆 < ( 𝐵 ‘ 𝑍 ) ) |
21 |
|
hoidmvlelem3.p |
⊢ 𝑃 = ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) |
22 |
|
hoidmvlelem3.i |
⊢ ( 𝜑 → ∀ 𝑒 ∈ ( ℝ ↑m 𝑌 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑌 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) |
23 |
|
hoidmvlelem3.i2 |
⊢ ( 𝜑 → X 𝑘 ∈ 𝑊 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
24 |
|
hoidmvlelem3.o |
⊢ 𝑂 = ( 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ↦ ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ) |
25 |
|
1nn |
⊢ 1 ∈ ℕ |
26 |
25
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → 1 ∈ ℕ ) |
27 |
|
0le0 |
⊢ 0 ≤ 0 |
28 |
27
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → 0 ≤ 0 ) |
29 |
16
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → 𝐺 = ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ) |
30 |
|
fveq2 |
⊢ ( 𝑌 = ∅ → ( 𝐿 ‘ 𝑌 ) = ( 𝐿 ‘ ∅ ) ) |
31 |
|
reseq2 |
⊢ ( 𝑌 = ∅ → ( 𝐴 ↾ 𝑌 ) = ( 𝐴 ↾ ∅ ) ) |
32 |
|
res0 |
⊢ ( 𝐴 ↾ ∅ ) = ∅ |
33 |
32
|
a1i |
⊢ ( 𝑌 = ∅ → ( 𝐴 ↾ ∅ ) = ∅ ) |
34 |
31 33
|
eqtrd |
⊢ ( 𝑌 = ∅ → ( 𝐴 ↾ 𝑌 ) = ∅ ) |
35 |
|
reseq2 |
⊢ ( 𝑌 = ∅ → ( 𝐵 ↾ 𝑌 ) = ( 𝐵 ↾ ∅ ) ) |
36 |
|
res0 |
⊢ ( 𝐵 ↾ ∅ ) = ∅ |
37 |
36
|
a1i |
⊢ ( 𝑌 = ∅ → ( 𝐵 ↾ ∅ ) = ∅ ) |
38 |
35 37
|
eqtrd |
⊢ ( 𝑌 = ∅ → ( 𝐵 ↾ 𝑌 ) = ∅ ) |
39 |
30 34 38
|
oveq123d |
⊢ ( 𝑌 = ∅ → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) = ( ∅ ( 𝐿 ‘ ∅ ) ∅ ) ) |
40 |
39
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) = ( ∅ ( 𝐿 ‘ ∅ ) ∅ ) ) |
41 |
|
f0 |
⊢ ∅ : ∅ ⟶ ℝ |
42 |
41
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ∅ : ∅ ⟶ ℝ ) |
43 |
1 42 42
|
hoidmv0val |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( ∅ ( 𝐿 ‘ ∅ ) ∅ ) = 0 ) |
44 |
29 40 43
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → 𝐺 = 0 ) |
45 |
|
nfcvd |
⊢ ( 𝜑 → Ⅎ 𝑗 ( 𝑃 ‘ 1 ) ) |
46 |
|
nfv |
⊢ Ⅎ 𝑗 𝜑 |
47 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 = 1 ) → 𝑗 = 1 ) |
48 |
47
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑗 = 1 ) → ( 𝑃 ‘ 𝑗 ) = ( 𝑃 ‘ 1 ) ) |
49 |
|
1red |
⊢ ( 𝜑 → 1 ∈ ℝ ) |
50 |
|
rge0ssre |
⊢ ( 0 [,) +∞ ) ⊆ ℝ |
51 |
|
id |
⊢ ( 𝜑 → 𝜑 ) |
52 |
25
|
a1i |
⊢ ( 𝜑 → 1 ∈ ℕ ) |
53 |
25
|
elexi |
⊢ 1 ∈ V |
54 |
|
eleq1 |
⊢ ( 𝑗 = 1 → ( 𝑗 ∈ ℕ ↔ 1 ∈ ℕ ) ) |
55 |
54
|
anbi2d |
⊢ ( 𝑗 = 1 → ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ↔ ( 𝜑 ∧ 1 ∈ ℕ ) ) ) |
56 |
|
fveq2 |
⊢ ( 𝑗 = 1 → ( 𝑃 ‘ 𝑗 ) = ( 𝑃 ‘ 1 ) ) |
57 |
56
|
eleq1d |
⊢ ( 𝑗 = 1 → ( ( 𝑃 ‘ 𝑗 ) ∈ ( 0 [,) +∞ ) ↔ ( 𝑃 ‘ 1 ) ∈ ( 0 [,) +∞ ) ) ) |
58 |
55 57
|
imbi12d |
⊢ ( 𝑗 = 1 → ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) ∈ ( 0 [,) +∞ ) ) ↔ ( ( 𝜑 ∧ 1 ∈ ℕ ) → ( 𝑃 ‘ 1 ) ∈ ( 0 [,) +∞ ) ) ) ) |
59 |
|
id |
⊢ ( 𝑗 ∈ ℕ → 𝑗 ∈ ℕ ) |
60 |
|
ovexd |
⊢ ( 𝑗 ∈ ℕ → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ∈ V ) |
61 |
21
|
fvmpt2 |
⊢ ( ( 𝑗 ∈ ℕ ∧ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ∈ V ) → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) |
62 |
59 60 61
|
syl2anc |
⊢ ( 𝑗 ∈ ℕ → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) |
63 |
62
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) |
64 |
5
|
a1i |
⊢ ( 𝜑 → 𝑊 = ( 𝑌 ∪ { 𝑍 } ) ) |
65 |
4
|
eldifad |
⊢ ( 𝜑 → 𝑍 ∈ 𝑋 ) |
66 |
|
snssi |
⊢ ( 𝑍 ∈ 𝑋 → { 𝑍 } ⊆ 𝑋 ) |
67 |
65 66
|
syl |
⊢ ( 𝜑 → { 𝑍 } ⊆ 𝑋 ) |
68 |
3 67
|
unssd |
⊢ ( 𝜑 → ( 𝑌 ∪ { 𝑍 } ) ⊆ 𝑋 ) |
69 |
64 68
|
eqsstrd |
⊢ ( 𝜑 → 𝑊 ⊆ 𝑋 ) |
70 |
2 69
|
ssfid |
⊢ ( 𝜑 → 𝑊 ∈ Fin ) |
71 |
|
ssun1 |
⊢ 𝑌 ⊆ ( 𝑌 ∪ { 𝑍 } ) |
72 |
5
|
eqcomi |
⊢ ( 𝑌 ∪ { 𝑍 } ) = 𝑊 |
73 |
71 72
|
sseqtri |
⊢ 𝑌 ⊆ 𝑊 |
74 |
73
|
a1i |
⊢ ( 𝜑 → 𝑌 ⊆ 𝑊 ) |
75 |
70 74
|
ssfid |
⊢ ( 𝜑 → 𝑌 ∈ Fin ) |
76 |
75
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑌 ∈ Fin ) |
77 |
|
iftrue |
⊢ ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ) |
78 |
77
|
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 ) ) |
79 |
10
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐶 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑊 ) ) |
80 |
|
elmapi |
⊢ ( ( 𝐶 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑊 ) → ( 𝐶 ‘ 𝑗 ) : 𝑊 ⟶ ℝ ) |
81 |
79 80
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐶 ‘ 𝑗 ) : 𝑊 ⟶ ℝ ) |
82 |
71 5
|
sseqtrri |
⊢ 𝑌 ⊆ 𝑊 |
83 |
82
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑌 ⊆ 𝑊 ) |
84 |
81 83
|
fssresd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) |
85 |
|
reex |
⊢ ℝ ∈ V |
86 |
85
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ℝ ∈ V ) |
87 |
70 74
|
ssexd |
⊢ ( 𝜑 → 𝑌 ∈ V ) |
88 |
87
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑌 ∈ V ) |
89 |
86 88
|
elmapd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) ↔ ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) ) |
90 |
84 89
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) ) |
91 |
90
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) ) |
92 |
78 91
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ ( ℝ ↑m 𝑌 ) ) |
93 |
|
iffalse |
⊢ ( ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = 𝐹 ) |
94 |
93
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = 𝐹 ) |
95 |
|
0red |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑌 ) → 0 ∈ ℝ ) |
96 |
95 9
|
fmptd |
⊢ ( 𝜑 → 𝐹 : 𝑌 ⟶ ℝ ) |
97 |
85
|
a1i |
⊢ ( 𝜑 → ℝ ∈ V ) |
98 |
97 75
|
elmapd |
⊢ ( 𝜑 → ( 𝐹 ∈ ( ℝ ↑m 𝑌 ) ↔ 𝐹 : 𝑌 ⟶ ℝ ) ) |
99 |
96 98
|
mpbird |
⊢ ( 𝜑 → 𝐹 ∈ ( ℝ ↑m 𝑌 ) ) |
100 |
99
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → 𝐹 ∈ ( ℝ ↑m 𝑌 ) ) |
101 |
94 100
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ ( ℝ ↑m 𝑌 ) ) |
102 |
92 101
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ ( ℝ ↑m 𝑌 ) ) |
103 |
102 11
|
fmptd |
⊢ ( 𝜑 → 𝐽 : ℕ ⟶ ( ℝ ↑m 𝑌 ) ) |
104 |
103
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐽 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑌 ) ) |
105 |
|
elmapi |
⊢ ( ( 𝐽 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑌 ) → ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ) |
106 |
104 105
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ) |
107 |
|
iftrue |
⊢ ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ) |
108 |
107
|
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 ) ) |
109 |
12
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐷 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑊 ) ) |
110 |
|
elmapi |
⊢ ( ( 𝐷 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑊 ) → ( 𝐷 ‘ 𝑗 ) : 𝑊 ⟶ ℝ ) |
111 |
109 110
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐷 ‘ 𝑗 ) : 𝑊 ⟶ ℝ ) |
112 |
111 83
|
fssresd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) |
113 |
86 88
|
elmapd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) ↔ ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) ) |
114 |
112 113
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) ) |
115 |
114
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) ) |
116 |
108 115
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ ( ℝ ↑m 𝑌 ) ) |
117 |
|
iffalse |
⊢ ( ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = 𝐹 ) |
118 |
117
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = 𝐹 ) |
119 |
118 100
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ ( ℝ ↑m 𝑌 ) ) |
120 |
116 119
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ ( ℝ ↑m 𝑌 ) ) |
121 |
120 13
|
fmptd |
⊢ ( 𝜑 → 𝐾 : ℕ ⟶ ( ℝ ↑m 𝑌 ) ) |
122 |
121
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐾 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑌 ) ) |
123 |
|
elmapi |
⊢ ( ( 𝐾 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑌 ) → ( 𝐾 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ) |
124 |
122 123
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐾 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ) |
125 |
1 76 106 124
|
hoidmvcl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ∈ ( 0 [,) +∞ ) ) |
126 |
63 125
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) ∈ ( 0 [,) +∞ ) ) |
127 |
53 58 126
|
vtocl |
⊢ ( ( 𝜑 ∧ 1 ∈ ℕ ) → ( 𝑃 ‘ 1 ) ∈ ( 0 [,) +∞ ) ) |
128 |
51 52 127
|
syl2anc |
⊢ ( 𝜑 → ( 𝑃 ‘ 1 ) ∈ ( 0 [,) +∞ ) ) |
129 |
50 128
|
sseldi |
⊢ ( 𝜑 → ( 𝑃 ‘ 1 ) ∈ ℝ ) |
130 |
129
|
recnd |
⊢ ( 𝜑 → ( 𝑃 ‘ 1 ) ∈ ℂ ) |
131 |
45 46 48 49 130
|
sumsnd |
⊢ ( 𝜑 → Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) = ( 𝑃 ‘ 1 ) ) |
132 |
131
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) = ( 𝑃 ‘ 1 ) ) |
133 |
|
fveq2 |
⊢ ( 𝑗 = 1 → ( 𝐽 ‘ 𝑗 ) = ( 𝐽 ‘ 1 ) ) |
134 |
|
fveq2 |
⊢ ( 𝑗 = 1 → ( 𝐾 ‘ 𝑗 ) = ( 𝐾 ‘ 1 ) ) |
135 |
133 134
|
oveq12d |
⊢ ( 𝑗 = 1 → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) = ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 1 ) ) ) |
136 |
|
ovex |
⊢ ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 1 ) ) ∈ V |
137 |
135 21 136
|
fvmpt |
⊢ ( 1 ∈ ℕ → ( 𝑃 ‘ 1 ) = ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 1 ) ) ) |
138 |
25 137
|
ax-mp |
⊢ ( 𝑃 ‘ 1 ) = ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 1 ) ) |
139 |
138
