Step |
Hyp |
Ref |
Expression |
1 |
|
lflnegcl.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lflnegcl.r |
⊢ 𝑅 = ( Scalar ‘ 𝑊 ) |
3 |
|
lflnegcl.i |
⊢ 𝐼 = ( invg ‘ 𝑅 ) |
4 |
|
lflnegcl.n |
⊢ 𝑁 = ( 𝑥 ∈ 𝑉 ↦ ( 𝐼 ‘ ( 𝐺 ‘ 𝑥 ) ) ) |
5 |
|
lflnegcl.f |
⊢ 𝐹 = ( LFnl ‘ 𝑊 ) |
6 |
|
lflnegcl.w |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
7 |
|
lflnegcl.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
8 |
2
|
lmodring |
⊢ ( 𝑊 ∈ LMod → 𝑅 ∈ Ring ) |
9 |
6 8
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
10 |
|
ringgrp |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp ) |
11 |
9 10
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Grp ) |
12 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → 𝑅 ∈ Grp ) |
13 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → 𝑊 ∈ LMod ) |
14 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → 𝐺 ∈ 𝐹 ) |
15 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → 𝑥 ∈ 𝑉 ) |
16 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
17 |
2 16 1 5
|
lflcl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑥 ∈ 𝑉 ) → ( 𝐺 ‘ 𝑥 ) ∈ ( Base ‘ 𝑅 ) ) |
18 |
13 14 15 17
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → ( 𝐺 ‘ 𝑥 ) ∈ ( Base ‘ 𝑅 ) ) |
19 |
16 3
|
grpinvcl |
⊢ ( ( 𝑅 ∈ Grp ∧ ( 𝐺 ‘ 𝑥 ) ∈ ( Base ‘ 𝑅 ) ) → ( 𝐼 ‘ ( 𝐺 ‘ 𝑥 ) ) ∈ ( Base ‘ 𝑅 ) ) |
20 |
12 18 19
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → ( 𝐼 ‘ ( 𝐺 ‘ 𝑥 ) ) ∈ ( Base ‘ 𝑅 ) ) |
21 |
20 4
|
fmptd |
⊢ ( 𝜑 → 𝑁 : 𝑉 ⟶ ( Base ‘ 𝑅 ) ) |
22 |
|
ringabl |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Abel ) |
23 |
9 22
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Abel ) |
24 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝑅 ∈ Abel ) |
25 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝑅 ∈ Ring ) |
26 |
|
simpr1 |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝑘 ∈ ( Base ‘ 𝑅 ) ) |
27 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝑊 ∈ LMod ) |
28 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝐺 ∈ 𝐹 ) |
29 |
|
simpr2 |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝑦 ∈ 𝑉 ) |
30 |
2 16 1 5
|
lflcl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑦 ∈ 𝑉 ) → ( 𝐺 ‘ 𝑦 ) ∈ ( Base ‘ 𝑅 ) ) |
31 |
27 28 29 30
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ ( Base ‘ 𝑅 ) ) |
32 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
33 |
16 32
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ ( Base ‘ 𝑅 ) ) → ( 𝑘 ( .r ‘ 𝑅 ) ( 𝐺 ‘ 𝑦 ) ) ∈ ( Base ‘ 𝑅 ) ) |
34 |
25 26 31 33
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝑘 ( .r ‘ 𝑅 ) ( 𝐺 ‘ 𝑦 ) ) ∈ ( Base ‘ 𝑅 ) ) |
