| Step |
Hyp |
Ref |
Expression |
| 1 |
|
qustgp.h |
⊢ 𝐻 = ( 𝐺 /s ( 𝐺 ~QG 𝑌 ) ) |
| 2 |
|
qustgpopn.x |
⊢ 𝑋 = ( Base ‘ 𝐺 ) |
| 3 |
|
qustgpopn.j |
⊢ 𝐽 = ( TopOpen ‘ 𝐺 ) |
| 4 |
|
qustgpopn.k |
⊢ 𝐾 = ( TopOpen ‘ 𝐻 ) |
| 5 |
|
qustgpopn.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ [ 𝑥 ] ( 𝐺 ~QG 𝑌 ) ) |
| 6 |
|
imassrn |
⊢ ( 𝐹 “ 𝑆 ) ⊆ ran 𝐹 |
| 7 |
1
|
a1i |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → 𝐻 = ( 𝐺 /s ( 𝐺 ~QG 𝑌 ) ) ) |
| 8 |
2
|
a1i |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → 𝑋 = ( Base ‘ 𝐺 ) ) |
| 9 |
|
ovex |
⊢ ( 𝐺 ~QG 𝑌 ) ∈ V |
| 10 |
9
|
a1i |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → ( 𝐺 ~QG 𝑌 ) ∈ V ) |
| 11 |
|
simp1 |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → 𝐺 ∈ TopGrp ) |
| 12 |
7 8 5 10 11
|
quslem |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → 𝐹 : 𝑋 –onto→ ( 𝑋 / ( 𝐺 ~QG 𝑌 ) ) ) |
| 13 |
|
forn |
⊢ ( 𝐹 : 𝑋 –onto→ ( 𝑋 / ( 𝐺 ~QG 𝑌 ) ) → ran 𝐹 = ( 𝑋 / ( 𝐺 ~QG 𝑌 ) ) ) |
| 14 |
12 13
|
syl |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → ran 𝐹 = ( 𝑋 / ( 𝐺 ~QG 𝑌 ) ) ) |
| 15 |
6 14
|
sseqtrid |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → ( 𝐹 “ 𝑆 ) ⊆ ( 𝑋 / ( 𝐺 ~QG 𝑌 ) ) ) |
| 16 |
|
eceq1 |
⊢ ( 𝑥 = 𝑦 → [ 𝑥 ] ( 𝐺 ~QG 𝑌 ) = [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) ) |
| 17 |
16
|
cbvmptv |
⊢ ( 𝑥 ∈ 𝑋 ↦ [ 𝑥 ] ( 𝐺 ~QG 𝑌 ) ) = ( 𝑦 ∈ 𝑋 ↦ [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) ) |
| 18 |
5 17
|
eqtri |
⊢ 𝐹 = ( 𝑦 ∈ 𝑋 ↦ [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) ) |
| 19 |
18
|
mptpreima |
⊢ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) = { 𝑦 ∈ 𝑋 ∣ [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) ∈ ( 𝐹 “ 𝑆 ) } |
| 20 |
19
|
reqabi |
⊢ ( 𝑦 ∈ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ↔ ( 𝑦 ∈ 𝑋 ∧ [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) ∈ ( 𝐹 “ 𝑆 ) ) ) |
| 21 |
5
|
funmpt2 |
⊢ Fun 𝐹 |
| 22 |
|
fvelima |
⊢ ( ( Fun 𝐹 ∧ [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) ∈ ( 𝐹 “ 𝑆 ) ) → ∃ 𝑧 ∈ 𝑆 ( 𝐹 ‘ 𝑧 ) = [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) ) |
| 23 |
21 22
|
mpan |
⊢ ( [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) ∈ ( 𝐹 “ 𝑆 ) → ∃ 𝑧 ∈ 𝑆 ( 𝐹 ‘ 𝑧 ) = [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) ) |
| 24 |
3 2
|
tgptopon |
⊢ ( 𝐺 ∈ TopGrp → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
| 25 |
11 24
|
syl |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
| 26 |
|
simp3 |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → 𝑆 ∈ 𝐽 ) |
| 27 |
|
toponss |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑆 ∈ 𝐽 ) → 𝑆 ⊆ 𝑋 ) |
| 28 |
25 26 27
|
syl2anc |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → 𝑆 ⊆ 𝑋 ) |
| 29 |
28
|
adantr |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) → 𝑆 ⊆ 𝑋 ) |
| 30 |
29
|
sselda |
⊢ ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → 𝑧 ∈ 𝑋 ) |
| 31 |
|
eceq1 |
⊢ ( 𝑥 = 𝑧 → [ 𝑥 ] ( 𝐺 ~QG 𝑌 ) = [ 𝑧 ] ( 𝐺 ~QG 𝑌 ) ) |
| 32 |
|
ecexg |
⊢ ( ( 𝐺 ~QG 𝑌 ) ∈ V → [ 𝑧 ] ( 𝐺 ~QG 𝑌 ) ∈ V ) |
| 33 |
9 32
|
ax-mp |
⊢ [ 𝑧 ] ( 𝐺 ~QG 𝑌 ) ∈ V |
| 34 |
31 5 33
|
fvmpt |
⊢ ( 𝑧 ∈ 𝑋 → ( 𝐹 ‘ 𝑧 ) = [ 𝑧 ] ( 𝐺 ~QG 𝑌 ) ) |
| 35 |
30 34
|
syl |
⊢ ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → ( 𝐹 ‘ 𝑧 ) = [ 𝑧 ] ( 𝐺 ~QG 𝑌 ) ) |