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( 𝑃 ‘ 1 ) = ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 1 ) ) ) |
140 |
30
|
oveqd |
⊢ ( 𝑌 = ∅ → ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 1 ) ) = ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ ∅ ) ( 𝐾 ‘ 1 ) ) ) |
141 |
140
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 1 ) ) = ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ ∅ ) ( 𝐾 ‘ 1 ) ) ) |
142 |
133
|
feq1d |
⊢ ( 𝑗 = 1 → ( ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ↔ ( 𝐽 ‘ 1 ) : 𝑌 ⟶ ℝ ) ) |
143 |
55 142
|
imbi12d |
⊢ ( 𝑗 = 1 → ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ) ↔ ( ( 𝜑 ∧ 1 ∈ ℕ ) → ( 𝐽 ‘ 1 ) : 𝑌 ⟶ ℝ ) ) ) |
144 |
84
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) |
145 |
78
|
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 ) ) |
146 |
144 145
|
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 ) |
147 |
96
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → 𝐹 : 𝑌 ⟶ ℝ ) |
148 |
94
|
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 ) ) |
149 |
147 148
|
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 ) |
150 |
146 149
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ ) |
151 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑗 ∈ ℕ ) |
152 |
|
fvex |
⊢ ( 𝐶 ‘ 𝑗 ) ∈ V |
153 |
152
|
resex |
⊢ ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ∈ V |
154 |
78 153
|
eqeltrdi |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V ) |
155 |
99
|
elexd |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
156 |
155
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝐹 ∈ V ) |
157 |
156
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → 𝐹 ∈ V ) |
158 |
94 157
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V ) |
159 |
154 158
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V ) |
160 |
11
|
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 ) ) |
161 |
151 159 160
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐽 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ) |
162 |
161
|
feq1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ↔ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ ) ) |
163 |
150 162
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ) |
164 |
53 143 163
|
vtocl |
⊢ ( ( 𝜑 ∧ 1 ∈ ℕ ) → ( 𝐽 ‘ 1 ) : 𝑌 ⟶ ℝ ) |
165 |
51 52 164
|
syl2anc |
⊢ ( 𝜑 → ( 𝐽 ‘ 1 ) : 𝑌 ⟶ ℝ ) |
166 |
165
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( 𝐽 ‘ 1 ) : 𝑌 ⟶ ℝ ) |
167 |
|
id |
⊢ ( 𝑌 = ∅ → 𝑌 = ∅ ) |
168 |
167
|
eqcomd |
⊢ ( 𝑌 = ∅ → ∅ = 𝑌 ) |
169 |
168
|
feq2d |
⊢ ( 𝑌 = ∅ → ( ( 𝐽 ‘ 1 ) : ∅ ⟶ ℝ ↔ ( 𝐽 ‘ 1 ) : 𝑌 ⟶ ℝ ) ) |
170 |
169
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( ( 𝐽 ‘ 1 ) : ∅ ⟶ ℝ ↔ ( 𝐽 ‘ 1 ) : 𝑌 ⟶ ℝ ) ) |
171 |
166 170
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( 𝐽 ‘ 1 ) : ∅ ⟶ ℝ ) |
172 |
134
|
feq1d |
⊢ ( 𝑗 = 1 → ( ( 𝐾 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ↔ ( 𝐾 ‘ 1 ) : 𝑌 ⟶ ℝ ) ) |
173 |
55 172
|
imbi12d |
⊢ ( 𝑗 = 1 → ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐾 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ) ↔ ( ( 𝜑 ∧ 1 ∈ ℕ ) → ( 𝐾 ‘ 1 ) : 𝑌 ⟶ ℝ ) ) ) |
174 |
53 173 124
|
vtocl |
⊢ ( ( 𝜑 ∧ 1 ∈ ℕ ) → ( 𝐾 ‘ 1 ) : 𝑌 ⟶ ℝ ) |
175 |
51 52 174
|
syl2anc |
⊢ ( 𝜑 → ( 𝐾 ‘ 1 ) : 𝑌 ⟶ ℝ ) |
176 |
175
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( 𝐾 ‘ 1 ) : 𝑌 ⟶ ℝ ) |
177 |
168
|
feq2d |
⊢ ( 𝑌 = ∅ → ( ( 𝐾 ‘ 1 ) : ∅ ⟶ ℝ ↔ ( 𝐾 ‘ 1 ) : 𝑌 ⟶ ℝ ) ) |
178 |
177
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( ( 𝐾 ‘ 1 ) : ∅ ⟶ ℝ ↔ ( 𝐾 ‘ 1 ) : 𝑌 ⟶ ℝ ) ) |
179 |
176 178
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( 𝐾 ‘ 1 ) : ∅ ⟶ ℝ ) |
180 |
1 171 179
|
hoidmv0val |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ ∅ ) ( 𝐾 ‘ 1 ) ) = 0 ) |
181 |
141 180
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( ( 𝐽 ‘ 1 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 1 ) ) = 0 ) |
182 |
132 139 181
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) = 0 ) |
183 |
182
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) ) = ( ( 1 + 𝐸 ) · 0 ) ) |
184 |
17
|
rpred |
⊢ ( 𝜑 → 𝐸 ∈ ℝ ) |
185 |
49 184
|
readdcld |
⊢ ( 𝜑 → ( 1 + 𝐸 ) ∈ ℝ ) |
186 |
185
|
recnd |
⊢ ( 𝜑 → ( 1 + 𝐸 ) ∈ ℂ ) |
187 |
186
|
mul01d |
⊢ ( 𝜑 → ( ( 1 + 𝐸 ) · 0 ) = 0 ) |
188 |
187
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( ( 1 + 𝐸 ) · 0 ) = 0 ) |
189 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → 0 = 0 ) |
190 |
183 188 189
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) ) = 0 ) |
191 |
44 190
|
breq12d |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ( 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) ) ↔ 0 ≤ 0 ) ) |
192 |
28 191
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) ) ) |
193 |
|
oveq2 |
⊢ ( 𝑚 = 1 → ( 1 ... 𝑚 ) = ( 1 ... 1 ) ) |
194 |
25
|
nnzi |
⊢ 1 ∈ ℤ |
195 |
|
fzsn |
⊢ ( 1 ∈ ℤ → ( 1 ... 1 ) = { 1 } ) |
196 |
194 195
|
ax-mp |
⊢ ( 1 ... 1 ) = { 1 } |
197 |
196
|
a1i |
⊢ ( 𝑚 = 1 → ( 1 ... 1 ) = { 1 } ) |
198 |
193 197
|
eqtrd |
⊢ ( 𝑚 = 1 → ( 1 ... 𝑚 ) = { 1 } ) |
199 |
198
|
sumeq1d |
⊢ ( 𝑚 = 1 → Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) = Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) ) |
200 |
199
|
oveq2d |
⊢ ( 𝑚 = 1 → ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) = ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) ) ) |
201 |
200
|
breq2d |
⊢ ( 𝑚 = 1 → ( 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ↔ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) ) ) ) |
202 |
201
|
rspcev |
⊢ ( ( 1 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ { 1 } ( 𝑃 ‘ 𝑗 ) ) ) → ∃ 𝑚 ∈ ℕ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) |
203 |
26 192 202
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑌 = ∅ ) → ∃ 𝑚 ∈ ℕ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) |
204 |
|
simpl |
⊢ ( ( 𝜑 ∧ ¬ 𝑌 = ∅ ) → 𝜑 ) |
205 |
|
neqne |
⊢ ( ¬ 𝑌 = ∅ → 𝑌 ≠ ∅ ) |
206 |
205
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ 𝑌 = ∅ ) → 𝑌 ≠ ∅ ) |
207 |
|
nfv |
⊢ Ⅎ 𝑗 ( 𝜑 ∧ 𝑌 ≠ ∅ ) |
208 |
194
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → 1 ∈ ℤ ) |
209 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
210 |
126
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) ∈ ( 0 [,) +∞ ) ) |
211 |
82
|
a1i |
⊢ ( 𝜑 → 𝑌 ⊆ 𝑊 ) |
212 |
6 211
|
fssresd |
⊢ ( 𝜑 → ( 𝐴 ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) |
213 |
7 211
|
fssresd |
⊢ ( 𝜑 → ( 𝐵 ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) |
214 |
1 75 212 213
|
hoidmvcl |
⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ∈ ( 0 [,) +∞ ) ) |
215 |
50 214
|
sseldi |
⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ∈ ℝ ) |
216 |
16 215
|
eqeltrid |
⊢ ( 𝜑 → 𝐺 ∈ ℝ ) |
217 |
|
0red |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
218 |
|
1rp |
⊢ 1 ∈ ℝ+ |
219 |
218
|
a1i |
⊢ ( 𝜑 → 1 ∈ ℝ+ ) |
220 |
219 17
|
jca |
⊢ ( 𝜑 → ( 1 ∈ ℝ+ ∧ 𝐸 ∈ ℝ+ ) ) |
221 |
|
rpaddcl |
⊢ ( ( 1 ∈ ℝ+ ∧ 𝐸 ∈ ℝ+ ) → ( 1 + 𝐸 ) ∈ ℝ+ ) |
222 |
220 221
|
syl |
⊢ ( 𝜑 → ( 1 + 𝐸 ) ∈ ℝ+ ) |
223 |
|
rpgt0 |
⊢ ( ( 1 + 𝐸 ) ∈ ℝ+ → 0 < ( 1 + 𝐸 ) ) |
224 |
222 223
|
syl |
⊢ ( 𝜑 → 0 < ( 1 + 𝐸 ) ) |
225 |
217 224
|
gtned |
⊢ ( 𝜑 → ( 1 + 𝐸 ) ≠ 0 ) |
226 |
216 185 225
|
redivcld |
⊢ ( 𝜑 → ( 𝐺 / ( 1 + 𝐸 ) ) ∈ ℝ ) |
227 |
226
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 𝐺 / ( 1 + 𝐸 ) ) ∈ ℝ ) |
228 |
226
|
ltpnfd |
⊢ ( 𝜑 → ( 𝐺 / ( 1 + 𝐸 ) ) < +∞ ) |
229 |
228
|
adantr |
⊢ ( ( 𝜑 ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ( 𝐺 / ( 1 + 𝐸 ) ) < +∞ ) |
230 |
|
id |
⊢ ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) |
231 |
230
|
eqcomd |
⊢ ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ → +∞ = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ) |
232 |
231
|
adantl |
⊢ ( ( 𝜑 ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → +∞ = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ) |
233 |
229 232
|
breqtrd |
⊢ ( ( 𝜑 ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ( 𝐺 / ( 1 + 𝐸 ) ) < ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ) |
234 |
233
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ( 𝐺 / ( 1 + 𝐸 ) ) < ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ) |
235 |
|
simpl |
⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ( 𝜑 ∧ 𝑌 ≠ ∅ ) ) |
236 |
|
simpr |
⊢ ( ( 𝜑 ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) |
237 |
|
nnex |
⊢ ℕ ∈ V |
238 |
237
|
a1i |
⊢ ( ( 𝜑 ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ℕ ∈ V ) |
239 |
|
icossicc |
⊢ ( 0 [,) +∞ ) ⊆ ( 0 [,] +∞ ) |
240 |
239 126
|
sseldi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) ∈ ( 0 [,] +∞ ) ) |
241 |
|
eqid |
⊢ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) = ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) |
242 |
240 241
|
fmptd |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) : ℕ ⟶ ( 0 [,] +∞ ) ) |
243 |
242
