Step |
Hyp |
Ref |
Expression |
1 |
|
amgm2d.0 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ+ ) |
2 |
|
amgm2d.1 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ+ ) |
3 |
|
eqid |
⊢ ( mulGrp ‘ ℂfld ) = ( mulGrp ‘ ℂfld ) |
4 |
|
fzofi |
⊢ ( 0 ..^ 2 ) ∈ Fin |
5 |
4
|
a1i |
⊢ ( 𝜑 → ( 0 ..^ 2 ) ∈ Fin ) |
6 |
|
2nn |
⊢ 2 ∈ ℕ |
7 |
|
lbfzo0 |
⊢ ( 0 ∈ ( 0 ..^ 2 ) ↔ 2 ∈ ℕ ) |
8 |
6 7
|
mpbir |
⊢ 0 ∈ ( 0 ..^ 2 ) |
9 |
8
|
ne0ii |
⊢ ( 0 ..^ 2 ) ≠ ∅ |
10 |
9
|
a1i |
⊢ ( 𝜑 → ( 0 ..^ 2 ) ≠ ∅ ) |
11 |
1 2
|
s2cld |
⊢ ( 𝜑 → 〈“ 𝐴 𝐵 ”〉 ∈ Word ℝ+ ) |
12 |
|
wrdf |
⊢ ( 〈“ 𝐴 𝐵 ”〉 ∈ Word ℝ+ → 〈“ 𝐴 𝐵 ”〉 : ( 0 ..^ ( ♯ ‘ 〈“ 𝐴 𝐵 ”〉 ) ) ⟶ ℝ+ ) |
13 |
|
s2len |
⊢ ( ♯ ‘ 〈“ 𝐴 𝐵 ”〉 ) = 2 |
14 |
13
|
eqcomi |
⊢ 2 = ( ♯ ‘ 〈“ 𝐴 𝐵 ”〉 ) |
15 |
14
|
oveq2i |
⊢ ( 0 ..^ 2 ) = ( 0 ..^ ( ♯ ‘ 〈“ 𝐴 𝐵 ”〉 ) ) |
16 |
15
|
feq2i |
⊢ ( 〈“ 𝐴 𝐵 ”〉 : ( 0 ..^ 2 ) ⟶ ℝ+ ↔ 〈“ 𝐴 𝐵 ”〉 : ( 0 ..^ ( ♯ ‘ 〈“ 𝐴 𝐵 ”〉 ) ) ⟶ ℝ+ ) |
17 |
12 16
|
sylibr |
⊢ ( 〈“ 𝐴 𝐵 ”〉 ∈ Word ℝ+ → 〈“ 𝐴 𝐵 ”〉 : ( 0 ..^ 2 ) ⟶ ℝ+ ) |
18 |
11 17
|
syl |
⊢ ( 𝜑 → 〈“ 𝐴 𝐵 ”〉 : ( 0 ..^ 2 ) ⟶ ℝ+ ) |
19 |
3 5 10 18
|
amgmlem |
⊢ ( 𝜑 → ( ( ( mulGrp ‘ ℂfld ) Σg 〈“ 𝐴 𝐵 ”〉 ) ↑𝑐 ( 1 / ( ♯ ‘ ( 0 ..^ 2 ) ) ) ) ≤ ( ( ℂfld Σg 〈“ 𝐴 𝐵 ”〉 ) / ( ♯ ‘ ( 0 ..^ 2 ) ) ) ) |
20 |
|
cnring |
⊢ ℂfld ∈ Ring |
21 |
3
|
ringmgp |
⊢ ( ℂfld ∈ Ring → ( mulGrp ‘ ℂfld ) ∈ Mnd ) |
22 |
20 21
|
mp1i |
⊢ ( 𝜑 → ( mulGrp ‘ ℂfld ) ∈ Mnd ) |
23 |
1
|
rpcnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
24 |
2
|
rpcnd |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
25 |
|
cnfldbas |
⊢ ℂ = ( Base ‘ ℂfld ) |
26 |
3 25
|
mgpbas |
⊢ ℂ = ( Base ‘ ( mulGrp ‘ ℂfld ) ) |
27 |
|
cnfldmul |
⊢ · = ( .r ‘ ℂfld ) |
28 |
3 27
|
mgpplusg |
⊢ · = ( +g ‘ ( mulGrp ‘ ℂfld ) ) |
29 |
26 28
|
gsumws2 |
⊢ ( ( ( mulGrp ‘ ℂfld ) ∈ Mnd ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( mulGrp ‘ ℂfld ) Σg 〈“ 𝐴 𝐵 ”〉 ) = ( 𝐴 · 𝐵 ) ) |
30 |
22 23 24 29
|
syl3anc |
⊢ ( 𝜑 → ( ( mulGrp ‘ ℂfld ) Σg 〈“ 𝐴 𝐵 ”〉 ) = ( 𝐴 · 𝐵 ) ) |
31 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
32 |
|
hashfzo0 |
⊢ ( 2 ∈ ℕ0 → ( ♯ ‘ ( 0 ..^ 2 ) ) = 2 ) |
33 |
31 32
|
mp1i |
⊢ ( 𝜑 → ( ♯ ‘ ( 0 ..^ 2 ) ) = 2 ) |
34 |
33
|
oveq2d |
⊢ ( 𝜑 → ( 1 / ( ♯ ‘ ( 0 ..^ 2 ) ) ) = ( 1 / 2 ) ) |
35 |
30 34
|
oveq12d |
⊢ ( 𝜑 → ( ( ( mulGrp ‘ ℂfld ) Σg 〈“ 𝐴 𝐵 ”〉 ) ↑𝑐 ( 1 / ( ♯ ‘ ( 0 ..^ 2 ) ) ) ) = ( ( 𝐴 · 𝐵 ) ↑𝑐 ( 1 / 2 ) ) ) |
36 |
|
ringmnd |
⊢ ( ℂfld ∈ Ring → ℂfld ∈ Mnd ) |
37 |
20 36
|
mp1i |
⊢ ( 𝜑 → ℂfld ∈ Mnd ) |
38 |
|
cnfldadd |
⊢ + = ( +g ‘ ℂfld ) |
39 |
25 38
|
gsumws2 |
⊢ ( ( ℂfld ∈ Mnd ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℂfld Σg 〈“ 𝐴 𝐵 ”〉 ) = ( 𝐴 + 𝐵 ) ) |
40 |
37 23 24 39
|
syl3anc |
⊢ ( 𝜑 → ( ℂfld Σg 〈“ 𝐴 𝐵 ”〉 ) = ( 𝐴 + 𝐵 ) ) |
41 |
40 33
|
oveq12d |
⊢ ( 𝜑 → ( ( ℂfld Σg 〈“ 𝐴 𝐵 ”〉 ) / ( ♯ ‘ ( 0 ..^ 2 ) ) ) = ( ( 𝐴 + 𝐵 ) / 2 ) ) |
42 |
19 35 41
|
3brtr3d |
⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) ↑𝑐 ( 1 / 2 ) ) ≤ ( ( 𝐴 + 𝐵 ) / 2 ) ) |