| 36 |
35
|
eqeq1d |
⊢ ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → ( ( 𝐹 ‘ 𝑧 ) = [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) ↔ [ 𝑧 ] ( 𝐺 ~QG 𝑌 ) = [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) ) ) |
| 37 |
|
eqcom |
⊢ ( [ 𝑧 ] ( 𝐺 ~QG 𝑌 ) = [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) ↔ [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) = [ 𝑧 ] ( 𝐺 ~QG 𝑌 ) ) |
| 38 |
36 37
|
bitrdi |
⊢ ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → ( ( 𝐹 ‘ 𝑧 ) = [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) ↔ [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) = [ 𝑧 ] ( 𝐺 ~QG 𝑌 ) ) ) |
| 39 |
|
nsgsubg |
⊢ ( 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) → 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ) |
| 40 |
39
|
3ad2ant2 |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ) |
| 41 |
40
|
ad2antrr |
⊢ ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ) |
| 42 |
|
eqid |
⊢ ( 𝐺 ~QG 𝑌 ) = ( 𝐺 ~QG 𝑌 ) |
| 43 |
2 42
|
eqger |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐺 ~QG 𝑌 ) Er 𝑋 ) |
| 44 |
41 43
|
syl |
⊢ ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → ( 𝐺 ~QG 𝑌 ) Er 𝑋 ) |
| 45 |
|
simplr |
⊢ ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → 𝑦 ∈ 𝑋 ) |
| 46 |
44 45
|
erth |
⊢ ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → ( 𝑦 ( 𝐺 ~QG 𝑌 ) 𝑧 ↔ [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) = [ 𝑧 ] ( 𝐺 ~QG 𝑌 ) ) ) |
| 47 |
11
|
ad2antrr |
⊢ ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → 𝐺 ∈ TopGrp ) |
| 48 |
2
|
subgss |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → 𝑌 ⊆ 𝑋 ) |
| 49 |
41 48
|
syl |
⊢ ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → 𝑌 ⊆ 𝑋 ) |
| 50 |
|
eqid |
⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 ) |
| 51 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
| 52 |
2 50 51 42
|
eqgval |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ⊆ 𝑋 ) → ( 𝑦 ( 𝐺 ~QG 𝑌 ) 𝑧 ↔ ( 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ) ) |
| 53 |
47 49 52
|
syl2anc |
⊢ ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → ( 𝑦 ( 𝐺 ~QG 𝑌 ) 𝑧 ↔ ( 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ) ) |
| 54 |
38 46 53
|
3bitr2d |
⊢ ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → ( ( 𝐹 ‘ 𝑧 ) = [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) ↔ ( 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ) ) |
| 55 |
|
eqid |
⊢ ( oppg ‘ 𝐺 ) = ( oppg ‘ 𝐺 ) |
| 56 |
|
eqid |
⊢ ( +g ‘ ( oppg ‘ 𝐺 ) ) = ( +g ‘ ( oppg ‘ 𝐺 ) ) |
| 57 |
51 55 56
|
oppgplus |
⊢ ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ( +g ‘ ( oppg ‘ 𝐺 ) ) 𝑎 ) = ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) |
| 58 |
57
|
mpteq2i |
⊢ ( 𝑎 ∈ 𝑋 ↦ ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ( +g ‘ ( oppg ‘ 𝐺 ) ) 𝑎 ) ) = ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
| 59 |
47
|
adantr |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → 𝐺 ∈ TopGrp ) |
| 60 |
55
|
oppgtgp |
⊢ ( 𝐺 ∈ TopGrp → ( oppg ‘ 𝐺 ) ∈ TopGrp ) |
| 61 |
59 60
|
syl |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → ( oppg ‘ 𝐺 ) ∈ TopGrp ) |
| 62 |
49
|
sselda |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑋 ) |
| 63 |
|
eqid |
⊢ ( 𝑎 ∈ 𝑋 ↦ ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ( +g ‘ ( oppg ‘ 𝐺 ) ) 𝑎 ) ) = ( 𝑎 ∈ 𝑋 ↦ ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ( +g ‘ ( oppg ‘ 𝐺 ) ) 𝑎 ) ) |
| 64 |
55 2
|
oppgbas |
⊢ 𝑋 = ( Base ‘ ( oppg ‘ 𝐺 ) ) |
| 65 |
55 3
|
oppgtopn |
⊢ 𝐽 = ( TopOpen ‘ ( oppg ‘ 𝐺 ) ) |
| 66 |
63 64 56 65
|
tgplacthmeo |
⊢ ( ( ( oppg ‘ 𝐺 ) ∈ TopGrp ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑋 ) → ( 𝑎 ∈ 𝑋 ↦ ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ( +g ‘ ( oppg ‘ 𝐺 ) ) 𝑎 ) ) ∈ ( 𝐽 Homeo 𝐽 ) ) |
| 67 |
61 62 66
|
syl2anc |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → ( 𝑎 ∈ 𝑋 ↦ ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ( +g ‘ ( oppg ‘ 𝐺 ) ) 𝑎 ) ) ∈ ( 𝐽 Homeo 𝐽 ) ) |
| 68 |
58 67
|
eqeltrrid |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) ∈ ( 𝐽 Homeo 𝐽 ) ) |
| 69 |
|
hmeocn |
⊢ ( ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) ∈ ( 𝐽 Homeo 𝐽 ) → ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) ∈ ( 𝐽 Cn 𝐽 ) ) |
| 70 |
68 69
|
syl |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) ∈ ( 𝐽 Cn 𝐽 ) ) |
| 71 |
26
|
ad3antrrr |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → 𝑆 ∈ 𝐽 ) |
| 72 |
|
cnima |
⊢ ( ( ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) ∈ ( 𝐽 Cn 𝐽 ) ∧ 𝑆 ∈ 𝐽 ) → ( ◡ ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) “ 𝑆 ) ∈ 𝐽 ) |
| 73 |
70 71 72
|
syl2anc |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → ( ◡ ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) “ 𝑆 ) ∈ 𝐽 ) |
| 74 |
45
|
adantr |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → 𝑦 ∈ 𝑋 ) |
| 75 |
|
tgpgrp |
⊢ ( 𝐺 ∈ TopGrp → 𝐺 ∈ Grp ) |
| 76 |
59 75
|
syl |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → 𝐺 ∈ Grp ) |
| 77 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
| 78 |
2 51 77 50
|
grprinv |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋 ) → ( 𝑦 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ) = ( 0g ‘ 𝐺 ) ) |
| 79 |
76 74 78
|
syl2anc |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → ( 𝑦 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ) = ( 0g ‘ 𝐺 ) ) |
| 80 |
79
|
oveq1d |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → ( ( 𝑦 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ) ( +g ‘ 𝐺 ) 𝑧 ) = ( ( 0g ‘ 𝐺 ) ( +g ‘ 𝐺 ) 𝑧 ) ) |
| 81 |
2 50
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ∈ 𝑋 ) |
| 82 |
76 74 81
|
syl2anc |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ∈ 𝑋 ) |
| 83 |
30
|
adantr |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → 𝑧 ∈ 𝑋 ) |
| 84 |
2 51
|
grpass |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑦 ∈ 𝑋 ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ) ) → ( ( 𝑦 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ) ( +g ‘ 𝐺 ) 𝑧 ) = ( 𝑦 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
| 85 |
76 74 82 83 84
|
syl13anc |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → ( ( 𝑦 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ) ( +g ‘ 𝐺 ) 𝑧 ) = ( 𝑦 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
| 86 |
2 51 77
|
grplid |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋 ) → ( ( 0g ‘ 𝐺 ) ( +g ‘ 𝐺 ) 𝑧 ) = 𝑧 ) |
| 87 |
76 83 86
|