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) : ℕ ⟶ ( 0 [,] +∞ ) ) |
244 |
238 243
|
sge0repnf |
⊢ ( ( 𝜑 ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ↔ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) ) |
245 |
236 244
|
mpbird |
⊢ ( ( 𝜑 ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) |
246 |
245
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) |
247 |
227
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) → ( 𝐺 / ( 1 + 𝐸 ) ) ∈ ℝ ) |
248 |
216
|
adantr |
⊢ ( ( 𝜑 ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) → 𝐺 ∈ ℝ ) |
249 |
248
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) → 𝐺 ∈ ℝ ) |
250 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) |
251 |
49 17
|
ltaddrpd |
⊢ ( 𝜑 → 1 < ( 1 + 𝐸 ) ) |
252 |
251
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → 1 < ( 1 + 𝐸 ) ) |
253 |
75
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → 𝑌 ∈ Fin ) |
254 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → 𝑌 ≠ ∅ ) |
255 |
212
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 𝐴 ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) |
256 |
213
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 𝐵 ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) |
257 |
1 253 254 255 256
|
hoidmvn0val |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) = ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) [,) ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) ) ) ) |
258 |
16
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → 𝐺 = ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ) |
259 |
|
fvres |
⊢ ( 𝑘 ∈ 𝑌 → ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) = ( 𝐴 ‘ 𝑘 ) ) |
260 |
|
fvres |
⊢ ( 𝑘 ∈ 𝑌 → ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) = ( 𝐵 ‘ 𝑘 ) ) |
261 |
259 260
|
oveq12d |
⊢ ( 𝑘 ∈ 𝑌 → ( ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) [,) ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
262 |
261
|
fveq2d |
⊢ ( 𝑘 ∈ 𝑌 → ( vol ‘ ( ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) [,) ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) ) ) = ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ) |
263 |
262
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( vol ‘ ( ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) [,) ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) ) ) = ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ) |
264 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → 𝐴 : 𝑊 ⟶ ℝ ) |
265 |
|
elun1 |
⊢ ( 𝑘 ∈ 𝑌 → 𝑘 ∈ ( 𝑌 ∪ { 𝑍 } ) ) |
266 |
265 5
|
eleqtrrdi |
⊢ ( 𝑘 ∈ 𝑌 → 𝑘 ∈ 𝑊 ) |
267 |
266
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → 𝑘 ∈ 𝑊 ) |
268 |
264 267
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( 𝐴 ‘ 𝑘 ) ∈ ℝ ) |
269 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → 𝐵 : 𝑊 ⟶ ℝ ) |
270 |
269 267
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( 𝐵 ‘ 𝑘 ) ∈ ℝ ) |
271 |
|
volico |
⊢ ( ( ( 𝐴 ‘ 𝑘 ) ∈ ℝ ∧ ( 𝐵 ‘ 𝑘 ) ∈ ℝ ) → ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = if ( ( 𝐴 ‘ 𝑘 ) < ( 𝐵 ‘ 𝑘 ) , ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) , 0 ) ) |
272 |
268 270 271
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( vol ‘ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = if ( ( 𝐴 ‘ 𝑘 ) < ( 𝐵 ‘ 𝑘 ) , ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) , 0 ) ) |
273 |
267 8
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( 𝐴 ‘ 𝑘 ) < ( 𝐵 ‘ 𝑘 ) ) |
274 |
273
|
iftrued |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → if ( ( 𝐴 ‘ 𝑘 ) < ( 𝐵 ‘ 𝑘 ) , ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) , 0 ) = ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) |
275 |
263 272 274
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( vol ‘ ( ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) [,) ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) ) ) = ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) |
276 |
275
|
prodeq2dv |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) [,) ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑌 ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) |
277 |
276
|
eqcomd |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝑌 ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) = ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) [,) ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) ) ) ) |
278 |
277
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ∏ 𝑘 ∈ 𝑌 ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) = ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) [,) ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) ) ) ) |
279 |
257 258 278
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → 𝐺 = ∏ 𝑘 ∈ 𝑌 ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) |
280 |
|
difrp |
⊢ ( ( ( 𝐴 ‘ 𝑘 ) ∈ ℝ ∧ ( 𝐵 ‘ 𝑘 ) ∈ ℝ ) → ( ( 𝐴 ‘ 𝑘 ) < ( 𝐵 ‘ 𝑘 ) ↔ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ ℝ+ ) ) |
281 |
268 270 280
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐴 ‘ 𝑘 ) < ( 𝐵 ‘ 𝑘 ) ↔ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ ℝ+ ) ) |
282 |
273 281
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ ℝ+ ) |
283 |
75 282
|
fprodrpcl |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝑌 ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ ℝ+ ) |
284 |
283
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ∏ 𝑘 ∈ 𝑌 ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ ℝ+ ) |
285 |
279 284
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → 𝐺 ∈ ℝ+ ) |
286 |
222
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 1 + 𝐸 ) ∈ ℝ+ ) |
287 |
285 286
|
ltdivgt1 |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 1 < ( 1 + 𝐸 ) ↔ ( 𝐺 / ( 1 + 𝐸 ) ) < 𝐺 ) ) |
288 |
252 287
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 𝐺 / ( 1 + 𝐸 ) ) < 𝐺 ) |
289 |
288
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) → ( 𝐺 / ( 1 + 𝐸 ) ) < 𝐺 ) |
290 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → X 𝑘 ∈ 𝑊 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
291 |
|
fvexd |
⊢ ( 𝜑 → ( 𝑥 ‘ 𝑘 ) ∈ V ) |
292 |
19
|
elexd |
⊢ ( 𝜑 → 𝑆 ∈ V ) |
293 |
291 292
|
ifcld |
⊢ ( 𝜑 → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ V ) |
294 |
293
|
ralrimivw |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝑊 if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ V ) |
295 |
294
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ∀ 𝑘 ∈ 𝑊 if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ V ) |
296 |
|
eqid |
⊢ ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) = ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) |
297 |
296
|
fnmpt |
⊢ ( ∀ 𝑘 ∈ 𝑊 if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ V → ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) Fn 𝑊 ) |
298 |
295 297
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) Fn 𝑊 ) |
299 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
300 |
|
mptexg |
⊢ ( 𝑊 ∈ Fin → ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ∈ V ) |
301 |
70 300
|
syl |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ∈ V ) |
302 |
301
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ∈ V ) |
303 |
24
|
fvmpt2 |
⊢ ( ( 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∧ ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ∈ V ) → ( 𝑂 ‘ 𝑥 ) = ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ) |
304 |
299 302 303
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( 𝑂 ‘ 𝑥 ) = ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ) |
305 |
304
|
fneq1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( ( 𝑂 ‘ 𝑥 ) Fn 𝑊 ↔ ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) Fn 𝑊 ) ) |
306 |
298 305
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( 𝑂 ‘ 𝑥 ) Fn 𝑊 ) |
307 |
|
nfv |
⊢ Ⅎ 𝑘 𝜑 |
308 |
|
nfcv |
⊢ Ⅎ 𝑘 𝑥 |
309 |
|
nfixp1 |
⊢ Ⅎ 𝑘 X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) |
310 |
308 309
|
nfel |
⊢ Ⅎ 𝑘 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) |
311 |
307 310
|
nfan |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
312 |
304
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) = ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑘 ) ) |
313 |
312
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) = ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑘 ) ) |
314 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → 𝑘 ∈ 𝑊 ) |
315 |
293
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ V ) |
316 |
296
|
fvmpt2 |
⊢ ( ( 𝑘 ∈ 𝑊 ∧ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ V ) → ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑘 ) = if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) |
317 |
314 315 316
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑘 ) = if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) |
