Step |
Hyp |
Ref |
Expression |
1 |
|
fllogbd.b |
⊢ ( 𝜑 → 𝐵 ∈ ( ℤ≥ ‘ 2 ) ) |
2 |
|
fllogbd.x |
⊢ ( 𝜑 → 𝑋 ∈ ℝ+ ) |
3 |
|
fllogbd.e |
⊢ 𝐸 = ( ⌊ ‘ ( 𝐵 logb 𝑋 ) ) |
4 |
|
relogbzcl |
⊢ ( ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑋 ∈ ℝ+ ) → ( 𝐵 logb 𝑋 ) ∈ ℝ ) |
5 |
1 2 4
|
syl2anc |
⊢ ( 𝜑 → ( 𝐵 logb 𝑋 ) ∈ ℝ ) |
6 |
|
flle |
⊢ ( ( 𝐵 logb 𝑋 ) ∈ ℝ → ( ⌊ ‘ ( 𝐵 logb 𝑋 ) ) ≤ ( 𝐵 logb 𝑋 ) ) |
7 |
5 6
|
syl |
⊢ ( 𝜑 → ( ⌊ ‘ ( 𝐵 logb 𝑋 ) ) ≤ ( 𝐵 logb 𝑋 ) ) |
8 |
3 7
|
eqbrtrid |
⊢ ( 𝜑 → 𝐸 ≤ ( 𝐵 logb 𝑋 ) ) |
9 |
|
eluzelz |
⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) → 𝐵 ∈ ℤ ) |
10 |
1 9
|
syl |
⊢ ( 𝜑 → 𝐵 ∈ ℤ ) |
11 |
10
|
zred |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
12 |
|
eluz2b1 |
⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ↔ ( 𝐵 ∈ ℤ ∧ 1 < 𝐵 ) ) |
13 |
12
|
simprbi |
⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) → 1 < 𝐵 ) |
14 |
1 13
|
syl |
⊢ ( 𝜑 → 1 < 𝐵 ) |
15 |
5
|
flcld |
⊢ ( 𝜑 → ( ⌊ ‘ ( 𝐵 logb 𝑋 ) ) ∈ ℤ ) |
16 |
3 15
|
eqeltrid |
⊢ ( 𝜑 → 𝐸 ∈ ℤ ) |
17 |
16
|
zred |
⊢ ( 𝜑 → 𝐸 ∈ ℝ ) |
18 |
11 14 17 5
|
cxpled |
⊢ ( 𝜑 → ( 𝐸 ≤ ( 𝐵 logb 𝑋 ) ↔ ( 𝐵 ↑𝑐 𝐸 ) ≤ ( 𝐵 ↑𝑐 ( 𝐵 logb 𝑋 ) ) ) ) |
19 |
8 18
|
mpbid |
⊢ ( 𝜑 → ( 𝐵 ↑𝑐 𝐸 ) ≤ ( 𝐵 ↑𝑐 ( 𝐵 logb 𝑋 ) ) ) |
20 |
10
|
zcnd |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
21 |
|
eluz2nn |
⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) → 𝐵 ∈ ℕ ) |
22 |
1 21
|
syl |
⊢ ( 𝜑 → 𝐵 ∈ ℕ ) |
23 |
22
|
nnne0d |
⊢ ( 𝜑 → 𝐵 ≠ 0 ) |
24 |
20 23 16
|
cxpexpzd |
⊢ ( 𝜑 → ( 𝐵 ↑𝑐 𝐸 ) = ( 𝐵 ↑ 𝐸 ) ) |
25 |
|
eluz2cnn0n1 |
⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) → 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ) |
26 |
1 25
|
syl |
⊢ ( 𝜑 → 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ) |
27 |
|
rpcnne0 |
⊢ ( 𝑋 ∈ ℝ+ → ( 𝑋 ∈ ℂ ∧ 𝑋 ≠ 0 ) ) |
28 |
|
eldifsn |
⊢ ( 𝑋 ∈ ( ℂ ∖ { 0 } ) ↔ ( 𝑋 ∈ ℂ ∧ 𝑋 ≠ 0 ) ) |
29 |
27 28
|
sylibr |
⊢ ( 𝑋 ∈ ℝ+ → 𝑋 ∈ ( ℂ ∖ { 0 } ) ) |
30 |
2 29
|
syl |
⊢ ( 𝜑 → 𝑋 ∈ ( ℂ ∖ { 0 } ) ) |
31 |
|
cxplogb |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝐵 ↑𝑐 ( 𝐵 logb 𝑋 ) ) = 𝑋 ) |
32 |
26 30 31
|
syl2anc |
⊢ ( 𝜑 → ( 𝐵 ↑𝑐 ( 𝐵 logb 𝑋 ) ) = 𝑋 ) |
33 |
19 24 32
|
3brtr3d |
⊢ ( 𝜑 → ( 𝐵 ↑ 𝐸 ) ≤ 𝑋 ) |
34 |
|
flltp1 |
⊢ ( ( 𝐵 logb 𝑋 ) ∈ ℝ → ( 𝐵 logb 𝑋 ) < ( ( ⌊ ‘ ( 𝐵 logb 𝑋 ) ) + 1 ) ) |
35 |
5 34
|
syl |
⊢ ( 𝜑 → ( 𝐵 logb 𝑋 ) < ( ( ⌊ ‘ ( 𝐵 logb 𝑋 ) ) + 1 ) ) |
36 |
3
|
a1i |
⊢ ( 𝜑 → 𝐸 = ( ⌊ ‘ ( 𝐵 logb 𝑋 ) ) ) |
37 |
36
|
oveq1d |
⊢ ( 𝜑 → ( 𝐸 + 1 ) = ( ( ⌊ ‘ ( 𝐵 logb 𝑋 ) ) + 1 ) ) |
38 |
35 37
|
breqtrrd |
⊢ ( 𝜑 → ( 𝐵 logb 𝑋 ) < ( 𝐸 + 1 ) ) |
39 |
16
|
peano2zd |
⊢ ( 𝜑 → ( 𝐸 + 1 ) ∈ ℤ ) |
40 |
39
|
zred |
⊢ ( 𝜑 → ( 𝐸 + 1 ) ∈ ℝ ) |
41 |
11 14 5 40
|
cxpltd |
⊢ ( 𝜑 → ( ( 𝐵 logb 𝑋 ) < ( 𝐸 + 1 ) ↔ ( 𝐵 ↑𝑐 ( 𝐵 logb 𝑋 ) ) < ( 𝐵 ↑𝑐 ( 𝐸 + 1 ) ) ) ) |
42 |
38 41
|
mpbid |
⊢ ( 𝜑 → ( 𝐵 ↑𝑐 ( 𝐵 logb 𝑋 ) ) < ( 𝐵 ↑𝑐 ( 𝐸 + 1 ) ) ) |
43 |
20 23 39
|
cxpexpzd |
⊢ ( 𝜑 → ( 𝐵 ↑𝑐 ( 𝐸 + 1 ) ) = ( 𝐵 ↑ ( 𝐸 + 1 ) ) ) |
44 |
42 32 43
|
3brtr3d |
⊢ ( 𝜑 → 𝑋 < ( 𝐵 ↑ ( 𝐸 + 1 ) ) ) |
45 |
33 44
|
jca |
⊢ ( 𝜑 → ( ( 𝐵 ↑ 𝐸 ) ≤ 𝑋 ∧ 𝑋 < ( 𝐵 ↑ ( 𝐸 + 1 ) ) ) ) |