syl2anc |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → ( ( 0g ‘ 𝐺 ) ( +g ‘ 𝐺 ) 𝑧 ) = 𝑧 ) |
| 88 |
80 85 87
|
3eqtr3d |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → ( 𝑦 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) = 𝑧 ) |
| 89 |
|
simplr |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → 𝑧 ∈ 𝑆 ) |
| 90 |
88 89
|
eqeltrd |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → ( 𝑦 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ∈ 𝑆 ) |
| 91 |
|
oveq1 |
⊢ ( 𝑎 = 𝑦 → ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) = ( 𝑦 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
| 92 |
91
|
eleq1d |
⊢ ( 𝑎 = 𝑦 → ( ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ∈ 𝑆 ↔ ( 𝑦 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ∈ 𝑆 ) ) |
| 93 |
|
eqid |
⊢ ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) = ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
| 94 |
93
|
mptpreima |
⊢ ( ◡ ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) “ 𝑆 ) = { 𝑎 ∈ 𝑋 ∣ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ∈ 𝑆 } |
| 95 |
92 94
|
elrab2 |
⊢ ( 𝑦 ∈ ( ◡ ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) “ 𝑆 ) ↔ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ∈ 𝑆 ) ) |
| 96 |
74 90 95
|
sylanbrc |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → 𝑦 ∈ ( ◡ ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) “ 𝑆 ) ) |
| 97 |
|
ecexg |
⊢ ( ( 𝐺 ~QG 𝑌 ) ∈ V → [ 𝑥 ] ( 𝐺 ~QG 𝑌 ) ∈ V ) |
| 98 |
9 97
|
ax-mp |
⊢ [ 𝑥 ] ( 𝐺 ~QG 𝑌 ) ∈ V |
| 99 |
98 5
|
fnmpti |
⊢ 𝐹 Fn 𝑋 |
| 100 |
29
|
ad3antrrr |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → 𝑆 ⊆ 𝑋 ) |
| 101 |
|
fnfvima |
⊢ ( ( 𝐹 Fn 𝑋 ∧ 𝑆 ⊆ 𝑋 ∧ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ∈ 𝑆 ) → ( 𝐹 ‘ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) ∈ ( 𝐹 “ 𝑆 ) ) |
| 102 |
101
|
3expia |
⊢ ( ( 𝐹 Fn 𝑋 ∧ 𝑆 ⊆ 𝑋 ) → ( ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ∈ 𝑆 → ( 𝐹 ‘ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) ∈ ( 𝐹 “ 𝑆 ) ) ) |
| 103 |
99 100 102
|
sylancr |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → ( ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ∈ 𝑆 → ( 𝐹 ‘ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) ∈ ( 𝐹 “ 𝑆 ) ) ) |
| 104 |
76
|
adantr |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → 𝐺 ∈ Grp ) |
| 105 |
|
simpr |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → 𝑎 ∈ 𝑋 ) |
| 106 |
62
|
adantr |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑋 ) |
| 107 |
2 51
|
grpcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑎 ∈ 𝑋 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑋 ) → ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ∈ 𝑋 ) |
| 108 |
104 105 106 107
|
syl3anc |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ∈ 𝑋 ) |
| 109 |
|
eceq1 |
⊢ ( 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) → [ 𝑥 ] ( 𝐺 ~QG 𝑌 ) = [ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ] ( 𝐺 ~QG 𝑌 ) ) |
| 110 |
109 5 98
|
fvmpt3i |
⊢ ( ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ∈ 𝑋 → ( 𝐹 ‘ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) = [ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ] ( 𝐺 ~QG 𝑌 ) ) |
| 111 |
108 110
|
