Step |
Hyp |
Ref |
Expression |
1 |
|
opprqus.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
opprqus.o |
⊢ 𝑂 = ( oppr ‘ 𝑅 ) |
3 |
|
opprqus.q |
⊢ 𝑄 = ( 𝑅 /s ( 𝑅 ~QG 𝐼 ) ) |
4 |
|
opprqus.i |
⊢ ( 𝜑 → 𝐼 ∈ ( NrmSGrp ‘ 𝑅 ) ) |
5 |
4
|
elfvexd |
⊢ ( 𝜑 → 𝑅 ∈ V ) |
6 |
|
nsgsubg |
⊢ ( 𝐼 ∈ ( NrmSGrp ‘ 𝑅 ) → 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ) |
7 |
1
|
subgss |
⊢ ( 𝐼 ∈ ( SubGrp ‘ 𝑅 ) → 𝐼 ⊆ 𝐵 ) |
8 |
4 6 7
|
3syl |
⊢ ( 𝜑 → 𝐼 ⊆ 𝐵 ) |
9 |
1 2 3 5 8
|
opprqusbas |
⊢ ( 𝜑 → ( Base ‘ ( oppr ‘ 𝑄 ) ) = ( Base ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) ) |
10 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) → ( Base ‘ ( oppr ‘ 𝑄 ) ) = ( Base ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) ) |
11 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) ∧ 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) → 𝐼 ∈ ( NrmSGrp ‘ 𝑅 ) ) |
12 |
|
eqid |
⊢ ( Base ‘ 𝑄 ) = ( Base ‘ 𝑄 ) |
13 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) → 𝑒 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) |
14 |
|
eqid |
⊢ ( oppr ‘ 𝑄 ) = ( oppr ‘ 𝑄 ) |
15 |
14 12
|
opprbas |
⊢ ( Base ‘ 𝑄 ) = ( Base ‘ ( oppr ‘ 𝑄 ) ) |
16 |
15
|
eqcomi |
⊢ ( Base ‘ ( oppr ‘ 𝑄 ) ) = ( Base ‘ 𝑄 ) |
17 |
13 16
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) → 𝑒 ∈ ( Base ‘ 𝑄 ) ) |
18 |
17
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) ∧ 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) → 𝑒 ∈ ( Base ‘ 𝑄 ) ) |
19 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) → 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) |
20 |
19 16
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) → 𝑥 ∈ ( Base ‘ 𝑄 ) ) |
21 |
20
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) ∧ 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) → 𝑥 ∈ ( Base ‘ 𝑄 ) ) |
22 |
1 2 3 11 12 18 21
|
opprqusplusg |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) ∧ 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) → ( 𝑒 ( +g ‘ ( oppr ‘ 𝑄 ) ) 𝑥 ) = ( 𝑒 ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) 𝑥 ) ) |
23 |
22
|
eqeq1d |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) ∧ 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) → ( ( 𝑒 ( +g ‘ ( oppr ‘ 𝑄 ) ) 𝑥 ) = 𝑥 ↔ ( 𝑒 ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) 𝑥 ) = 𝑥 ) ) |
24 |
1 2 3 11 12 21 18
|
opprqusplusg |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) ∧ 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) → ( 𝑥 ( +g ‘ ( oppr ‘ 𝑄 ) ) 𝑒 ) = ( 𝑥 ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) 𝑒 ) ) |
25 |
24
|
eqeq1d |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) ∧ 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) → ( ( 𝑥 ( +g ‘ ( oppr ‘ 𝑄 ) ) 𝑒 ) = 𝑥 ↔ ( 𝑥 ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) 𝑒 ) = 𝑥 ) ) |
26 |
23 25
|
anbi12d |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) ∧ 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) → ( ( ( 𝑒 ( +g ‘ ( oppr ‘ 𝑄 ) ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ ( oppr ‘ 𝑄 ) ) 𝑒 ) = 𝑥 ) ↔ ( ( 𝑒 ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) 𝑒 ) = 𝑥 ) ) ) |
27 |
10 26
|
raleqbidva |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ) → ( ∀ 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ( ( 𝑒 ( +g ‘ ( oppr ‘ 𝑄 ) ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ ( oppr ‘ 𝑄 ) ) 𝑒 ) = 𝑥 ) ↔ ∀ 𝑥 ∈ ( Base ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) ( ( 𝑒 ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) 𝑒 ) = 𝑥 ) ) ) |
