Step |
Hyp |
Ref |
Expression |
1 |
|
nmoco.1 |
⊢ 𝑁 = ( 𝑆 normOp 𝑈 ) |
2 |
|
nmoco.2 |
⊢ 𝐿 = ( 𝑇 normOp 𝑈 ) |
3 |
|
nmoco.3 |
⊢ 𝑀 = ( 𝑆 normOp 𝑇 ) |
4 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
5 |
|
eqid |
⊢ ( norm ‘ 𝑆 ) = ( norm ‘ 𝑆 ) |
6 |
|
eqid |
⊢ ( norm ‘ 𝑈 ) = ( norm ‘ 𝑈 ) |
7 |
|
eqid |
⊢ ( 0g ‘ 𝑆 ) = ( 0g ‘ 𝑆 ) |
8 |
|
nghmrcl1 |
⊢ ( 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) → 𝑆 ∈ NrmGrp ) |
9 |
8
|
adantl |
⊢ ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) → 𝑆 ∈ NrmGrp ) |
10 |
|
nghmrcl2 |
⊢ ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) → 𝑈 ∈ NrmGrp ) |
11 |
10
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) → 𝑈 ∈ NrmGrp ) |
12 |
|
nghmghm |
⊢ ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) → 𝐹 ∈ ( 𝑇 GrpHom 𝑈 ) ) |
13 |
|
nghmghm |
⊢ ( 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) → 𝐺 ∈ ( 𝑆 GrpHom 𝑇 ) ) |
14 |
|
ghmco |
⊢ ( ( 𝐹 ∈ ( 𝑇 GrpHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 GrpHom 𝑇 ) ) → ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑆 GrpHom 𝑈 ) ) |
15 |
12 13 14
|
syl2an |
⊢ ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) → ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑆 GrpHom 𝑈 ) ) |
16 |
2
|
nghmcl |
⊢ ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) → ( 𝐿 ‘ 𝐹 ) ∈ ℝ ) |
17 |
3
|
nghmcl |
⊢ ( 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) → ( 𝑀 ‘ 𝐺 ) ∈ ℝ ) |
18 |
|
remulcl |
⊢ ( ( ( 𝐿 ‘ 𝐹 ) ∈ ℝ ∧ ( 𝑀 ‘ 𝐺 ) ∈ ℝ ) → ( ( 𝐿 ‘ 𝐹 ) · ( 𝑀 ‘ 𝐺 ) ) ∈ ℝ ) |
19 |
16 17 18
|
syl2an |
⊢ ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) → ( ( 𝐿 ‘ 𝐹 ) · ( 𝑀 ‘ 𝐺 ) ) ∈ ℝ ) |
20 |
|
nghmrcl1 |
⊢ ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) → 𝑇 ∈ NrmGrp ) |
21 |
2
|
nmoge0 |
⊢ ( ( 𝑇 ∈ NrmGrp ∧ 𝑈 ∈ NrmGrp ∧ 𝐹 ∈ ( 𝑇 GrpHom 𝑈 ) ) → 0 ≤ ( 𝐿 ‘ 𝐹 ) ) |
22 |
20 10 12 21
|
syl3anc |
⊢ ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) → 0 ≤ ( 𝐿 ‘ 𝐹 ) ) |
23 |
16 22
|
jca |
⊢ ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) → ( ( 𝐿 ‘ 𝐹 ) ∈ ℝ ∧ 0 ≤ ( 𝐿 ‘ 𝐹 ) ) ) |
24 |
|
nghmrcl2 |
⊢ ( 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) → 𝑇 ∈ NrmGrp ) |
25 |
3
|
nmoge0 |
⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐺 ∈ ( 𝑆 GrpHom 𝑇 ) ) → 0 ≤ ( 𝑀 ‘ 𝐺 ) ) |
26 |
8 24 13 25
|
syl3anc |
⊢ ( 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) → 0 ≤ ( 𝑀 ‘ 𝐺 ) ) |
27 |
17 26
|
jca |
⊢ ( 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) → ( ( 𝑀 ‘ 𝐺 ) ∈ ℝ ∧ 0 ≤ ( 𝑀 ‘ 𝐺 ) ) ) |
28 |
|
mulge0 |
⊢ ( ( ( ( 𝐿 ‘ 𝐹 ) ∈ ℝ ∧ 0 ≤ ( 𝐿 ‘ 𝐹 ) ) ∧ ( ( 𝑀 ‘ 𝐺 ) ∈ ℝ ∧ 0 ≤ ( 𝑀 ‘ 𝐺 ) ) ) → 0 ≤ ( ( 𝐿 ‘ 𝐹 ) · ( 𝑀 ‘ 𝐺 ) ) ) |
29 |
23 27 28
|
syl2an |
⊢ ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) → 0 ≤ ( ( 𝐿 ‘ 𝐹 ) · ( 𝑀 ‘ 𝐺 ) ) ) |
30 |
10
|
ad2antrr |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → 𝑈 ∈ NrmGrp ) |
31 |
12
|
ad2antrr |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → 𝐹 ∈ ( 𝑇 GrpHom 𝑈 ) ) |
