Step |
Hyp |
Ref |
Expression |
1 |
|
pgpfac1.k |
⊢ 𝐾 = ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) |
2 |
|
pgpfac1.s |
⊢ 𝑆 = ( 𝐾 ‘ { 𝐴 } ) |
3 |
|
pgpfac1.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
4 |
|
pgpfac1.o |
⊢ 𝑂 = ( od ‘ 𝐺 ) |
5 |
|
pgpfac1.e |
⊢ 𝐸 = ( gEx ‘ 𝐺 ) |
6 |
|
pgpfac1.z |
⊢ 0 = ( 0g ‘ 𝐺 ) |
7 |
|
pgpfac1.l |
⊢ ⊕ = ( LSSum ‘ 𝐺 ) |
8 |
|
pgpfac1.p |
⊢ ( 𝜑 → 𝑃 pGrp 𝐺 ) |
9 |
|
pgpfac1.g |
⊢ ( 𝜑 → 𝐺 ∈ Abel ) |
10 |
|
pgpfac1.n |
⊢ ( 𝜑 → 𝐵 ∈ Fin ) |
11 |
|
pgpfac1.oe |
⊢ ( 𝜑 → ( 𝑂 ‘ 𝐴 ) = 𝐸 ) |
12 |
|
pgpfac1.ab |
⊢ ( 𝜑 → 𝐴 ∈ 𝐵 ) |
13 |
|
ablgrp |
⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp ) |
14 |
3
|
subgid |
⊢ ( 𝐺 ∈ Grp → 𝐵 ∈ ( SubGrp ‘ 𝐺 ) ) |
15 |
9 13 14
|
3syl |
⊢ ( 𝜑 → 𝐵 ∈ ( SubGrp ‘ 𝐺 ) ) |
16 |
|
eleq1 |
⊢ ( 𝑠 = 𝑢 → ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ↔ 𝑢 ∈ ( SubGrp ‘ 𝐺 ) ) ) |
17 |
|
eleq2 |
⊢ ( 𝑠 = 𝑢 → ( 𝐴 ∈ 𝑠 ↔ 𝐴 ∈ 𝑢 ) ) |
18 |
16 17
|
anbi12d |
⊢ ( 𝑠 = 𝑢 → ( ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑠 ) ↔ ( 𝑢 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑢 ) ) ) |
19 |
|
eqeq2 |
⊢ ( 𝑠 = 𝑢 → ( ( 𝑆 ⊕ 𝑡 ) = 𝑠 ↔ ( 𝑆 ⊕ 𝑡 ) = 𝑢 ) ) |
20 |
19
|
anbi2d |
⊢ ( 𝑠 = 𝑢 → ( ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ↔ ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑢 ) ) ) |
21 |
20
|
rexbidv |
⊢ ( 𝑠 = 𝑢 → ( ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ↔ ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑢 ) ) ) |
22 |
18 21
|
imbi12d |
⊢ ( 𝑠 = 𝑢 → ( ( ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ↔ ( ( 𝑢 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑢 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑢 ) ) ) ) |
23 |
22
|
imbi2d |
⊢ ( 𝑠 = 𝑢 → ( ( 𝜑 → ( ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ↔ ( 𝜑 → ( ( 𝑢 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑢 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑢 ) ) ) ) ) |
24 |
|
eleq1 |
⊢ ( 𝑠 = 𝐵 → ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ↔ 𝐵 ∈ ( SubGrp ‘ 𝐺 ) ) ) |
25 |
|
eleq2 |
⊢ ( 𝑠 = 𝐵 → ( 𝐴 ∈ 𝑠 ↔ 𝐴 ∈ 𝐵 ) ) |
26 |
24 25
|
anbi12d |
⊢ ( 𝑠 = 𝐵 → ( ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑠 ) ↔ ( 𝐵 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝐵 ) ) ) |
27 |
|
eqeq2 |
⊢ ( 𝑠 = 𝐵 → ( ( 𝑆 ⊕ 𝑡 ) = 𝑠 ↔ ( 𝑆 ⊕ 𝑡 ) = 𝐵 ) ) |
28 |
27
|
anbi2d |
⊢ ( 𝑠 = 𝐵 → ( ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ↔ ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝐵 ) ) ) |
29 |
28
|
rexbidv |
⊢ ( 𝑠 = 𝐵 → ( ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ↔ ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝐵 ) ) ) |
30 |
26 29
|
imbi12d |
⊢ ( 𝑠 = 𝐵 → ( ( ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ↔ ( ( 𝐵 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝐵 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝐵 ) ) ) ) |
31 |
30
|
imbi2d |
⊢ ( 𝑠 = 𝐵 → ( ( 𝜑 → ( ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ↔ ( 𝜑 → ( ( 𝐵 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝐵 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝐵 ) ) ) ) ) |
32 |
|
bi2.04 |
⊢ ( ( 𝑠 ⊊ 𝑢 → ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐴 ∈ 𝑠 → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) ↔ ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑠 ⊊ 𝑢 → ( 𝐴 ∈ 𝑠 → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) ) |
33 |
|
impexp |
⊢ ( ( ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ↔ ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐴 ∈ 𝑠 → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) |
34 |
33
|
imbi2i |
⊢ ( ( 𝑠 ⊊ 𝑢 → ( ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ↔ ( 𝑠 ⊊ 𝑢 → ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐴 ∈ 𝑠 → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) ) |
35 |
|
impexp |
⊢ ( ( ( 𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ↔ ( 𝑠 ⊊ 𝑢 → ( 𝐴 ∈ 𝑠 → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) |
36 |
35
|
imbi2i |
⊢ ( ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) → ( ( 𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ↔ ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑠 ⊊ 𝑢 → ( 𝐴 ∈ 𝑠 → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) ) |
37 |
32 34 36
|
3bitr4i |
⊢ ( ( 𝑠 ⊊ 𝑢 → ( ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ↔ ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) → ( ( 𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) |
38 |
37
|
imbi2i |
