Step |
Hyp |
Ref |
Expression |
1 |
|
cznrng.y |
⊢ 𝑌 = ( ℤ/nℤ ‘ 𝑁 ) |
2 |
|
cznrng.b |
⊢ 𝐵 = ( Base ‘ 𝑌 ) |
3 |
|
cznrng.x |
⊢ 𝑋 = ( 𝑌 sSet 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 〉 ) |
4 |
|
cznrng.0 |
⊢ 0 = ( 0g ‘ 𝑌 ) |
5 |
|
nnnn0 |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℕ0 ) |
6 |
1
|
zncrng |
⊢ ( 𝑁 ∈ ℕ0 → 𝑌 ∈ CRing ) |
7 |
5 6
|
syl |
⊢ ( 𝑁 ∈ ℕ → 𝑌 ∈ CRing ) |
8 |
|
crngring |
⊢ ( 𝑌 ∈ CRing → 𝑌 ∈ Ring ) |
9 |
2 4
|
ring0cl |
⊢ ( 𝑌 ∈ Ring → 0 ∈ 𝐵 ) |
10 |
|
eleq1a |
⊢ ( 0 ∈ 𝐵 → ( 𝐶 = 0 → 𝐶 ∈ 𝐵 ) ) |
11 |
9 10
|
syl |
⊢ ( 𝑌 ∈ Ring → ( 𝐶 = 0 → 𝐶 ∈ 𝐵 ) ) |
12 |
8 11
|
syl |
⊢ ( 𝑌 ∈ CRing → ( 𝐶 = 0 → 𝐶 ∈ 𝐵 ) ) |
13 |
7 12
|
syl |
⊢ ( 𝑁 ∈ ℕ → ( 𝐶 = 0 → 𝐶 ∈ 𝐵 ) ) |
14 |
13
|
imp |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) → 𝐶 ∈ 𝐵 ) |
15 |
1 2 3
|
cznabel |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐶 ∈ 𝐵 ) → 𝑋 ∈ Abel ) |
16 |
15
|
adantlr |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) → 𝑋 ∈ Abel ) |
17 |
|
eqid |
⊢ ( mulGrp ‘ 𝑋 ) = ( mulGrp ‘ 𝑋 ) |
18 |
1 2 3
|
cznrnglem |
⊢ 𝐵 = ( Base ‘ 𝑋 ) |
19 |
17 18
|
mgpbas |
⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑋 ) ) |
20 |
3
|
fveq2i |
⊢ ( mulGrp ‘ 𝑋 ) = ( mulGrp ‘ ( 𝑌 sSet 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 〉 ) ) |
21 |
1
|
fvexi |
⊢ 𝑌 ∈ V |
22 |
2
|
fvexi |
⊢ 𝐵 ∈ V |
23 |
22 22
|
mpoex |
⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ∈ V |
24 |
|
mulrid |
⊢ .r = Slot ( .r ‘ ndx ) |
25 |
24
|
setsid |
⊢ ( ( 𝑌 ∈ V ∧ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ∈ V ) → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) = ( .r ‘ ( 𝑌 sSet 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 〉 ) ) ) |
26 |
21 23 25
|
mp2an |
⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) = ( .r ‘ ( 𝑌 sSet 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 〉 ) ) |
27 |
20 26
|
mgpplusg |
⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) = ( +g ‘ ( mulGrp ‘ 𝑋 ) ) |
28 |
27
|
eqcomi |
⊢ ( +g ‘ ( mulGrp ‘ 𝑋 ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) |
29 |
|
ne0i |
⊢ ( 𝐶 ∈ 𝐵 → 𝐵 ≠ ∅ ) |
30 |
29
|
adantl |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) → 𝐵 ≠ ∅ ) |
31 |
|
simpr |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) → 𝐶 ∈ 𝐵 ) |
32 |
19 28 30 31
|
copissgrp |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) → ( mulGrp ‘ 𝑋 ) ∈ Smgrp ) |
33 |
|
oveq1 |
⊢ ( 𝐶 = 0 → ( 𝐶 ( +g ‘ 𝑌 ) 𝐶 ) = ( 0 ( +g ‘ 𝑌 ) 𝐶 ) ) |
34 |
33
|
ad3antlr |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → ( 𝐶 ( +g ‘ 𝑌 ) 𝐶 ) = ( 0 ( +g ‘ 𝑌 ) 𝐶 ) ) |
35 |
7 8
|
syl |
⊢ ( 𝑁 ∈ ℕ → 𝑌 ∈ Ring ) |
36 |
|
ringmnd |
⊢ ( 𝑌 ∈ Ring → 𝑌 ∈ Mnd ) |
37 |
35 36
|
syl |
⊢ ( 𝑁 ∈ ℕ → 𝑌 ∈ Mnd ) |
38 |
37
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) → 𝑌 ∈ Mnd ) |
39 |
38
|
anim1i |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) → ( 𝑌 ∈ Mnd ∧ 𝐶 ∈ 𝐵 ) ) |
40 |
39
|
adantr |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → ( 𝑌 ∈ Mnd ∧ 𝐶 ∈ 𝐵 ) ) |
41 |
|
eqid |
⊢ ( +g ‘ 𝑌 ) = ( +g ‘ 𝑌 ) |
42 |
2 41 4
|
mndlid |
⊢ ( ( 𝑌 ∈ Mnd ∧ 𝐶 ∈ 𝐵 ) → ( 0 ( +g ‘ 𝑌 ) 𝐶 ) = 𝐶 ) |
43 |
40 42
|
syl |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → ( 0 ( +g ‘ 𝑌 ) 𝐶 ) = 𝐶 ) |
44 |
34 43
|
eqtrd |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → ( 𝐶 ( +g ‘ 𝑌 ) 𝐶 ) = 𝐶 ) |
45 |
|