318 |
317
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) → ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑘 ) = if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) |
319 |
313 318
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) = if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) |
320 |
|
iftrue |
⊢ ( 𝑘 ∈ 𝑌 → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) = ( 𝑥 ‘ 𝑘 ) ) |
321 |
320
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) ∧ 𝑘 ∈ 𝑌 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) = ( 𝑥 ‘ 𝑘 ) ) |
322 |
|
vex |
⊢ 𝑥 ∈ V |
323 |
322
|
elixp |
⊢ ( 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ↔ ( 𝑥 Fn 𝑌 ∧ ∀ 𝑘 ∈ 𝑌 ( 𝑥 ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ) |
324 |
323
|
simprbi |
⊢ ( 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) → ∀ 𝑘 ∈ 𝑌 ( 𝑥 ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
325 |
324
|
adantr |
⊢ ( ( 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ∀ 𝑘 ∈ 𝑌 ( 𝑥 ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
326 |
|
simpr |
⊢ ( ( 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → 𝑘 ∈ 𝑌 ) |
327 |
|
rspa |
⊢ ( ( ∀ 𝑘 ∈ 𝑌 ( 𝑥 ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑥 ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
328 |
325 326 327
|
syl2anc |
⊢ ( ( 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑥 ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
329 |
328
|
ad4ant24 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑥 ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
330 |
321 329
|
eqeltrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) ∧ 𝑘 ∈ 𝑌 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
331 |
|
snidg |
⊢ ( 𝑍 ∈ ( 𝑋 ∖ 𝑌 ) → 𝑍 ∈ { 𝑍 } ) |
332 |
4 331
|
syl |
⊢ ( 𝜑 → 𝑍 ∈ { 𝑍 } ) |
333 |
|
elun2 |
⊢ ( 𝑍 ∈ { 𝑍 } → 𝑍 ∈ ( 𝑌 ∪ { 𝑍 } ) ) |
334 |
332 333
|
syl |
⊢ ( 𝜑 → 𝑍 ∈ ( 𝑌 ∪ { 𝑍 } ) ) |
335 |
72
|
a1i |
⊢ ( 𝜑 → ( 𝑌 ∪ { 𝑍 } ) = 𝑊 ) |
336 |
334 335
|
eleqtrd |
⊢ ( 𝜑 → 𝑍 ∈ 𝑊 ) |
337 |
6 336
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) ∈ ℝ ) |
338 |
337
|
rexrd |
⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) ∈ ℝ* ) |
339 |
7 336
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐵 ‘ 𝑍 ) ∈ ℝ ) |
340 |
339
|
rexrd |
⊢ ( 𝜑 → ( 𝐵 ‘ 𝑍 ) ∈ ℝ* ) |
341 |
|
iccssxr |
⊢ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ⊆ ℝ* |
342 |
|
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 ) ) |
343 |
18 342
|
eqsstri |
⊢ 𝑈 ⊆ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) |
344 |
343 19
|
sseldi |
⊢ ( 𝜑 → 𝑆 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ) |
345 |
341 344
|
sseldi |
⊢ ( 𝜑 → 𝑆 ∈ ℝ* ) |
346 |
|
iccgelb |
⊢ ( ( ( 𝐴 ‘ 𝑍 ) ∈ ℝ* ∧ ( 𝐵 ‘ 𝑍 ) ∈ ℝ* ∧ 𝑆 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ) → ( 𝐴 ‘ 𝑍 ) ≤ 𝑆 ) |
347 |
338 340 344 346
|
syl3anc |
⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) ≤ 𝑆 ) |
348 |
338 340 345 347 20
|
elicod |
⊢ ( 𝜑 → 𝑆 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) |
349 |
348
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ¬ 𝑘 ∈ 𝑌 ) → 𝑆 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) |
350 |
|
iffalse |
⊢ ( ¬ 𝑘 ∈ 𝑌 → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) = 𝑆 ) |
351 |
350
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ¬ 𝑘 ∈ 𝑌 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) = 𝑆 ) |
352 |
5
|
eleq2i |
⊢ ( 𝑘 ∈ 𝑊 ↔ 𝑘 ∈ ( 𝑌 ∪ { 𝑍 } ) ) |
353 |
352
|
biimpi |
⊢ ( 𝑘 ∈ 𝑊 → 𝑘 ∈ ( 𝑌 ∪ { 𝑍 } ) ) |
354 |
353
|
adantr |
⊢ ( ( 𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌 ) → 𝑘 ∈ ( 𝑌 ∪ { 𝑍 } ) ) |
355 |
|
simpr |
⊢ ( ( 𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌 ) → ¬ 𝑘 ∈ 𝑌 ) |
356 |
|
elunnel1 |
⊢ ( ( 𝑘 ∈ ( 𝑌 ∪ { 𝑍 } ) ∧ ¬ 𝑘 ∈ 𝑌 ) → 𝑘 ∈ { 𝑍 } ) |
357 |
354 355 356
|
syl2anc |
⊢ ( ( 𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌 ) → 𝑘 ∈ { 𝑍 } ) |
358 |
|
elsni |
⊢ ( 𝑘 ∈ { 𝑍 } → 𝑘 = 𝑍 ) |
359 |
357 358
|
syl |
⊢ ( ( 𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌 ) → 𝑘 = 𝑍 ) |
360 |
|
fveq2 |
⊢ ( 𝑘 = 𝑍 → ( 𝐴 ‘ 𝑘 ) = ( 𝐴 ‘ 𝑍 ) ) |
361 |
|
fveq2 |
⊢ ( 𝑘 = 𝑍 → ( 𝐵 ‘ 𝑘 ) = ( 𝐵 ‘ 𝑍 ) ) |
362 |
360 361
|
oveq12d |
⊢ ( 𝑘 = 𝑍 → ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) |
363 |
359 362
|
syl |
⊢ ( ( 𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌 ) → ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) |
364 |
363
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ¬ 𝑘 ∈ 𝑌 ) → ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) |
365 |
351 364
|
eleq12d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ¬ 𝑘 ∈ 𝑌 ) → ( if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ↔ 𝑆 ∈ ( ( 𝐴 ‘ 𝑍 ) [,) ( 𝐵 ‘ 𝑍 ) ) ) ) |
366 |
349 365
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ¬ 𝑘 ∈ 𝑌 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
367 |
366
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) ∧ ¬ 𝑘 ∈ 𝑌 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
368 |
330 367
|
pm2.61dan |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
369 |
319 368
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑊 ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
370 |
369
|
ex |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( 𝑘 ∈ 𝑊 → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ) |
371 |
311 370
|
ralrimi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
372 |
306 371
|
jca |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( ( 𝑂 ‘ 𝑥 ) Fn 𝑊 ∧ ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ) |
373 |
|
fvex |
⊢ ( 𝑂 ‘ 𝑥 ) ∈ V |
374 |
373
|
elixp |
⊢ ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ↔ ( ( 𝑂 ‘ 𝑥 ) Fn 𝑊 ∧ ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ) |
375 |
372 374
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
376 |
290 375
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( 𝑂 ‘ 𝑥 ) ∈ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
377 |
|
eliun |
⊢ ( ( 𝑂 ‘ 𝑥 ) ∈ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ ∃ 𝑗 ∈ ℕ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
378 |
376 377
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ∃ 𝑗 ∈ ℕ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
379 |
|
ixpfn |
⊢ ( 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) → 𝑥 Fn 𝑌 ) |
380 |
379
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → 𝑥 Fn 𝑌 ) |
381 |
380
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → 𝑥 Fn 𝑌 ) |
382 |
|
nfv |
⊢ Ⅎ 𝑘 𝑗 ∈ ℕ |
383 |
311 382
|
nfan |
⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) |
384 |
|
nfcv |
⊢ Ⅎ 𝑘 ( 𝑂 ‘ 𝑥 ) |
385 |
|
nfixp1 |
⊢ Ⅎ 𝑘 X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) |
386 |
384 385
|
nfel |
⊢ Ⅎ 𝑘 ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) |
387 |
383 386
|
nfan |
⊢ Ⅎ 𝑘 ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
388 |
312
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) = ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑘 ) ) |
389 |
293
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ∈ V ) |
390 |
267 389 316
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑘 ) = if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) |
391 |
390
|
3adant2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑘 ) = if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) |
392 |
320
|
3ad2ant3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) = ( 𝑥 ‘ 𝑘 ) ) |
393 |
388 391 392
|
3eqtrrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑥 ‘ 𝑘 ) = ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ) |
394 |
393
|
ad5ant125 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑥 ‘ 𝑘 ) = ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ) |
395 |
|
simpl |
⊢ ( ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
396 |
373
|
elixp |
⊢ ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ ( ( 𝑂 ‘ 𝑥 ) Fn 𝑊 ∧ ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
397 |
395 396
|
sylib |
⊢ ( ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝑂 ‘ 𝑥 ) Fn 𝑊 ∧ ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
398 |
397
|
simprd |
⊢ ( ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
399 |
266
|
adantl |
⊢ ( ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → 𝑘 ∈ 𝑊 ) |
400 |
|
rspa |
⊢ ( ( ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑊 ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
401 |
398 399 400
|
syl2anc |
⊢ ( ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
402 |
401
|
adantll |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
403 |
394 402
|
eqeltrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑥 ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
404 |
51
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → 𝜑 ) |
405 |
59
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → 𝑗 ∈ ℕ ) |
406 |
304