syl |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → ( 𝐹 ‘ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) = [ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ] ( 𝐺 ~QG 𝑌 ) ) |
| 112 |
44
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → ( 𝐺 ~QG 𝑌 ) Er 𝑋 ) |
| 113 |
2 51 77 50
|
grplinv |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑎 ∈ 𝑋 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ( +g ‘ 𝐺 ) 𝑎 ) = ( 0g ‘ 𝐺 ) ) |
| 114 |
104 105 113
|
syl2anc |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ( +g ‘ 𝐺 ) 𝑎 ) = ( 0g ‘ 𝐺 ) ) |
| 115 |
114
|
oveq1d |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ( +g ‘ 𝐺 ) 𝑎 ) ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) = ( ( 0g ‘ 𝐺 ) ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
| 116 |
2 50
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑎 ∈ 𝑋 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ∈ 𝑋 ) |
| 117 |
104 105 116
|
syl2anc |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ∈ 𝑋 ) |
| 118 |
2 51
|
grpass |
⊢ ( ( 𝐺 ∈ Grp ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑋 ) ) → ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ( +g ‘ 𝐺 ) 𝑎 ) ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) = ( ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ( +g ‘ 𝐺 ) ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) ) |
| 119 |
104 117 105 106 118
|
syl13anc |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ( +g ‘ 𝐺 ) 𝑎 ) ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) = ( ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ( +g ‘ 𝐺 ) ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) ) |
| 120 |
2 51 77
|
grplid |
⊢ ( ( 𝐺 ∈ Grp ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑋 ) → ( ( 0g ‘ 𝐺 ) ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) = ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) |
| 121 |
104 106 120
|
syl2anc |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → ( ( 0g ‘ 𝐺 ) ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) = ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) |
| 122 |
115 119 121
|
3eqtr3d |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ( +g ‘ 𝐺 ) ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) = ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) |
| 123 |
|
simplr |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) |
| 124 |
122 123
|
eqeltrd |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ( +g ‘ 𝐺 ) ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) ∈ 𝑌 ) |
| 125 |
49
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → 𝑌 ⊆ 𝑋 ) |
| 126 |
2 50 51 42
|
eqgval |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ) → ( 𝑎 ( 𝐺 ~QG 𝑌 ) ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ↔ ( 𝑎 ∈ 𝑋 ∧ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ∈ 𝑋 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ( +g ‘ 𝐺 ) ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) ∈ 𝑌 ) ) ) |
| 127 |
104 125 126
|
syl2anc |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → ( 𝑎 ( 𝐺 ~QG 𝑌 ) ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ↔ ( 𝑎 ∈ 𝑋 ∧ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ∈ 𝑋 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ( +g ‘ 𝐺 ) ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) ∈ 𝑌 ) ) ) |