32 |
|
eqid |
⊢ ( Base ‘ 𝑇 ) = ( Base ‘ 𝑇 ) |
33 |
|
eqid |
⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 ) |
34 |
32 33
|
ghmf |
⊢ ( 𝐹 ∈ ( 𝑇 GrpHom 𝑈 ) → 𝐹 : ( Base ‘ 𝑇 ) ⟶ ( Base ‘ 𝑈 ) ) |
35 |
31 34
|
syl |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → 𝐹 : ( Base ‘ 𝑇 ) ⟶ ( Base ‘ 𝑈 ) ) |
36 |
13
|
ad2antlr |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → 𝐺 ∈ ( 𝑆 GrpHom 𝑇 ) ) |
37 |
4 32
|
ghmf |
⊢ ( 𝐺 ∈ ( 𝑆 GrpHom 𝑇 ) → 𝐺 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ) |
38 |
36 37
|
syl |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → 𝐺 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ) |
39 |
|
simprl |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → 𝑥 ∈ ( Base ‘ 𝑆 ) ) |
40 |
38 39
|
ffvelrnd |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( 𝐺 ‘ 𝑥 ) ∈ ( Base ‘ 𝑇 ) ) |
41 |
35 40
|
ffvelrnd |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑥 ) ) ∈ ( Base ‘ 𝑈 ) ) |
42 |
33 6
|
nmcl |
⊢ ( ( 𝑈 ∈ NrmGrp ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑥 ) ) ∈ ( Base ‘ 𝑈 ) ) → ( ( norm ‘ 𝑈 ) ‘ ( 𝐹 ‘ ( 𝐺 ‘ 𝑥 ) ) ) ∈ ℝ ) |
43 |
30 41 42
|
syl2anc |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( ( norm ‘ 𝑈 ) ‘ ( 𝐹 ‘ ( 𝐺 ‘ 𝑥 ) ) ) ∈ ℝ ) |
44 |
16
|
ad2antrr |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( 𝐿 ‘ 𝐹 ) ∈ ℝ ) |
45 |
20
|
ad2antrr |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → 𝑇 ∈ NrmGrp ) |
46 |
|
eqid |
⊢ ( norm ‘ 𝑇 ) = ( norm ‘ 𝑇 ) |
47 |
32 46
|
nmcl |
⊢ ( ( 𝑇 ∈ NrmGrp ∧ ( 𝐺 ‘ 𝑥 ) ∈ ( Base ‘ 𝑇 ) ) → ( ( norm ‘ 𝑇 ) ‘ ( 𝐺 ‘ 𝑥 ) ) ∈ ℝ ) |
48 |
45 40 47
|
syl2anc |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( ( norm ‘ 𝑇 ) ‘ ( 𝐺 ‘ 𝑥 ) ) ∈ ℝ ) |
49 |
44 48
|
remulcld |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( ( 𝐿 ‘ 𝐹 ) · ( ( norm ‘ 𝑇 ) ‘ ( 𝐺 ‘ 𝑥 ) ) ) ∈ ℝ ) |
50 |
17
|
ad2antlr |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( 𝑀 ‘ 𝐺 ) ∈ ℝ ) |
51 |
4 5
|
nmcl |
⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ∈ ℝ ) |
52 |
8 51
|
sylan |
⊢ ( ( 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ∈ ℝ ) |
53 |
52
|
ad2ant2lr |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ∈ ℝ ) |
54 |
50 53
|
remulcld |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( ( 𝑀 ‘ 𝐺 ) · ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ) ∈ ℝ ) |
55 |
44 54
|
remulcld |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( ( 𝐿 ‘ 𝐹 ) · ( ( 𝑀 ‘ 𝐺 ) · ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ) ) ∈ ℝ ) |
56 |
|
simpll |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ) |
57 |
2 32 46 6
|
nmoi |
⊢ ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ ( 𝐺 ‘ 𝑥 ) ∈ ( Base ‘ 𝑇 ) ) → ( ( norm ‘ 𝑈 ) ‘ ( 𝐹 ‘ ( 𝐺 ‘ 𝑥 ) ) ) ≤ ( ( 𝐿 ‘ 𝐹 ) · ( ( norm ‘ 𝑇 ) ‘ ( 𝐺 ‘ 𝑥 ) ) ) ) |
58 |
56 40 57
|
syl2anc |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( ( norm ‘ 𝑈 ) ‘ ( 𝐹 ‘ ( 𝐺 ‘ 𝑥 ) ) ) ≤ ( ( 𝐿 ‘ 𝐹 ) · ( ( norm ‘ 𝑇 ) ‘ ( 𝐺 ‘ 𝑥 ) ) ) ) |