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑍 ) = ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑍 ) ) |
407 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) = ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ) |
408 |
|
eleq1 |
⊢ ( 𝑘 = 𝑍 → ( 𝑘 ∈ 𝑌 ↔ 𝑍 ∈ 𝑌 ) ) |
409 |
|
fveq2 |
⊢ ( 𝑘 = 𝑍 → ( 𝑥 ‘ 𝑘 ) = ( 𝑥 ‘ 𝑍 ) ) |
410 |
408 409
|
ifbieq1d |
⊢ ( 𝑘 = 𝑍 → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) = if ( 𝑍 ∈ 𝑌 , ( 𝑥 ‘ 𝑍 ) , 𝑆 ) ) |
411 |
410
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 = 𝑍 ) → if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) = if ( 𝑍 ∈ 𝑌 , ( 𝑥 ‘ 𝑍 ) , 𝑆 ) ) |
412 |
|
fvexd |
⊢ ( 𝜑 → ( 𝑥 ‘ 𝑍 ) ∈ V ) |
413 |
412 292
|
ifcld |
⊢ ( 𝜑 → if ( 𝑍 ∈ 𝑌 , ( 𝑥 ‘ 𝑍 ) , 𝑆 ) ∈ V ) |
414 |
407 411 336 413
|
fvmptd |
⊢ ( 𝜑 → ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑍 ) = if ( 𝑍 ∈ 𝑌 , ( 𝑥 ‘ 𝑍 ) , 𝑆 ) ) |
415 |
414
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( ( 𝑘 ∈ 𝑊 ↦ if ( 𝑘 ∈ 𝑌 , ( 𝑥 ‘ 𝑘 ) , 𝑆 ) ) ‘ 𝑍 ) = if ( 𝑍 ∈ 𝑌 , ( 𝑥 ‘ 𝑍 ) , 𝑆 ) ) |
416 |
4
|
eldifbd |
⊢ ( 𝜑 → ¬ 𝑍 ∈ 𝑌 ) |
417 |
416
|
iffalsed |
⊢ ( 𝜑 → if ( 𝑍 ∈ 𝑌 , ( 𝑥 ‘ 𝑍 ) , 𝑆 ) = 𝑆 ) |
418 |
417
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → if ( 𝑍 ∈ 𝑌 , ( 𝑥 ‘ 𝑍 ) , 𝑆 ) = 𝑆 ) |
419 |
406 415 418
|
3eqtrrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → 𝑆 = ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑍 ) ) |
420 |
419
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → 𝑆 = ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑍 ) ) |
421 |
404 336
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → 𝑍 ∈ 𝑊 ) |
422 |
396
|
simprbi |
⊢ ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) → ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
423 |
422
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
424 |
|
fveq2 |
⊢ ( 𝑘 = 𝑍 → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) = ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑍 ) ) |
425 |
|
fveq2 |
⊢ ( 𝑘 = 𝑍 → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) |
426 |
|
fveq2 |
⊢ ( 𝑘 = 𝑍 → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) |
427 |
425 426
|
oveq12d |
⊢ ( 𝑘 = 𝑍 → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
428 |
424 427
|
eleq12d |
⊢ ( 𝑘 = 𝑍 → ( ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑍 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) ) |
429 |
428
|
rspcva |
⊢ ( ( 𝑍 ∈ 𝑊 ∧ ∀ 𝑘 ∈ 𝑊 ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑘 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑍 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
430 |
421 423 429
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → ( ( 𝑂 ‘ 𝑥 ) ‘ 𝑍 ) ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
431 |
420 430
|
eqeltrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
432 |
161
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝐽 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ) |
433 |
77
|
3ad2ant3 |
⊢ ( ( 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 ) ) |
434 |
432 433
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝐽 ‘ 𝑗 ) = ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ) |
435 |
434
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) ) |
436 |
404 405 431 435
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) ) |
437 |
436
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) ) |
438 |
|
fvres |
⊢ ( 𝑘 ∈ 𝑌 → ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) ) |
439 |
438
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) ) |
440 |
437 439
|
eqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) ) |
441 |
120
|
elexd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V ) |
442 |
13
|
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 ) ) |
443 |
151 441 442
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐾 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ) |
444 |
443
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝐾 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ) |
445 |
107
|
3ad2ant3 |
⊢ ( ( 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 ) ) |
446 |
444 445
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝐾 ‘ 𝑗 ) = ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ) |
447 |
446
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) ) |
448 |
404 405 431 447
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) ) |
449 |
448
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) ) |
450 |
|
fvres |
⊢ ( 𝑘 ∈ 𝑌 → ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) |
451 |
450
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) |
452 |
449 451
|
eqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) |
453 |
440 452
|
oveq12d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
454 |
453
|
eqcomd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
455 |
403 454
|
eleqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑥 ‘ 𝑘 ) ∈ ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
456 |
455
|
ex |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → ( 𝑘 ∈ 𝑌 → ( 𝑥 ‘ 𝑘 ) ∈ ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
457 |
387 456
|
ralrimi |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → ∀ 𝑘 ∈ 𝑌 ( 𝑥 ‘ 𝑘 ) ∈ ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
458 |
381 457
|
jca |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → ( 𝑥 Fn 𝑌 ∧ ∀ 𝑘 ∈ 𝑌 ( 𝑥 ‘ 𝑘 ) ∈ ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
459 |
322
|
elixp |
⊢ ( 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ ( 𝑥 Fn 𝑌 ∧ ∀ 𝑘 ∈ 𝑌 ( 𝑥 ‘ 𝑘 ) ∈ ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
460 |
458 459
|
sylibr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) → 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
461 |
460
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) → 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
462 |
461
|
reximdva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ( ∃ 𝑗 ∈ ℕ ( 𝑂 ‘ 𝑥 ) ∈ X 𝑘 ∈ 𝑊 ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) → ∃ 𝑗 ∈ ℕ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
463 |
378 462
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → ∃ 𝑗 ∈ ℕ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
464 |
|
eliun |
⊢ ( 𝑥 ∈ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ ∃ 𝑗 ∈ ℕ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
465 |
463 464
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) → 𝑥 ∈ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
466 |
465
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) 𝑥 ∈ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
467 |
|
dfss3 |
⊢ ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ ∀ 𝑥 ∈ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) 𝑥 ∈ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
468 |
466 467
|
sylibr |
⊢ ( 𝜑 → X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
469 |
|
ovexd |
⊢ ( 𝜑 → ( ℝ ↑m 𝑌 ) ∈ V ) |
470 |
237
|
a1i |
⊢ ( 𝜑 → ℕ ∈ V ) |
471 |
469 470
|
elmapd |
⊢ ( 𝜑 → ( 𝐾 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ↔ 𝐾 : ℕ ⟶ ( ℝ ↑m 𝑌 ) ) ) |
472 |
121 471
|
mpbird |
⊢ ( 𝜑 → 𝐾 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ) |
473 |
469 470
|
elmapd |
⊢ ( 𝜑 → ( 𝐽 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ↔ 𝐽 : ℕ ⟶ ( ℝ ↑m 𝑌 ) ) ) |
474 |
103 473
|
mpbird |
⊢ ( 𝜑 → 𝐽 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ) |
475 |
97 87
|
elmapd |
⊢ ( 𝜑 → ( ( 𝐵 ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) ↔ ( 𝐵 ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) ) |
476 |
213 475
|
mpbird |
⊢ ( 𝜑 → ( 𝐵 ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) ) |
477 |
97 87
|
elmapd |
⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) ↔ ( 𝐴 ↾ 𝑌 ) : 𝑌 ⟶ ℝ ) ) |
478 |
212 477
|
mpbird |
⊢ ( 𝜑 → ( 𝐴 ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) ) |
479 |
|
fveq1 |
⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → ( 𝑒 ‘ 𝑘 ) = ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) ) |
480 |
479
|
adantr |
⊢ ( ( 𝑒 = ( 𝐴 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑒 ‘ 𝑘 ) = ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) ) |
481 |
259
|
adantl |
⊢ ( ( 𝑒 = ( 𝐴 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐴 ↾ 𝑌 ) ‘ 𝑘 ) = ( 𝐴 ‘ 𝑘 ) ) |
482 |
480 481
|
eqtrd |
⊢ ( ( 𝑒 = ( 𝐴 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑒 ‘ 𝑘 ) = ( 𝐴 ‘ 𝑘 ) ) |
483 |
482
|
oveq1d |
⊢ ( ( 𝑒 = ( 𝐴 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ) |
484 |
483
|
ixpeq2dva |
⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → X 𝑘 ∈ 𝑌 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ) |
485 |
484
|
sseq1d |
⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → ( X 𝑘 ∈ 𝑌 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
486 |
|
oveq1 |
⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → ( 𝑒 ( 𝐿 ‘ 𝑌 ) 𝑓 ) = ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ) |
487 |
486
|
breq1d |
⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → ( ( 𝑒 ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ↔ ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) |
488 |
485 487
|
imbi12d |
⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → ( ( X 𝑘 ∈ 𝑌 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ) |
489 |
488
|
ralbidv |
⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → ( ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ) |
490 |
489
|
ralbidv |
⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → ( ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ) |
491 |
490
|
ralbidv |
⊢ ( 𝑒 = ( 𝐴 ↾ 𝑌 ) → ( ∀ 𝑓 ∈ ( ℝ ↑m 𝑌 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑓 ∈ ( ℝ ↑m 𝑌 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ) |
492 |
491
|
rspcva |
⊢ ( ( ( 𝐴 ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) ∧ ∀ 𝑒 ∈ ( ℝ ↑m 𝑌 ) ∀ 𝑓 ∈ ( ℝ ↑m 𝑌 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝑒 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( 𝑒 ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) → ∀ 𝑓 ∈ ( ℝ ↑m 𝑌 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) |
493 |
478 22 492
|
syl2anc |
⊢ ( 𝜑 → ∀ 𝑓 ∈ ( ℝ ↑m 𝑌 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) |
494 |
|
fveq1 |
⊢ ( 𝑓 = ( 𝐵 ↾ 𝑌 ) → ( 𝑓 ‘ 𝑘 ) = ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) ) |
495 |
494
|
adantr |
⊢ ( ( 𝑓 = ( 𝐵 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑓 ‘ 𝑘 ) = ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) ) |
496 |
260
|
adantl |
⊢ ( ( 𝑓 = ( 𝐵 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐵 ↾ 𝑌 ) ‘ 𝑘 ) = ( 𝐵 ‘ 𝑘 ) ) |
497 |
495 496
|
eqtrd |
⊢ ( ( 𝑓 = ( 𝐵 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝑓 ‘ 𝑘 ) = ( 𝐵 ‘ 𝑘 ) ) |
498 |
497
|
oveq2d |
⊢ ( ( 𝑓 = ( 𝐵 ↾ 𝑌 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
499 |
498
|
ixpeq2dva |
⊢ ( 𝑓 = ( 𝐵 ↾ 𝑌 ) → X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
500 |
499
|
sseq1d |
⊢ ( 𝑓 = ( 𝐵 ↾ 𝑌 ) → ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
501 |
|
oveq2 |
⊢ ( 𝑓 = ( 𝐵 ↾ 𝑌 ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) = ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ) |
502 |
501
|
breq1d |
⊢ ( 𝑓 = ( 𝐵 ↾ 𝑌 ) → ( ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ↔ ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) |
503 |
500 502
|
imbi12d |
⊢ ( 𝑓 = ( 𝐵 ↾ 𝑌 ) → ( ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ) |
504 |
503
|
ralbidv |
⊢ ( 𝑓 = ( 𝐵 ↾ 𝑌 ) → ( ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ) |
505 |
504
|
ralbidv |
⊢ ( 𝑓 = ( 𝐵 ↾ 𝑌 ) → ( ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ) |
506 |
505
|
rspcva |
⊢ ( ( ( 𝐵 ↾ 𝑌 ) ∈ ( ℝ ↑m 𝑌 ) ∧ ∀ 𝑓 ∈ ( ℝ ↑m 𝑌 ) ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝑓 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) 𝑓 ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) → ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) |
507 |
476 493 506
|
syl2anc |
⊢ ( 𝜑 → ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) |
508 |
|
fveq1 |
⊢ ( 𝑔 = 𝐽 → ( 𝑔 ‘ 𝑗 ) = ( 𝐽 ‘ 𝑗 ) ) |
509 |
508
|
fveq1d |
⊢ ( 𝑔 = 𝐽 → ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) ) |
510 |
509
|
oveq1d |
⊢ ( 𝑔 = 𝐽 → ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
511 |
510
|
ixpeq2dv |
⊢ ( 𝑔 = 𝐽 → X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
512 |
511
|
iuneq2d |
⊢ ( 𝑔 = 𝐽 → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
513 |
512
|
sseq2d |
⊢ ( 𝑔 = 𝐽 → ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
514 |
508
|
oveq1d |
⊢ ( 𝑔 = 𝐽 → ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) |
515 |
514
|
mpteq2dv |
⊢ ( 𝑔 = 𝐽 → ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) |
516 |
515
|
fveq2d |
⊢ ( 𝑔 = 𝐽 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) |
517 |
516
|
breq2d |
⊢ ( 𝑔 = 𝐽 → ( ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ↔ ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) |
518 |
513 517
|
imbi12d |
⊢ ( 𝑔 = 𝐽 → ( ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ) |
519 |
518
|
ralbidv |
⊢ ( 𝑔 = 𝐽 → ( ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) ) |
520 |
519
|
rspcva |
⊢ ( ( 𝐽 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∧ ∀ 𝑔 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝑔 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) → ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) |
521 |
474 507 520
|
syl2anc |
⊢ ( 𝜑 → ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) |
522 |
|
fveq1 |
⊢ ( ℎ = 𝐾 → ( ℎ ‘ 𝑗 ) = ( 𝐾 ‘ 𝑗 ) ) |
523 |
522
|
fveq1d |
⊢ ( ℎ = 𝐾 → ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) |
524 |
523
|
oveq2d |
⊢ ( ℎ = 𝐾 → ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
525 |
524
|
ixpeq2dv |
⊢ ( ℎ = 𝐾 → X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
526 |
525
|
iuneq2d |
⊢ ( ℎ = 𝐾 → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
527 |
526
|
sseq2d |
⊢ ( ℎ = 𝐾 → ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) ↔ X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
528 |
522
|
oveq2d |
⊢ ( ℎ = 𝐾 → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) |
529 |
528
|
mpteq2dv |
⊢ ( ℎ = 𝐾 → ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) |
530 |
529
|
fveq2d |
⊢ ( ℎ = 𝐾 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) ) |
531 |
530
|
breq2d |
⊢ ( ℎ = 𝐾 → ( ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ↔ ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) ) ) |
532 |
527 531
|
imbi12d |
⊢ ( ℎ = 𝐾 → ( ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ↔ ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) ) ) ) |
533 |
532
|
rspcva |
⊢ ( ( 𝐾 ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ∧ ∀ ℎ ∈ ( ( ℝ ↑m 𝑌 ) ↑m ℕ ) ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ℎ ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( ℎ ‘ 𝑗 ) ) ) ) ) ) → ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) ) ) |
534 |
472 521 533
|
syl2anc |
⊢ ( 𝜑 → ( X 𝑘 ∈ 𝑌 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑌 ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) ) ) |
535 |
468 534
|
mpd |
⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) ) |
536 |
|
idd |
⊢ ( 𝜑 → ( ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) ) ) |
537 |
535 536
|
mpd |
⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) ) |
538 |
537
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) ) |
539 |
62
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) |
540 |
539
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) |
541 |
540
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) ) |
542 |
258 541
|
breq12d |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 𝐺 ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ↔ ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ) ) ) |
543 |
538 542
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → 𝐺 ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ) |
544 |
543
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) → 𝐺 ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ) |
545 |
247 249 250 289 544
|
ltletrd |
⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ ) → ( 𝐺 / ( 1 + 𝐸 ) ) < ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ) |
546 |
235 246 545
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ ¬ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) = +∞ ) → ( 𝐺 / ( 1 + 𝐸 ) ) < ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ) |
547 |
234 546
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( 𝐺 / ( 1 + 𝐸 ) ) < ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝑃 ‘ 𝑗 ) ) ) ) |
548 |
207 208 209 210 227 547
|
sge0uzfsumgt |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ∃ 𝑚 ∈ ℕ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) |
549 |
226
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ( 𝐺 / ( 1 + 𝐸 ) ) ∈ ℝ ) |
550 |
|
fzfid |
⊢ ( 𝜑 → ( 1 ... 𝑚 ) ∈ Fin ) |
551 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑚 ) ) → 𝜑 ) |
552 |
|
elfznn |
⊢ ( 𝑗 ∈ ( 1 ... 𝑚 ) → 𝑗 ∈ ℕ ) |
553 |
552
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑚 ) ) → 𝑗 ∈ ℕ ) |
554 |
50 126
|
sseldi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) ∈ ℝ ) |
555 |
551 553 554
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑚 ) ) → ( 𝑃 ‘ 𝑗 ) ∈ ℝ ) |
556 |
550 555
|
fsumrecl |
⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ∈ ℝ ) |
557 |
556
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ∈ ℝ ) |
558 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) |
559 |
549 557 558
|
ltled |
⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ( 𝐺 / ( 1 + 𝐸 ) ) ≤ Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) |
560 |
216
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → 𝐺 ∈ ℝ ) |
561 |
222
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ( 1 + 𝐸 ) ∈ ℝ+ ) |
562 |
560 557 561
|
ledivmuld |
⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ( ( 𝐺 / ( 1 + 𝐸 ) ) ≤ Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ↔ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) ) |
563 |
559 562
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) |
564 |
563
|
ex |
⊢ ( 𝜑 → ( ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) ) |
565 |
564
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) ) |
566 |
565
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) ) |
567 |
566
|
reximdva |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ( ∃ 𝑚 ∈ ℕ ( 𝐺 / ( 1 + 𝐸 ) ) < Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) → ∃ 𝑚 ∈ ℕ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) ) |
568 |
548 567
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ ∅ ) → ∃ 𝑚 ∈ ℕ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) |
569 |
204 206 568
|
syl2anc |
⊢ ( ( 𝜑 ∧ ¬ 𝑌 = ∅ ) → ∃ 𝑚 ∈ ℕ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) |
570 |
203 569
|
pm2.61dan |
⊢ ( 𝜑 → ∃ 𝑚 ∈ ℕ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) |
571 |
2
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝑋 ∈ Fin ) |
572 |
3
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝑌 ⊆ 𝑋 ) |
573 |
4
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝑍 ∈ ( 𝑋 ∖ 𝑌 ) ) |
574 |
6
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝐴 : 𝑊 ⟶ ℝ ) |
575 |
7
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝐵 : 𝑊 ⟶ ℝ ) |
576 |
10
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝐶 : ℕ ⟶ ( ℝ ↑m 𝑊 ) ) |
577 |
|
eqid |
⊢ ( 𝑦 ∈ 𝑌 ↦ 0 ) = ( 𝑦 ∈ 𝑌 ↦ 0 ) |
578 |
|
eqid |
⊢ ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) = ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) |
579 |
12
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝐷 : ℕ ⟶ ( ℝ ↑m 𝑊 ) ) |
580 |
|
eqid |
⊢ ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) = ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) |
581 |
|
fveq2 |
⊢ ( 𝑖 = 𝑗 → ( 𝐶 ‘ 𝑖 ) = ( 𝐶 ‘ 𝑗 ) ) |
582 |
|
fveq2 |
⊢ ( 𝑖 = 𝑗 → ( 𝐷 ‘ 𝑖 ) = ( 𝐷 ‘ 𝑗 ) ) |
583 |
581 582
|
oveq12d |
⊢ ( 𝑖 = 𝑗 → ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑖 ) ) = ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑗 ) ) ) |
584 |
583
|
cbvmptv |
⊢ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑖 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑗 ) ) ) |
585 |
584
|
fveq2i |
⊢ ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑖 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑗 ) ) ) ) |
586 |
585 14
|
eqeltrid |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑖 ) ) ) ) ∈ ℝ ) |
587 |
586
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑖 ) ) ) ) ∈ ℝ ) |
588 |
|
eleq1w |
⊢ ( 𝑗 = 𝑖 → ( 𝑗 ∈ 𝑌 ↔ 𝑖 ∈ 𝑌 ) ) |
589 |
|
fveq2 |
⊢ ( 𝑗 = 𝑖 → ( 𝑐 ‘ 𝑗 ) = ( 𝑐 ‘ 𝑖 ) ) |
590 |
589
|
breq1d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 ↔ ( 𝑐 ‘ 𝑖 ) ≤ 𝑥 ) ) |
591 |
590 589
|
ifbieq1d |
⊢ ( 𝑗 = 𝑖 → if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) = if ( ( 𝑐 ‘ 𝑖 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑖 ) , 𝑥 ) ) |
592 |
588 589 591
|
ifbieq12d |
⊢ ( 𝑗 = 𝑖 → if ( 𝑗 ∈ 𝑌 , ( 𝑐 ‘ 𝑗 ) , if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) ) = if ( 𝑖 ∈ 𝑌 , ( 𝑐 ‘ 𝑖 ) , if ( ( 𝑐 ‘ 𝑖 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑖 ) , 𝑥 ) ) ) |
593 |
592
|
cbvmptv |
⊢ ( 𝑗 ∈ 𝑊 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑐 ‘ 𝑗 ) , if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) ) ) = ( 𝑖 ∈ 𝑊 ↦ if ( 𝑖 ∈ 𝑌 , ( 𝑐 ‘ 𝑖 ) , if ( ( 𝑐 ‘ 𝑖 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑖 ) , 𝑥 ) ) ) |
594 |
593
|
mpteq2i |
⊢ ( 𝑐 ∈ ( ℝ ↑m 𝑊 ) ↦ ( 𝑗 ∈ 𝑊 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑐 ‘ 𝑗 ) , if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) ) ) ) = ( 𝑐 ∈ ( ℝ ↑m 𝑊 ) ↦ ( 𝑖 ∈ 𝑊 ↦ if ( 𝑖 ∈ 𝑌 , ( 𝑐 ‘ 𝑖 ) , if ( ( 𝑐 ‘ 𝑖 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑖 ) , 𝑥 ) ) ) ) |
595 |
594
|
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 ) ↦ ( 𝑖 ∈ 𝑊 ↦ if ( 𝑖 ∈ 𝑌 , ( 𝑐 ‘ 𝑖 ) , if ( ( 𝑐 ‘ 𝑖 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑖 ) , 𝑥 ) ) ) ) ) |
596 |
15 595
|
eqtri |
⊢ 𝐻 = ( 𝑥 ∈ ℝ ↦ ( 𝑐 ∈ ( ℝ ↑m 𝑊 ) ↦ ( 𝑖 ∈ 𝑊 ↦ if ( 𝑖 ∈ 𝑌 , ( 𝑐 ‘ 𝑖 ) , if ( ( 𝑐 ‘ 𝑖 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑖 ) , 𝑥 ) ) ) ) ) |
597 |
17
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝐸 ∈ ℝ+ ) |
598 |
|
fveq2 |
⊢ ( 𝑗 = 𝑖 → ( 𝐶 ‘ 𝑗 ) = ( 𝐶 ‘ 𝑖 ) ) |
599 |
|
fveq2 |
⊢ ( 𝑗 = 𝑖 → ( 𝐷 ‘ 𝑗 ) = ( 𝐷 ‘ 𝑖 ) ) |
600 |
599
|
fveq2d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) = ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) ) |
601 |
598 600
|
oveq12d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) = ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) ) ) |
602 |
601
|
cbvmptv |
⊢ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) = ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) ) ) |
603 |
602
|
fveq2i |
⊢ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) = ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) ) ) ) |
604 |
603
|
oveq2i |
⊢ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) = ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) ) ) ) ) |
605 |
604
|
breq2i |
⊢ ( ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ↔ ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) ) ) ) ) ) |
606 |
605
|
rabbii |
⊢ { 𝑧 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) } = { 𝑧 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) ) ) ) ) } |
607 |
18 606
|
eqtri |
⊢ 𝑈 = { 𝑧 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑖 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑖 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑖 ) ) ) ) ) ) } |
608 |
19
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝑆 ∈ 𝑈 ) |
609 |
20
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝑆 < ( 𝐵 ‘ 𝑍 ) ) |
610 |
|
eqid |
⊢ ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) = ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) |
611 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝑚 ∈ ℕ ) |
612 |
|
id |
⊢ ( 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) |
613 |
|
fveq2 |
⊢ ( 𝑗 = 𝑖 → ( 𝑃 ‘ 𝑗 ) = ( 𝑃 ‘ 𝑖 ) ) |
614 |
613
|
cbvsumv |
⊢ Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) = Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑖 ) |
615 |
614
|
oveq2i |
⊢ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) = ( ( 1 + 𝐸 ) · Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑖 ) ) |
616 |
615
|
a1i |
⊢ ( 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) = ( ( 1 + 𝐸 ) · Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑖 ) ) ) |
617 |
612 616
|
breqtrd |
⊢ ( 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑖 ) ) ) |
618 |
617
|
3ad2ant3 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑖 ) ) ) |
619 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑚 ) ) → 𝜑 ) |
620 |
|
elfznn |
⊢ ( 𝑖 ∈ ( 1 ... 𝑚 ) → 𝑖 ∈ ℕ ) |
621 |
620
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑚 ) ) → 𝑖 ∈ ℕ ) |
622 |
|
eleq1w |
⊢ ( 𝑗 = 𝑖 → ( 𝑗 ∈ ℕ ↔ 𝑖 ∈ ℕ ) ) |
623 |
|
fveq2 |
⊢ ( 𝑗 = 𝑖 → ( 𝐽 ‘ 𝑗 ) = ( 𝐽 ‘ 𝑖 ) ) |
624 |
|
fveq2 |
⊢ ( 𝑗 = 𝑖 → ( 𝐾 ‘ 𝑗 ) = ( 𝐾 ‘ 𝑖 ) ) |
625 |
623 624
|
oveq12d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) = ( ( 𝐽 ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑖 ) ) ) |
626 |
613 625
|
eqeq12d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ↔ ( 𝑃 ‘ 𝑖 ) = ( ( 𝐽 ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑖 ) ) ) ) |
627 |
622 626
|
imbi12d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝑗 ∈ ℕ → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ) ↔ ( 𝑖 ∈ ℕ → ( 𝑃 ‘ 𝑖 ) = ( ( 𝐽 ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑖 ) ) ) ) ) |
628 |
627 62
|
chvarvv |
⊢ ( 𝑖 ∈ ℕ → ( 𝑃 ‘ 𝑖 ) = ( ( 𝐽 ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑖 ) ) ) |
629 |
628
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( 𝑃 ‘ 𝑖 ) = ( ( 𝐽 ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑖 ) ) ) |
630 |
622
|
anbi2d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ↔ ( 𝜑 ∧ 𝑖 ∈ ℕ ) ) ) |
631 |
598
|
fveq1d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ) |
632 |
599
|
fveq1d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) |
633 |
631 632
|
oveq12d |
⊢ ( 𝑗 = 𝑖 → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) |
634 |
633
|
eleq2d |
⊢ ( 𝑗 = 𝑖 → ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ↔ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) ) |