| 128 |
105 108 124 127
|
mpbir3and |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → 𝑎 ( 𝐺 ~QG 𝑌 ) ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
| 129 |
112 128
|
erthi |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → [ 𝑎 ] ( 𝐺 ~QG 𝑌 ) = [ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ] ( 𝐺 ~QG 𝑌 ) ) |
| 130 |
111 129
|
eqtr4d |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → ( 𝐹 ‘ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) = [ 𝑎 ] ( 𝐺 ~QG 𝑌 ) ) |
| 131 |
130
|
eleq1d |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → ( ( 𝐹 ‘ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) ∈ ( 𝐹 “ 𝑆 ) ↔ [ 𝑎 ] ( 𝐺 ~QG 𝑌 ) ∈ ( 𝐹 “ 𝑆 ) ) ) |
| 132 |
103 131
|
sylibd |
⊢ ( ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ∧ 𝑎 ∈ 𝑋 ) → ( ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ∈ 𝑆 → [ 𝑎 ] ( 𝐺 ~QG 𝑌 ) ∈ ( 𝐹 “ 𝑆 ) ) ) |
| 133 |
132
|
ss2rabdv |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → { 𝑎 ∈ 𝑋 ∣ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ∈ 𝑆 } ⊆ { 𝑎 ∈ 𝑋 ∣ [ 𝑎 ] ( 𝐺 ~QG 𝑌 ) ∈ ( 𝐹 “ 𝑆 ) } ) |
| 134 |
|
eceq1 |
⊢ ( 𝑥 = 𝑎 → [ 𝑥 ] ( 𝐺 ~QG 𝑌 ) = [ 𝑎 ] ( 𝐺 ~QG 𝑌 ) ) |
| 135 |
134
|
cbvmptv |
⊢ ( 𝑥 ∈ 𝑋 ↦ [ 𝑥 ] ( 𝐺 ~QG 𝑌 ) ) = ( 𝑎 ∈ 𝑋 ↦ [ 𝑎 ] ( 𝐺 ~QG 𝑌 ) ) |
| 136 |
5 135
|
eqtri |
⊢ 𝐹 = ( 𝑎 ∈ 𝑋 ↦ [ 𝑎 ] ( 𝐺 ~QG 𝑌 ) ) |
| 137 |
136
|
mptpreima |
⊢ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) = { 𝑎 ∈ 𝑋 ∣ [ 𝑎 ] ( 𝐺 ~QG 𝑌 ) ∈ ( 𝐹 “ 𝑆 ) } |
| 138 |
133 94 137
|
3sstr4g |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → ( ◡ ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) “ 𝑆 ) ⊆ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ) |
| 139 |
|
eleq2 |
⊢ ( 𝑢 = ( ◡ ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) “ 𝑆 ) → ( 𝑦 ∈ 𝑢 ↔ 𝑦 ∈ ( ◡ ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) “ 𝑆 ) ) ) |
| 140 |
|
sseq1 |
⊢ ( 𝑢 = ( ◡ ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) “ 𝑆 ) → ( 𝑢 ⊆ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ↔ ( ◡ ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) “ 𝑆 ) ⊆ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ) ) |
| 141 |
139 140
|
anbi12d |
⊢ ( 𝑢 = ( ◡ ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) “ 𝑆 ) → ( ( 𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ) ↔ ( 𝑦 ∈ ( ◡ ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) “ 𝑆 ) ∧ ( ◡ ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) “ 𝑆 ) ⊆ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ) ) ) |
| 142 |
141
|
rspcev |
⊢ ( ( ( ◡ ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) “ 𝑆 ) ∈ 𝐽 ∧ ( 𝑦 ∈ ( ◡ ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) “ 𝑆 ) ∧ ( ◡ ( 𝑎 ∈ 𝑋 ↦ ( 𝑎 ( +g ‘ 𝐺 ) ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ) ) “ 𝑆 ) ⊆ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ) ) → ∃ 𝑢 ∈ 𝐽 ( 𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ) ) |
| 143 |
73 96 138 142
|
syl12anc |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → ∃ 𝑢 ∈ 𝐽 ( 𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ) ) |
| 144 |
143
|
3ad2antr3 |
⊢ ( ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ ( 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) ) → ∃ 𝑢 ∈ 𝐽 ( 𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ) ) |
| 145 |
144
|
ex |
⊢ ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → ( ( 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑌 ) → ∃ 𝑢 ∈ 𝐽 ( 𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ) ) ) |
| 146 |
54 145
|
sylbid |
⊢ ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → ( ( 𝐹 ‘ 𝑧 ) = [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) → ∃ 𝑢 ∈ 𝐽 ( 𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ) ) ) |
| 147 |
146
|
rexlimdva |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) → ( ∃ 𝑧 ∈ 𝑆 ( 𝐹 ‘ 𝑧 ) = [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) → ∃ 𝑢 ∈ 𝐽 ( 𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ) ) ) |
| 148 |
23 147
|
syl5 |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) ∧ 𝑦 ∈ 𝑋 ) → ( [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) ∈ ( 𝐹 “ 𝑆 ) → ∃ 𝑢 ∈ 𝐽 ( 𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ) ) ) |
| 149 |
148
|
expimpd |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → ( ( 𝑦 ∈ 𝑋 ∧ [ 𝑦 ] ( 𝐺 ~QG 𝑌 ) ∈ ( 𝐹 “ 𝑆 ) ) → ∃ 𝑢 ∈ 𝐽 ( 𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ) ) ) |
| 150 |
20 149
|
biimtrid |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → ( 𝑦 ∈ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) → ∃ 𝑢 ∈ 𝐽 ( 𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ) ) ) |
| 151 |
150
|
ralrimiv |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → ∀ 𝑦 ∈ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ∃ 𝑢 ∈ 𝐽 ( 𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ) ) |
| 152 |
|
topontop |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐽 ∈ Top ) |
| 153 |
|
eltop2 |
⊢ ( 𝐽 ∈ Top → ( ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ∈ 𝐽 ↔ ∀ 𝑦 ∈ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ∃ 𝑢 ∈ 𝐽 ( 𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ) ) ) |
| 154 |
25 152 153
|
3syl |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → ( ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ∈ 𝐽 ↔ ∀ 𝑦 ∈ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ∃ 𝑢 ∈ 𝐽 ( 𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ) ) ) |
| 155 |
151 154
|
mpbird |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ∈ 𝐽 ) |
| 156 |
|
elqtop3 |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 : 𝑋 –onto→ ( 𝑋 / ( 𝐺 ~QG 𝑌 ) ) ) → ( ( 𝐹 “ 𝑆 ) ∈ ( 𝐽 qTop 𝐹 ) ↔ ( ( 𝐹 “ 𝑆 ) ⊆ ( 𝑋 / ( 𝐺 ~QG 𝑌 ) ) ∧ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ∈ 𝐽 ) ) ) |
| 157 |
25 12 156
|
syl2anc |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → ( ( 𝐹 “ 𝑆 ) ∈ ( 𝐽 qTop 𝐹 ) ↔ ( ( 𝐹 “ 𝑆 ) ⊆ ( 𝑋 / ( 𝐺 ~QG 𝑌 ) ) ∧ ( ◡ 𝐹 “ ( 𝐹 “ 𝑆 ) ) ∈ 𝐽 ) ) ) |
| 158 |
15 155 157
|
mpbir2and |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → ( 𝐹 “ 𝑆 ) ∈ ( 𝐽 qTop 𝐹 ) ) |
| 159 |
7 8 5 10 11
|
qusval |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → 𝐻 = ( 𝐹 “s 𝐺 ) ) |
| 160 |
159 8 12 11 3 4
|
imastopn |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → 𝐾 = ( 𝐽 qTop 𝐹 ) ) |
| 161 |
158 160
|
eleqtrrd |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑌 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐽 ) → ( 𝐹 “ 𝑆 ) ∈ 𝐾 ) |