59 |
23
|
ad2antrr |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( ( 𝐿 ‘ 𝐹 ) ∈ ℝ ∧ 0 ≤ ( 𝐿 ‘ 𝐹 ) ) ) |
60 |
3 4 5 46
|
nmoi |
⊢ ( ( 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → ( ( norm ‘ 𝑇 ) ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ ( ( 𝑀 ‘ 𝐺 ) · ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ) ) |
61 |
60
|
ad2ant2lr |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( ( norm ‘ 𝑇 ) ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ ( ( 𝑀 ‘ 𝐺 ) · ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ) ) |
62 |
|
lemul2a |
⊢ ( ( ( ( ( norm ‘ 𝑇 ) ‘ ( 𝐺 ‘ 𝑥 ) ) ∈ ℝ ∧ ( ( 𝑀 ‘ 𝐺 ) · ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ) ∈ ℝ ∧ ( ( 𝐿 ‘ 𝐹 ) ∈ ℝ ∧ 0 ≤ ( 𝐿 ‘ 𝐹 ) ) ) ∧ ( ( norm ‘ 𝑇 ) ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ ( ( 𝑀 ‘ 𝐺 ) · ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ) ) → ( ( 𝐿 ‘ 𝐹 ) · ( ( norm ‘ 𝑇 ) ‘ ( 𝐺 ‘ 𝑥 ) ) ) ≤ ( ( 𝐿 ‘ 𝐹 ) · ( ( 𝑀 ‘ 𝐺 ) · ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ) ) ) |
63 |
48 54 59 61 62
|
syl31anc |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( ( 𝐿 ‘ 𝐹 ) · ( ( norm ‘ 𝑇 ) ‘ ( 𝐺 ‘ 𝑥 ) ) ) ≤ ( ( 𝐿 ‘ 𝐹 ) · ( ( 𝑀 ‘ 𝐺 ) · ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ) ) ) |
64 |
43 49 55 58 63
|
letrd |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( ( norm ‘ 𝑈 ) ‘ ( 𝐹 ‘ ( 𝐺 ‘ 𝑥 ) ) ) ≤ ( ( 𝐿 ‘ 𝐹 ) · ( ( 𝑀 ‘ 𝐺 ) · ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ) ) ) |
65 |
|
fvco3 |
⊢ ( ( 𝐺 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑥 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑥 ) ) ) |
66 |
38 39 65
|
syl2anc |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑥 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑥 ) ) ) |
67 |
66
|
fveq2d |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( ( norm ‘ 𝑈 ) ‘ ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑥 ) ) = ( ( norm ‘ 𝑈 ) ‘ ( 𝐹 ‘ ( 𝐺 ‘ 𝑥 ) ) ) ) |
68 |
44
|
recnd |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( 𝐿 ‘ 𝐹 ) ∈ ℂ ) |
69 |
50
|
recnd |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( 𝑀 ‘ 𝐺 ) ∈ ℂ ) |
70 |
53
|
recnd |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ∈ ℂ ) |
71 |
68 69 70
|
mulassd |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( ( ( 𝐿 ‘ 𝐹 ) · ( 𝑀 ‘ 𝐺 ) ) · ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ) = ( ( 𝐿 ‘ 𝐹 ) · ( ( 𝑀 ‘ 𝐺 ) · ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ) ) ) |
72 |
64 67 71
|
3brtr4d |
⊢ ( ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑆 ) ) ) → ( ( norm ‘ 𝑈 ) ‘ ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑥 ) ) ≤ ( ( ( 𝐿 ‘ 𝐹 ) · ( 𝑀 ‘ 𝐺 ) ) · ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ) ) |
73 |
1 4 5 6 7 9 11 15 19 29 72
|
nmolb2d |
⊢ ( ( 𝐹 ∈ ( 𝑇 NGHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 NGHom 𝑇 ) ) → ( 𝑁 ‘ ( 𝐹 ∘ 𝐺 ) ) ≤ ( ( 𝐿 ‘ 𝐹 ) · ( 𝑀 ‘ 𝐺 ) ) ) |