28 |
27
|
pm5.32da |
⊢ ( 𝜑 → ( ( 𝑒 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ∧ ∀ 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ( ( 𝑒 ( +g ‘ ( oppr ‘ 𝑄 ) ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ ( oppr ‘ 𝑄 ) ) 𝑒 ) = 𝑥 ) ) ↔ ( 𝑒 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ∧ ∀ 𝑥 ∈ ( Base ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) ( ( 𝑒 ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) 𝑒 ) = 𝑥 ) ) ) ) |
29 |
9
|
eleq2d |
⊢ ( 𝜑 → ( 𝑒 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ↔ 𝑒 ∈ ( Base ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) ) ) |
30 |
29
|
anbi1d |
⊢ ( 𝜑 → ( ( 𝑒 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ∧ ∀ 𝑥 ∈ ( Base ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) ( ( 𝑒 ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) 𝑒 ) = 𝑥 ) ) ↔ ( 𝑒 ∈ ( Base ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) ∧ ∀ 𝑥 ∈ ( Base ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) ( ( 𝑒 ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) 𝑒 ) = 𝑥 ) ) ) ) |
31 |
28 30
|
bitrd |
⊢ ( 𝜑 → ( ( 𝑒 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ∧ ∀ 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ( ( 𝑒 ( +g ‘ ( oppr ‘ 𝑄 ) ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ ( oppr ‘ 𝑄 ) ) 𝑒 ) = 𝑥 ) ) ↔ ( 𝑒 ∈ ( Base ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) ∧ ∀ 𝑥 ∈ ( Base ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) ( ( 𝑒 ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) 𝑒 ) = 𝑥 ) ) ) ) |
32 |
31
|
iotabidv |
⊢ ( 𝜑 → ( ℩ 𝑒 ( 𝑒 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ∧ ∀ 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ( ( 𝑒 ( +g ‘ ( oppr ‘ 𝑄 ) ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ ( oppr ‘ 𝑄 ) ) 𝑒 ) = 𝑥 ) ) ) = ( ℩ 𝑒 ( 𝑒 ∈ ( Base ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) ∧ ∀ 𝑥 ∈ ( Base ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) ( ( 𝑒 ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) 𝑒 ) = 𝑥 ) ) ) ) |
33 |
|
eqid |
⊢ ( +g ‘ 𝑄 ) = ( +g ‘ 𝑄 ) |
34 |
14 33
|
oppradd |
⊢ ( +g ‘ 𝑄 ) = ( +g ‘ ( oppr ‘ 𝑄 ) ) |
35 |
34
|
eqcomi |
⊢ ( +g ‘ ( oppr ‘ 𝑄 ) ) = ( +g ‘ 𝑄 ) |
36 |
|
eqid |
⊢ ( 0g ‘ 𝑄 ) = ( 0g ‘ 𝑄 ) |
37 |
14 36
|
oppr0 |
⊢ ( 0g ‘ 𝑄 ) = ( 0g ‘ ( oppr ‘ 𝑄 ) ) |
38 |
37
|
eqcomi |
⊢ ( 0g ‘ ( oppr ‘ 𝑄 ) ) = ( 0g ‘ 𝑄 ) |
39 |
16 35 38
|
grpidval |
⊢ ( 0g ‘ ( oppr ‘ 𝑄 ) ) = ( ℩ 𝑒 ( 𝑒 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ∧ ∀ 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑄 ) ) ( ( 𝑒 ( +g ‘ ( oppr ‘ 𝑄 ) ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ ( oppr ‘ 𝑄 ) ) 𝑒 ) = 𝑥 ) ) ) |
40 |
|
eqid |
⊢ ( Base ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) = ( Base ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) |
41 |
|
eqid |
⊢ ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) = ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) |
42 |
|
eqid |
⊢ ( 0g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) = ( 0g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) |
43 |
40 41 42
|
grpidval |
⊢ ( 0g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) = ( ℩ 𝑒 ( 𝑒 ∈ ( Base ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) ∧ ∀ 𝑥 ∈ ( Base ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) ( ( 𝑒 ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) 𝑒 ) = 𝑥 ) ) ) |
44 |
32 39 43
|
3eqtr4g |
⊢ ( 𝜑 → ( 0g ‘ ( oppr ‘ 𝑄 ) ) = ( 0g ‘ ( 𝑂 /s ( 𝑂 ~QG 𝐼 ) ) ) ) |