⊢ ( ( 𝜑 → ( 𝑠 ⊊ 𝑢 → ( ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) ↔ ( 𝜑 → ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) → ( ( 𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) ) |
39 |
|
bi2.04 |
⊢ ( ( 𝑠 ⊊ 𝑢 → ( 𝜑 → ( ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) ↔ ( 𝜑 → ( 𝑠 ⊊ 𝑢 → ( ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) ) |
40 |
|
bi2.04 |
⊢ ( ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝜑 → ( ( 𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) ↔ ( 𝜑 → ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) → ( ( 𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) ) |
41 |
38 39 40
|
3bitr4i |
⊢ ( ( 𝑠 ⊊ 𝑢 → ( 𝜑 → ( ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) ↔ ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝜑 → ( ( 𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) ) |
42 |
41
|
albii |
⊢ ( ∀ 𝑠 ( 𝑠 ⊊ 𝑢 → ( 𝜑 → ( ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) ↔ ∀ 𝑠 ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝜑 → ( ( 𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) ) |
43 |
|
df-ral |
⊢ ( ∀ 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ( 𝜑 → ( ( 𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ↔ ∀ 𝑠 ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝜑 → ( ( 𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) ) |
44 |
|
r19.21v |
⊢ ( ∀ 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ( 𝜑 → ( ( 𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ↔ ( 𝜑 → ∀ 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) |
45 |
42 43 44
|
3bitr2i |
⊢ ( ∀ 𝑠 ( 𝑠 ⊊ 𝑢 → ( 𝜑 → ( ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) ↔ ( 𝜑 → ∀ 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) |
46 |
|
psseq1 |
⊢ ( 𝑥 = 𝑠 → ( 𝑥 ⊊ 𝑢 ↔ 𝑠 ⊊ 𝑢 ) ) |
47 |
|
eleq2 |
⊢ ( 𝑥 = 𝑠 → ( 𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑠 ) ) |
48 |
46 47
|
anbi12d |
⊢ ( 𝑥 = 𝑠 → ( ( 𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥 ) ↔ ( 𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠 ) ) ) |
49 |
|
ineq2 |
⊢ ( 𝑦 = 𝑡 → ( 𝑆 ∩ 𝑦 ) = ( 𝑆 ∩ 𝑡 ) ) |
50 |
49
|
eqeq1d |
⊢ ( 𝑦 = 𝑡 → ( ( 𝑆 ∩ 𝑦 ) = { 0 } ↔ ( 𝑆 ∩ 𝑡 ) = { 0 } ) ) |
51 |
|
oveq2 |
⊢ ( 𝑦 = 𝑡 → ( 𝑆 ⊕ 𝑦 ) = ( 𝑆 ⊕ 𝑡 ) ) |
52 |
51
|
eqeq1d |
⊢ ( 𝑦 = 𝑡 → ( ( 𝑆 ⊕ 𝑦 ) = 𝑥 ↔ ( 𝑆 ⊕ 𝑡 ) = 𝑥 ) ) |
53 |
50 52
|
anbi12d |
⊢ ( 𝑦 = 𝑡 → ( ( ( 𝑆 ∩ 𝑦 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑦 ) = 𝑥 ) ↔ ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑥 ) ) ) |
54 |
53
|
cbvrexvw |
⊢ ( ∃ 𝑦 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑦 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑦 ) = 𝑥 ) ↔ ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑥 ) ) |
55 |
|
eqeq2 |
⊢ ( 𝑥 = 𝑠 → ( ( 𝑆 ⊕ 𝑡 ) = 𝑥 ↔ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) |
56 |
55
|
anbi2d |
⊢ ( 𝑥 = 𝑠 → ( ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑥 ) ↔ ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) |
57 |
56
|
rexbidv |
⊢ ( 𝑥 = 𝑠 → ( ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑥 ) ↔ ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) |
58 |
54 57
|
syl5bb |
⊢ ( 𝑥 = 𝑠 → ( ∃ 𝑦 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑦 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑦 ) = 𝑥 ) ↔ ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) |
59 |
48 58
|
imbi12d |
⊢ ( 𝑥 = 𝑠 → ( ( ( 𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥 ) → ∃ 𝑦 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑦 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑦 ) = 𝑥 ) ) ↔ ( ( 𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) |
60 |
59
|
cbvralvw |
⊢ ( ∀ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥 ) → ∃ 𝑦 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑦 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑦 ) = 𝑥 ) ) ↔ ∀ 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) |
61 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ ( ∀ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥 ) → ∃ 𝑦 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑦 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑦 ) = 𝑥 ) ) ∧ ( 𝑢 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑢 ) ) ) → 𝑃 pGrp 𝐺 ) |
62 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ( ∀ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥 ) → ∃ 𝑦 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑦 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑦 ) = 𝑥 ) ) ∧ ( 𝑢 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑢 ) ) ) → 𝐺 ∈ Abel ) |