35 |
|
simpr3 |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝑧 ∈ 𝑉 ) |
36 |
2 16 1 5
|
lflcl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑧 ∈ 𝑉 ) → ( 𝐺 ‘ 𝑧 ) ∈ ( Base ‘ 𝑅 ) ) |
37 |
27 28 35 36
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ ( Base ‘ 𝑅 ) ) |
38 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
39 |
16 38 3
|
ablinvadd |
⊢ ( ( 𝑅 ∈ Abel ∧ ( 𝑘 ( .r ‘ 𝑅 ) ( 𝐺 ‘ 𝑦 ) ) ∈ ( Base ‘ 𝑅 ) ∧ ( 𝐺 ‘ 𝑧 ) ∈ ( Base ‘ 𝑅 ) ) → ( 𝐼 ‘ ( ( 𝑘 ( .r ‘ 𝑅 ) ( 𝐺 ‘ 𝑦 ) ) ( +g ‘ 𝑅 ) ( 𝐺 ‘ 𝑧 ) ) ) = ( ( 𝐼 ‘ ( 𝑘 ( .r ‘ 𝑅 ) ( 𝐺 ‘ 𝑦 ) ) ) ( +g ‘ 𝑅 ) ( 𝐼 ‘ ( 𝐺 ‘ 𝑧 ) ) ) ) |
40 |
24 34 37 39
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝐼 ‘ ( ( 𝑘 ( .r ‘ 𝑅 ) ( 𝐺 ‘ 𝑦 ) ) ( +g ‘ 𝑅 ) ( 𝐺 ‘ 𝑧 ) ) ) = ( ( 𝐼 ‘ ( 𝑘 ( .r ‘ 𝑅 ) ( 𝐺 ‘ 𝑦 ) ) ) ( +g ‘ 𝑅 ) ( 𝐼 ‘ ( 𝐺 ‘ 𝑧 ) ) ) ) |
41 |
|
eqid |
⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 ) |
42 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 ) |
43 |
1 41 2 42 16 38 32 5
|
lfli |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝐺 ‘ ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ) = ( ( 𝑘 ( .r ‘ 𝑅 ) ( 𝐺 ‘ 𝑦 ) ) ( +g ‘ 𝑅 ) ( 𝐺 ‘ 𝑧 ) ) ) |
44 |
27 28 26 29 35 43
|
syl113anc |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝐺 ‘ ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ) = ( ( 𝑘 ( .r ‘ 𝑅 ) ( 𝐺 ‘ 𝑦 ) ) ( +g ‘ 𝑅 ) ( 𝐺 ‘ 𝑧 ) ) ) |
45 |
44
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝐼 ‘ ( 𝐺 ‘ ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ) ) = ( 𝐼 ‘ ( ( 𝑘 ( .r ‘ 𝑅 ) ( 𝐺 ‘ 𝑦 ) ) ( +g ‘ 𝑅 ) ( 𝐺 ‘ 𝑧 ) ) ) ) |
46 |
16 32 3 25 26 31
|
ringmneg2 |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝑘 ( .r ‘ 𝑅 ) ( 𝐼 ‘ ( 𝐺 ‘ 𝑦 ) ) ) = ( 𝐼 ‘ ( 𝑘 ( .r ‘ 𝑅 ) ( 𝐺 ‘ 𝑦 ) ) ) ) |
47 |
46
|
oveq1d |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( 𝑘 ( .r ‘ 𝑅 ) ( 𝐼 ‘ ( 𝐺 ‘ 𝑦 ) ) ) ( +g ‘ 𝑅 ) ( 𝐼 ‘ ( 𝐺 ‘ 𝑧 ) ) ) = ( ( 𝐼 ‘ ( 𝑘 ( .r ‘ 𝑅 ) ( 𝐺 ‘ 𝑦 ) ) ) ( +g ‘ 𝑅 ) ( 𝐼 ‘ ( 𝐺 ‘ 𝑧 ) ) ) ) |
48 |
40 45 47
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝐼 ‘ ( 𝐺 ‘ ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ) ) = ( ( 𝑘 ( .r ‘ 𝑅 ) ( 𝐼 ‘ ( 𝐺 ‘ 𝑦 ) ) ) ( +g ‘ 𝑅 ) ( 𝐼 ‘ ( 𝐺 ‘ 𝑧 ) ) ) ) |
49 |
1 2 42 16
|
lmodvscl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ) → ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ∈ 𝑉 ) |
50 |
27 26 29 49
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ∈ 𝑉 ) |
51 |
1 41
|
lmodvacl |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) → ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ∈ 𝑉 ) |
52 |
27 50 35 51