eqidd |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ) |
46 |
|
eqidd |
⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) ∧ ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) ) → 𝐶 = 𝐶 ) |
47 |
|
simpr1 |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → 𝑎 ∈ 𝐵 ) |
48 |
|
simpr2 |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → 𝑏 ∈ 𝐵 ) |
49 |
31
|
adantr |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → 𝐶 ∈ 𝐵 ) |
50 |
45 46 47 48 49
|
ovmpod |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑏 ) = 𝐶 ) |
51 |
|
eqidd |
⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) ∧ ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑐 ) ) → 𝐶 = 𝐶 ) |
52 |
|
simpr3 |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → 𝑐 ∈ 𝐵 ) |
53 |
45 51 47 52 49
|
ovmpod |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) = 𝐶 ) |
54 |
50 53
|
oveq12d |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → ( ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑏 ) ( +g ‘ 𝑌 ) ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ) = ( 𝐶 ( +g ‘ 𝑌 ) 𝐶 ) ) |
55 |
|
eqidd |
⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) ∧ ( 𝑥 = 𝑎 ∧ 𝑦 = ( 𝑏 ( +g ‘ 𝑌 ) 𝑐 ) ) ) → 𝐶 = 𝐶 ) |
56 |
35
|
ad3antrrr |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → 𝑌 ∈ Ring ) |
57 |
2 41
|
ringacl |
⊢ ( ( 𝑌 ∈ Ring ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) → ( 𝑏 ( +g ‘ 𝑌 ) 𝑐 ) ∈ 𝐵 ) |
58 |
56 48 52 57
|
syl3anc |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → ( 𝑏 ( +g ‘ 𝑌 ) 𝑐 ) ∈ 𝐵 ) |
59 |
45 55 47 58 49
|
ovmpod |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ( 𝑏 ( +g ‘ 𝑌 ) 𝑐 ) ) = 𝐶 ) |
60 |
44 54 59
|
3eqtr4rd |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ( 𝑏 ( +g ‘ 𝑌 ) 𝑐 ) ) = ( ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑏 ) ( +g ‘ 𝑌 ) ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ) ) |
61 |
|
eqidd |
⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) ∧ ( 𝑥 = 𝑏 ∧ 𝑦 = 𝑐 ) ) → 𝐶 = 𝐶 ) |
62 |
45 61 48 52 49
|
ovmpod |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → ( 𝑏 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) = 𝐶 ) |
63 |
53 62
|
oveq12d |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → ( ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ( +g ‘ 𝑌 ) ( 𝑏 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ) = ( 𝐶 ( +g ‘ 𝑌 ) 𝐶 ) ) |
64 |
|
eqidd |
⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) ∧ ( 𝑥 = ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ∧ 𝑦 = 𝑐 ) ) → 𝐶 = 𝐶 ) |
65 |
2 41
|
ringacl |
⊢ ( ( 𝑌 ∈ Ring ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) → ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ∈ 𝐵 ) |
66 |
56 47 48 65
|
syl3anc |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ∈ 𝐵 ) |
67 |
45 64 66 52 49
|
ovmpod |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → ( ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) = 𝐶 ) |
68 |
44 63 67
|
3eqtr4rd |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → ( ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) = ( ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ( +g ‘ 𝑌 ) ( 𝑏 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ) ) |
69 |
60 68
|