635 |
598
|
reseq1d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) = ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) ) |
636 |
634 635
|
ifbieq1d |
⊢ ( 𝑗 = 𝑖 → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) |
637 |
623 636
|
eqeq12d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝐽 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ↔ ( 𝐽 ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) ) |
638 |
630 637
|
imbi12d |
⊢ ( 𝑗 = 𝑖 → ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐽 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ) ↔ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( 𝐽 ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) ) ) |
639 |
638 161
|
chvarvv |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( 𝐽 ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) |
640 |
599
|
reseq1d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) = ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) ) |
641 |
634 640
|
ifbieq1d |
⊢ ( 𝑗 = 𝑖 → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) |
642 |
624 641
|
eqeq12d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝐾 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ↔ ( 𝐾 ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) ) |
643 |
630 642
|
imbi12d |
⊢ ( 𝑗 = 𝑖 → ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐾 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ) ↔ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( 𝐾 ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) ) ) |
644 |
643 443
|
chvarvv |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( 𝐾 ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) |
645 |
639 644
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝐽 ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑖 ) ) = ( if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ( 𝐿 ‘ 𝑌 ) if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) ) |
646 |
629 645
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( 𝑃 ‘ 𝑖 ) = ( if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ( 𝐿 ‘ 𝑌 ) if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) ) |
647 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → 𝑖 ∈ ℕ ) |
648 |
|
ovexd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ∈ V ) |
649 |
610
|
fvmpt2 |
⊢ ( ( 𝑖 ∈ ℕ ∧ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ∈ V ) → ( ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) ‘ 𝑖 ) = ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) |
650 |
647 648 649
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) ‘ 𝑖 ) = ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) |
651 |
|
fvex |
⊢ ( 𝐶 ‘ 𝑖 ) ∈ V |
652 |
651
|
resex |
⊢ ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) ∈ V |
653 |
652
|
a1i |
⊢ ( 𝜑 → ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) ∈ V ) |
654 |
9 155
|
eqeltrrid |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝑌 ↦ 0 ) ∈ V ) |
655 |
653 654
|
ifcld |
⊢ ( 𝜑 → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ∈ V ) |
656 |
655
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ∈ V ) |
657 |
578
|
fvmpt2 |
⊢ ( ( 𝑖 ∈ ℕ ∧ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ∈ V ) → ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) |
658 |
647 656 657
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) |
659 |
9
|
eqcomi |
⊢ ( 𝑦 ∈ 𝑌 ↦ 0 ) = 𝐹 |
660 |
|
ifeq2 |
⊢ ( ( 𝑦 ∈ 𝑌 ↦ 0 ) = 𝐹 → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) |
661 |
659 660
|
ax-mp |
⊢ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) |
662 |
661
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) |
663 |
658 662
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) |
664 |
|
fvex |
⊢ ( 𝐷 ‘ 𝑖 ) ∈ V |
665 |
664
|
resex |
⊢ ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) ∈ V |
666 |
665
|
a1i |
⊢ ( 𝜑 → ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) ∈ V ) |
667 |
666 654
|
ifcld |
⊢ ( 𝜑 → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ∈ V ) |
668 |
667
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ∈ V ) |
669 |
580
|
fvmpt2 |
⊢ ( ( 𝑖 ∈ ℕ ∧ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ∈ V ) → ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) |
670 |
647 668 669
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) |
671 |
|
biid |
⊢ ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ↔ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) |
672 |
671 659
|
ifbieq2i |
⊢ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) |
673 |
672
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) |
674 |
670 673
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) |
675 |
663 674
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) = ( if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ( 𝐿 ‘ 𝑌 ) if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) ) |
676 |
650 675
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) ‘ 𝑖 ) = ( if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ( 𝐿 ‘ 𝑌 ) if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , 𝐹 ) ) ) |
677 |
646 676
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( 𝑃 ‘ 𝑖 ) = ( ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) ‘ 𝑖 ) ) |
678 |
619 621 677
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑚 ) ) → ( 𝑃 ‘ 𝑖 ) = ( ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) ‘ 𝑖 ) ) |
679 |
678
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑚 ) ) → ( 𝑃 ‘ 𝑖 ) = ( ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) ‘ 𝑖 ) ) |
680 |
679
|
sumeq2dv |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑖 ) = Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) ‘ 𝑖 ) ) |
681 |
680
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → ( ( 1 + 𝐸 ) · Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑖 ) ) = ( ( 1 + 𝐸 ) · Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) ‘ 𝑖 ) ) ) |
682 |
618 681
|
breqtrd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑖 ∈ ( 1 ... 𝑚 ) ( ( 𝑖 ∈ ℕ ↦ ( ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ( 𝐿 ‘ 𝑌 ) ( ( 𝑖 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑖 ) ↾ 𝑌 ) , ( 𝑦 ∈ 𝑌 ↦ 0 ) ) ) ‘ 𝑖 ) ) ) ‘ 𝑖 ) ) ) |
683 |
|
fveq2 |
⊢ ( 𝑗 = ℎ → ( 𝐷 ‘ 𝑗 ) = ( 𝐷 ‘ ℎ ) ) |
684 |
683
|
fveq1d |
⊢ ( 𝑗 = ℎ → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) |
685 |
684
|
cbvmptv |
⊢ ( 𝑗 ∈ { 𝑖 ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( ℎ ∈ { 𝑖 ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) |
686 |
685
|
rneqi |
⊢ ran ( 𝑗 ∈ { 𝑖 ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ran ( ℎ ∈ { 𝑖 ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) |
687 |
|
fveq2 |
⊢ ( ℎ = 𝑖 → ( 𝐶 ‘ ℎ ) = ( 𝐶 ‘ 𝑖 ) ) |
688 |
687
|
fveq1d |
⊢ ( ℎ = 𝑖 → ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) = ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ) |
689 |
|
fveq2 |
⊢ ( ℎ = 𝑖 → ( 𝐷 ‘ ℎ ) = ( 𝐷 ‘ 𝑖 ) ) |
690 |
689
|
fveq1d |
⊢ ( ℎ = 𝑖 → ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) |
691 |
688 690
|
oveq12d |
⊢ ( ℎ = 𝑖 → ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) |
692 |
691
|
eleq2d |
⊢ ( ℎ = 𝑖 → ( 𝑆 ∈ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) ↔ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) ) |
693 |
692
|
cbvrabv |
⊢ { ℎ ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) } = { 𝑖 ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) } |
694 |
693
|
mpteq1i |
⊢ ( 𝑗 ∈ { ℎ ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( 𝑗 ∈ { 𝑖 ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) |
695 |
694
|
rneqi |
⊢ ran ( 𝑗 ∈ { ℎ ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ran ( 𝑗 ∈ { 𝑖 ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) |
696 |
695
|
uneq2i |
⊢ ( { ( 𝐵 ‘ 𝑍 ) } ∪ ran ( 𝑗 ∈ { ℎ ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) = ( { ( 𝐵 ‘ 𝑍 ) } ∪ ran ( 𝑗 ∈ { 𝑖 ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) |
697 |
|
eqid |
⊢ inf ( ( { ( 𝐵 ‘ 𝑍 ) } ∪ ran ( 𝑗 ∈ { ℎ ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) , ℝ , < ) = inf ( ( { ( 𝐵 ‘ 𝑍 ) } ∪ ran ( 𝑗 ∈ { ℎ ∈ ( 1 ... 𝑚 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ ℎ ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ ℎ ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) , ℝ , < ) |
698 |
1 571 572 573 5 574 575 576 577 578 579 580 587 596 16 597 607 608 609 610 611 682 686 696 697
|
hoidmvlelem2 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) ) → ∃ 𝑢 ∈ 𝑈 𝑆 < 𝑢 ) |
699 |
698
|
3exp |
⊢ ( 𝜑 → ( 𝑚 ∈ ℕ → ( 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ∃ 𝑢 ∈ 𝑈 𝑆 < 𝑢 ) ) ) |
700 |
699
|
rexlimdv |
⊢ ( 𝜑 → ( ∃ 𝑚 ∈ ℕ 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑚 ) ( 𝑃 ‘ 𝑗 ) ) → ∃ 𝑢 ∈ 𝑈 𝑆 < 𝑢 ) ) |
701 |
570 700
|
mpd |
⊢ ( 𝜑 → ∃ 𝑢 ∈ 𝑈 𝑆 < 𝑢 ) |