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ∈ 𝑉 ) |
53 |
|
2fveq3 |
⊢ ( 𝑥 = ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) → ( 𝐼 ‘ ( 𝐺 ‘ 𝑥 ) ) = ( 𝐼 ‘ ( 𝐺 ‘ ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ) ) ) |
54 |
|
fvex |
⊢ ( 𝐼 ‘ ( 𝐺 ‘ ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ) ) ∈ V |
55 |
53 4 54
|
fvmpt |
⊢ ( ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ∈ 𝑉 → ( 𝑁 ‘ ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ) = ( 𝐼 ‘ ( 𝐺 ‘ ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ) ) ) |
56 |
52 55
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝑁 ‘ ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ) = ( 𝐼 ‘ ( 𝐺 ‘ ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ) ) ) |
57 |
|
2fveq3 |
⊢ ( 𝑥 = 𝑦 → ( 𝐼 ‘ ( 𝐺 ‘ 𝑥 ) ) = ( 𝐼 ‘ ( 𝐺 ‘ 𝑦 ) ) ) |
58 |
|
fvex |
⊢ ( 𝐼 ‘ ( 𝐺 ‘ 𝑦 ) ) ∈ V |
59 |
57 4 58
|
fvmpt |
⊢ ( 𝑦 ∈ 𝑉 → ( 𝑁 ‘ 𝑦 ) = ( 𝐼 ‘ ( 𝐺 ‘ 𝑦 ) ) ) |
60 |
29 59
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝑁 ‘ 𝑦 ) = ( 𝐼 ‘ ( 𝐺 ‘ 𝑦 ) ) ) |
61 |
60
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝑘 ( .r ‘ 𝑅 ) ( 𝑁 ‘ 𝑦 ) ) = ( 𝑘 ( .r ‘ 𝑅 ) ( 𝐼 ‘ ( 𝐺 ‘ 𝑦 ) ) ) ) |
62 |
|
2fveq3 |
⊢ ( 𝑥 = 𝑧 → ( 𝐼 ‘ ( 𝐺 ‘ 𝑥 ) ) = ( 𝐼 ‘ ( 𝐺 ‘ 𝑧 ) ) ) |
63 |
|
fvex |
⊢ ( 𝐼 ‘ ( 𝐺 ‘ 𝑧 ) ) ∈ V |
64 |
62 4 63
|
fvmpt |
⊢ ( 𝑧 ∈ 𝑉 → ( 𝑁 ‘ 𝑧 ) = ( 𝐼 ‘ ( 𝐺 ‘ 𝑧 ) ) ) |
65 |
35 64
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝑁 ‘ 𝑧 ) = ( 𝐼 ‘ ( 𝐺 ‘ 𝑧 ) ) ) |
66 |
61 65
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( 𝑘 ( .r ‘ 𝑅 ) ( 𝑁 ‘ 𝑦 ) ) ( +g ‘ 𝑅 ) ( 𝑁 ‘ 𝑧 ) ) = ( ( 𝑘 ( .r ‘ 𝑅 ) ( 𝐼 ‘ ( 𝐺 ‘ 𝑦 ) ) ) ( +g ‘ 𝑅 ) ( 𝐼 ‘ ( 𝐺 ‘ 𝑧 ) ) ) ) |
67 |
48 56 66
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝑁 ‘ ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ) = ( ( 𝑘 ( .r ‘ 𝑅 ) ( 𝑁 ‘ 𝑦 ) ) ( +g ‘ 𝑅 ) ( 𝑁 ‘ 𝑧 ) ) ) |
68 |
67
|
ralrimivvva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ 𝑉 ∀ 𝑧 ∈ 𝑉 ( 𝑁 ‘ ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ) = ( ( 𝑘 ( .r ‘ 𝑅 ) ( 𝑁 ‘ 𝑦 ) ) ( +g ‘ 𝑅 ) ( 𝑁 ‘ 𝑧 ) ) ) |
69 |
1 41 2 42 16 38 32 5
|
islfl |
⊢ ( 𝑊 ∈ LMod → ( 𝑁 ∈ 𝐹 ↔ ( 𝑁 : 𝑉 ⟶ ( Base ‘ 𝑅 ) ∧ ∀ 𝑘 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ 𝑉 ∀ 𝑧 ∈ 𝑉 ( 𝑁 ‘ ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ) = ( ( 𝑘 ( .r ‘ 𝑅 ) ( 𝑁 ‘ 𝑦 ) ) ( +g ‘ 𝑅 ) ( 𝑁 ‘ 𝑧 ) ) ) ) ) |
70 |
6 69
|
syl |
⊢ ( 𝜑 → ( 𝑁 ∈ 𝐹 ↔ ( 𝑁 : 𝑉 ⟶ ( Base ‘ 𝑅 ) ∧ ∀ 𝑘 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ 𝑉 ∀ 𝑧 ∈ 𝑉 ( 𝑁 ‘ ( ( 𝑘 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ) = ( ( 𝑘 ( .r ‘ 𝑅 ) ( 𝑁 ‘ 𝑦 ) ) ( +g ‘ 𝑅 ) ( 𝑁 ‘ 𝑧 ) ) ) ) ) |
71 |
21 68 70
|
mpbir2and |
⊢ ( 𝜑 → 𝑁 ∈ 𝐹 ) |