jca |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → ( ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ( 𝑏 ( +g ‘ 𝑌 ) 𝑐 ) ) = ( ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑏 ) ( +g ‘ 𝑌 ) ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ) ∧ ( ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) = ( ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ( +g ‘ 𝑌 ) ( 𝑏 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ) ) ) |
70 |
69
|
ralrimivvva |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) → ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ 𝐵 ( ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ( 𝑏 ( +g ‘ 𝑌 ) 𝑐 ) ) = ( ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑏 ) ( +g ‘ 𝑌 ) ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ) ∧ ( ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) = ( ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ( +g ‘ 𝑌 ) ( 𝑏 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ) ) ) |
71 |
16 32 70
|
3jca |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵 ) → ( 𝑋 ∈ Abel ∧ ( mulGrp ‘ 𝑋 ) ∈ Smgrp ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ 𝐵 ( ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ( 𝑏 ( +g ‘ 𝑌 ) 𝑐 ) ) = ( ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑏 ) ( +g ‘ 𝑌 ) ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ) ∧ ( ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) = ( ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ( +g ‘ 𝑌 ) ( 𝑏 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ) ) ) ) |
72 |
14 71
|
mpdan |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) → ( 𝑋 ∈ Abel ∧ ( mulGrp ‘ 𝑋 ) ∈ Smgrp ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ 𝐵 ( ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ( 𝑏 ( +g ‘ 𝑌 ) 𝑐 ) ) = ( ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑏 ) ( +g ‘ 𝑌 ) ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ) ∧ ( ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) = ( ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ( +g ‘ 𝑌 ) ( 𝑏 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ) ) ) ) |
73 |
|
plusgid |
⊢ +g = Slot ( +g ‘ ndx ) |
74 |
|
plusgndxnmulrndx |
⊢ ( +g ‘ ndx ) ≠ ( .r ‘ ndx ) |
75 |
73 74
|
setsnid |
⊢ ( +g ‘ 𝑌 ) = ( +g ‘ ( 𝑌 sSet 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 〉 ) ) |
76 |
3
|
fveq2i |
⊢ ( +g ‘ 𝑋 ) = ( +g ‘ ( 𝑌 sSet 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 〉 ) ) |
77 |
75 76
|
eqtr4i |
⊢ ( +g ‘ 𝑌 ) = ( +g ‘ 𝑋 ) |
78 |
3
|
eqcomi |
⊢ ( 𝑌 sSet 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 〉 ) = 𝑋 |
79 |
78
|
fveq2i |
⊢ ( .r ‘ ( 𝑌 sSet 〈 ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 〉 ) ) = ( .r ‘ 𝑋 ) |
80 |
26 79
|
eqtri |
⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) = ( .r ‘ 𝑋 ) |
81 |
18 17 77 80
|
isrng |
⊢ ( 𝑋 ∈ Rng ↔ ( 𝑋 ∈ Abel ∧ ( mulGrp ‘ 𝑋 ) ∈ Smgrp ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ 𝐵 ( ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ( 𝑏 ( +g ‘ 𝑌 ) 𝑐 ) ) = ( ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑏 ) ( +g ‘ 𝑌 ) ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ) ∧ ( ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) = ( ( 𝑎 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ( +g ‘ 𝑌 ) ( 𝑏 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) 𝑐 ) ) ) ) ) |
82 |
72 81
|
sylibr |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐶 = 0 ) → 𝑋 ∈ Rng ) |