63 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ( ∀ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥 ) → ∃ 𝑦 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑦 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑦 ) = 𝑥 ) ) ∧ ( 𝑢 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑢 ) ) ) → 𝐵 ∈ Fin ) |
64 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ ( ∀ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥 ) → ∃ 𝑦 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑦 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑦 ) = 𝑥 ) ) ∧ ( 𝑢 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑢 ) ) ) → ( 𝑂 ‘ 𝐴 ) = 𝐸 ) |
65 |
|
simprrl |
⊢ ( ( 𝜑 ∧ ( ∀ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥 ) → ∃ 𝑦 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑦 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑦 ) = 𝑥 ) ) ∧ ( 𝑢 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑢 ) ) ) → 𝑢 ∈ ( SubGrp ‘ 𝐺 ) ) |
66 |
|
simprrr |
⊢ ( ( 𝜑 ∧ ( ∀ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥 ) → ∃ 𝑦 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑦 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑦 ) = 𝑥 ) ) ∧ ( 𝑢 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑢 ) ) ) → 𝐴 ∈ 𝑢 ) |
67 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( ∀ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥 ) → ∃ 𝑦 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑦 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑦 ) = 𝑥 ) ) ∧ ( 𝑢 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑢 ) ) ) → ∀ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥 ) → ∃ 𝑦 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑦 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑦 ) = 𝑥 ) ) ) |
68 |
67 60
|
sylib |
⊢ ( ( 𝜑 ∧ ( ∀ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥 ) → ∃ 𝑦 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑦 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑦 ) = 𝑥 ) ) ∧ ( 𝑢 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑢 ) ) ) → ∀ 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) |
69 |
1 2 3 4 5 6 7 61 62 63 64 65 66 68
|
pgpfac1lem5 |
⊢ ( ( 𝜑 ∧ ( ∀ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥 ) → ∃ 𝑦 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑦 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑦 ) = 𝑥 ) ) ∧ ( 𝑢 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑢 ) ) ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑢 ) ) |
70 |
69
|
exp32 |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑥 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑥 ) → ∃ 𝑦 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑦 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑦 ) = 𝑥 ) ) → ( ( 𝑢 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑢 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑢 ) ) ) ) |
71 |
60 70
|
syl5bir |
⊢ ( 𝜑 → ( ∀ 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) → ( ( 𝑢 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑢 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑢 ) ) ) ) |
72 |
71
|
a2i |
⊢ ( ( 𝜑 → ∀ 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑠 ⊊ 𝑢 ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) → ( 𝜑 → ( ( 𝑢 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑢 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑢 ) ) ) ) |
73 |
45 72
|
sylbi |
⊢ ( ∀ 𝑠 ( 𝑠 ⊊ 𝑢 → ( 𝜑 → ( ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) → ( 𝜑 → ( ( 𝑢 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑢 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑢 ) ) ) ) |
74 |
73
|
a1i |
⊢ ( 𝑢 ∈ Fin → ( ∀ 𝑠 ( 𝑠 ⊊ 𝑢 → ( 𝜑 → ( ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑠 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑠 ) ) ) ) → ( 𝜑 → ( ( 𝑢 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑢 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑢 ) ) ) ) ) |
75 |
23 31 74
|
findcard3 |
⊢ ( 𝐵 ∈ Fin → ( 𝜑 → ( ( 𝐵 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝐵 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝐵 ) ) ) ) |
76 |
10 75
|
mpcom |
⊢ ( 𝜑 → ( ( 𝐵 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝐵 ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝐵 ) ) ) |
77 |
15 12 76
|
mp2and |
⊢ ( 𝜑 → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝐵 ) ) |