Step |
Hyp |
Ref |
Expression |
1 |
|
mplcoe1.p |
⊢ 𝑃 = ( 𝐼 mPoly 𝑅 ) |
2 |
|
mplcoe1.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
3 |
|
mplcoe1.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
4 |
|
mplcoe1.o |
⊢ 1 = ( 1r ‘ 𝑅 ) |
5 |
|
mplcoe1.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
6 |
|
mplcoe2.g |
⊢ 𝐺 = ( mulGrp ‘ 𝑃 ) |
7 |
|
mplcoe2.m |
⊢ ↑ = ( .g ‘ 𝐺 ) |
8 |
|
mplcoe2.v |
⊢ 𝑉 = ( 𝐼 mVar 𝑅 ) |
9 |
|
mplcoe5.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
10 |
|
mplcoe5.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐷 ) |
11 |
|
mplcoe5.c |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐼 ∀ 𝑦 ∈ 𝐼 ( ( 𝑉 ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑥 ) ) = ( ( 𝑉 ‘ 𝑥 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑦 ) ) ) |
12 |
|
mplcoe5.s |
⊢ ( 𝜑 → 𝑆 ⊆ 𝐼 ) |
13 |
|
vex |
⊢ 𝑥 ∈ V |
14 |
|
eqid |
⊢ ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) = ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) |
15 |
14
|
elrnmpt |
⊢ ( 𝑥 ∈ V → ( 𝑥 ∈ ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ↔ ∃ 𝑘 ∈ 𝑆 𝑥 = ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ) |
16 |
13 15
|
mp1i |
⊢ ( 𝜑 → ( 𝑥 ∈ ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ↔ ∃ 𝑘 ∈ 𝑆 𝑥 = ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ) |
17 |
|
vex |
⊢ 𝑦 ∈ V |
18 |
14
|
elrnmpt |
⊢ ( 𝑦 ∈ V → ( 𝑦 ∈ ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ↔ ∃ 𝑘 ∈ 𝑆 𝑦 = ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ) |
19 |
17 18
|
mp1i |
⊢ ( 𝜑 → ( 𝑦 ∈ ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ↔ ∃ 𝑘 ∈ 𝑆 𝑦 = ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ) |
20 |
|
fveq2 |
⊢ ( 𝑘 = 𝑙 → ( 𝑌 ‘ 𝑘 ) = ( 𝑌 ‘ 𝑙 ) ) |
21 |
|
fveq2 |
⊢ ( 𝑘 = 𝑙 → ( 𝑉 ‘ 𝑘 ) = ( 𝑉 ‘ 𝑙 ) ) |
22 |
20 21
|
oveq12d |
⊢ ( 𝑘 = 𝑙 → ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) = ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) ) |
23 |
22
|
eqeq2d |
⊢ ( 𝑘 = 𝑙 → ( 𝑦 = ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ↔ 𝑦 = ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) ) ) |
24 |
23
|
cbvrexvw |
⊢ ( ∃ 𝑘 ∈ 𝑆 𝑦 = ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ↔ ∃ 𝑙 ∈ 𝑆 𝑦 = ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) ) |
25 |
|
eqid |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 ) |
26 |
|
eqid |
⊢ ( .r ‘ 𝑃 ) = ( .r ‘ 𝑃 ) |
27 |
6 26
|
mgpplusg |
⊢ ( .r ‘ 𝑃 ) = ( +g ‘ 𝐺 ) |
28 |
27
|
eqcomi |
⊢ ( +g ‘ 𝐺 ) = ( .r ‘ 𝑃 ) |
29 |
1
|
mplring |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ Ring ) → 𝑃 ∈ Ring ) |
30 |
5 9 29
|
syl2anc |
⊢ ( 𝜑 → 𝑃 ∈ Ring ) |
31 |
|
ringsrg |
⊢ ( 𝑃 ∈ Ring → 𝑃 ∈ SRing ) |
32 |
30 31
|
syl |
⊢ ( 𝜑 → 𝑃 ∈ SRing ) |
33 |
32
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) → 𝑃 ∈ SRing ) |
34 |
33
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) ∧ 𝑘 ∈ 𝑆 ) → 𝑃 ∈ SRing ) |
35 |
6 25
|
mgpbas |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝐺 ) |
36 |
6
|
ringmgp |
⊢ ( 𝑃 ∈ Ring → 𝐺 ∈ Mnd ) |
37 |
30 36
|
syl |
⊢ ( 𝜑 → 𝐺 ∈ Mnd ) |
38 |
37
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) → 𝐺 ∈ Mnd ) |
39 |
12
|
sseld |
⊢ ( 𝜑 → ( 𝑙 ∈ 𝑆 → 𝑙 ∈ 𝐼 ) ) |
40 |
39
|
imdistani |
⊢ ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) → ( 𝜑 ∧ 𝑙 ∈ 𝐼 ) ) |
41 |
2
|
psrbag |
⊢ ( 𝐼 ∈ 𝑊 → ( 𝑌 ∈ 𝐷 ↔ ( 𝑌 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝑌 “ ℕ ) ∈ Fin ) ) ) |
42 |
5 41
|
syl |
⊢ ( 𝜑 → ( 𝑌 ∈ 𝐷 ↔ ( 𝑌 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝑌 “ ℕ ) ∈ Fin ) ) ) |
43 |
10 42
|
mpbid |
⊢ ( 𝜑 → ( 𝑌 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝑌 “ ℕ ) ∈ Fin ) ) |
44 |
43
|
simpld |
⊢ ( 𝜑 → 𝑌 : 𝐼 ⟶ ℕ0 ) |
45 |
44
|
ffvelcdmda |
⊢ ( ( 𝜑 ∧ 𝑙 ∈ 𝐼 ) → ( 𝑌 ‘ 𝑙 ) ∈ ℕ0 ) |
46 |
40 45
|
syl |
⊢ ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) → ( 𝑌 ‘ 𝑙 ) ∈ ℕ0 ) |
47 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) → 𝐼 ∈ 𝑊 ) |
48 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) → 𝑅 ∈ Ring ) |
49 |
12
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) → 𝑙 ∈ 𝐼 ) |
50 |
1 8 25 47 48 49
|
mvrcl |
⊢ ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) → ( 𝑉 ‘ 𝑙 ) ∈ ( Base ‘ 𝑃 ) ) |
51 |
35 7 38 46 50
|
mulgnn0cld |
⊢ ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) → ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) ∈ ( Base ‘ 𝑃 ) ) |
52 |
51
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) ∧ 𝑘 ∈ 𝑆 ) → ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) ∈ ( Base ‘ 𝑃 ) ) |
53 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑆 ) → 𝐼 ∈ 𝑊 ) |
54 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑆 ) → 𝑅 ∈ Ring ) |
55 |
12
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑆 ) → 𝑘 ∈ 𝐼 ) |
56 |
1 8 25 53 54 55
|
mvrcl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑆 ) → ( 𝑉 ‘ 𝑘 ) ∈ ( Base ‘ 𝑃 ) ) |
57 |
56
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) ∧ 𝑘 ∈ 𝑆 ) → ( 𝑉 ‘ 𝑘 ) ∈ ( Base ‘ 𝑃 ) ) |
58 |
44
|
ffvelcdmda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( 𝑌 ‘ 𝑘 ) ∈ ℕ0 ) |
59 |
55 58
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑆 ) → ( 𝑌 ‘ 𝑘 ) ∈ ℕ0 ) |
60 |
59
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) ∧ 𝑘 ∈ 𝑆 ) → ( 𝑌 ‘ 𝑘 ) ∈ ℕ0 ) |
61 |
50
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) ∧ 𝑘 ∈ 𝑆 ) → ( 𝑉 ‘ 𝑙 ) ∈ ( Base ‘ 𝑃 ) ) |
62 |
46
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) ∧ 𝑘 ∈ 𝑆 ) → ( 𝑌 ‘ 𝑙 ) ∈ ℕ0 ) |
63 |
|
fveq2 |
⊢ ( 𝑥 = 𝑙 → ( 𝑉 ‘ 𝑥 ) = ( 𝑉 ‘ 𝑙 ) ) |
64 |
63
|
oveq2d |
⊢ ( 𝑥 = 𝑙 → ( ( 𝑉 ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑥 ) ) = ( ( 𝑉 ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑙 ) ) ) |
65 |
63
|
oveq1d |
⊢ ( 𝑥 = 𝑙 → ( ( 𝑉 ‘ 𝑥 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑦 ) ) = ( ( 𝑉 ‘ 𝑙 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑦 ) ) ) |
66 |
64 65
|
eqeq12d |
⊢ ( 𝑥 = 𝑙 → ( ( ( 𝑉 ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑥 ) ) = ( ( 𝑉 ‘ 𝑥 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑦 ) ) ↔ ( ( 𝑉 ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑙 ) ) = ( ( 𝑉 ‘ 𝑙 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑦 ) ) ) ) |
67 |
|
fveq2 |
⊢ ( 𝑦 = 𝑘 → ( 𝑉 ‘ 𝑦 ) = ( 𝑉 ‘ 𝑘 ) ) |
68 |
67
|
oveq1d |
⊢ ( 𝑦 = 𝑘 → ( ( 𝑉 ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑙 ) ) = ( ( 𝑉 ‘ 𝑘 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑙 ) ) ) |
69 |
67
|
oveq2d |
⊢ ( 𝑦 = 𝑘 → ( ( 𝑉 ‘ 𝑙 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑦 ) ) = ( ( 𝑉 ‘ 𝑙 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑘 ) ) ) |
70 |
68 69
|
eqeq12d |
⊢ ( 𝑦 = 𝑘 → ( ( ( 𝑉 ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑙 ) ) = ( ( 𝑉 ‘ 𝑙 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑦 ) ) ↔ ( ( 𝑉 ‘ 𝑘 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑙 ) ) = ( ( 𝑉 ‘ 𝑙 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑘 ) ) ) ) |
71 |
66 70
|
rspc2v |
⊢ ( ( 𝑙 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼 ) → ( ∀ 𝑥 ∈ 𝐼 ∀ 𝑦 ∈ 𝐼 ( ( 𝑉 ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑥 ) ) = ( ( 𝑉 ‘ 𝑥 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑦 ) ) → ( ( 𝑉 ‘ 𝑘 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑙 ) ) = ( ( 𝑉 ‘ 𝑙 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑘 ) ) ) ) |
72 |
49 55
|
anim12dan |
⊢ ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆 ) ) → ( 𝑙 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼 ) ) |
73 |
71 72
|
syl11 |
⊢ ( ∀ 𝑥 ∈ 𝐼 ∀ 𝑦 ∈ 𝐼 ( ( 𝑉 ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑥 ) ) = ( ( 𝑉 ‘ 𝑥 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑦 ) ) → ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆 ) ) → ( ( 𝑉 ‘ 𝑘 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑙 ) ) = ( ( 𝑉 ‘ 𝑙 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑘 ) ) ) ) |
74 |
73
|
expd |
⊢ ( ∀ 𝑥 ∈ 𝐼 ∀ 𝑦 ∈ 𝐼 ( ( 𝑉 ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑥 ) ) = ( ( 𝑉 ‘ 𝑥 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑦 ) ) → ( 𝜑 → ( ( 𝑙 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆 ) → ( ( 𝑉 ‘ 𝑘 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑙 ) ) = ( ( 𝑉 ‘ 𝑙 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑘 ) ) ) ) ) |
75 |
11 74
|
mpcom |
⊢ ( 𝜑 → ( ( 𝑙 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆 ) → ( ( 𝑉 ‘ 𝑘 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑙 ) ) = ( ( 𝑉 ‘ 𝑙 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑘 ) ) ) ) |
76 |
75
|
impl |
⊢ ( ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) ∧ 𝑘 ∈ 𝑆 ) → ( ( 𝑉 ‘ 𝑘 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑙 ) ) = ( ( 𝑉 ‘ 𝑙 ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑘 ) ) ) |
77 |
25 28 6 7 34 57 61 62 76
|
srgpcomp |
⊢ ( ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) ∧ 𝑘 ∈ 𝑆 ) → ( ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) ( +g ‘ 𝐺 ) ( 𝑉 ‘ 𝑘 ) ) = ( ( 𝑉 ‘ 𝑘 ) ( +g ‘ 𝐺 ) ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) ) ) |
78 |
25 28 6 7 34 52 57 60 77
|
srgpcomp |
⊢ ( ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) ∧ 𝑘 ∈ 𝑆 ) → ( ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ( +g ‘ 𝐺 ) ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) ) = ( ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) ( +g ‘ 𝐺 ) ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ) |
79 |
|
oveq12 |
⊢ ( ( 𝑥 = ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ∧ 𝑦 = ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ( +g ‘ 𝐺 ) ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) ) ) |
80 |
|
oveq12 |
⊢ ( ( 𝑦 = ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) ∧ 𝑥 = ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) → ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) ( +g ‘ 𝐺 ) ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ) |
81 |
80
|
ancoms |
⊢ ( ( 𝑥 = ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ∧ 𝑦 = ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) ) → ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) ( +g ‘ 𝐺 ) ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ) |
82 |
79 81
|
eqeq12d |
⊢ ( ( 𝑥 = ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ∧ 𝑦 = ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) ) → ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ↔ ( ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ( +g ‘ 𝐺 ) ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) ) = ( ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) ( +g ‘ 𝐺 ) ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ) ) |
83 |
78 82
|
syl5ibrcom |
⊢ ( ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) ∧ 𝑘 ∈ 𝑆 ) → ( ( 𝑥 = ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ∧ 𝑦 = ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) ) |
84 |
83
|
expd |
⊢ ( ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) ∧ 𝑘 ∈ 𝑆 ) → ( 𝑥 = ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) → ( 𝑦 = ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) ) ) |
85 |
84
|
rexlimdva |
⊢ ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) → ( ∃ 𝑘 ∈ 𝑆 𝑥 = ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) → ( 𝑦 = ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) ) ) |
86 |
85
|
com23 |
⊢ ( ( 𝜑 ∧ 𝑙 ∈ 𝑆 ) → ( 𝑦 = ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) → ( ∃ 𝑘 ∈ 𝑆 𝑥 = ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) ) ) |
87 |
86
|
rexlimdva |
⊢ ( 𝜑 → ( ∃ 𝑙 ∈ 𝑆 𝑦 = ( ( 𝑌 ‘ 𝑙 ) ↑ ( 𝑉 ‘ 𝑙 ) ) → ( ∃ 𝑘 ∈ 𝑆 𝑥 = ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) ) ) |
88 |
24 87
|
biimtrid |
⊢ ( 𝜑 → ( ∃ 𝑘 ∈ 𝑆 𝑦 = ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) → ( ∃ 𝑘 ∈ 𝑆 𝑥 = ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) ) ) |
89 |
19 88
|
sylbid |
⊢ ( 𝜑 → ( 𝑦 ∈ ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) → ( ∃ 𝑘 ∈ 𝑆 𝑥 = ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) ) ) |
90 |
89
|
com23 |
⊢ ( 𝜑 → ( ∃ 𝑘 ∈ 𝑆 𝑥 = ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) → ( 𝑦 ∈ ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) ) ) |
91 |
16 90
|
sylbid |
⊢ ( 𝜑 → ( 𝑥 ∈ ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) → ( 𝑦 ∈ ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) ) ) |
92 |
91
|
imp32 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ∧ 𝑦 ∈ ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) |
93 |
92
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ∀ 𝑦 ∈ ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) |
94 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
95 |
37
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑆 ) → 𝐺 ∈ Mnd ) |
96 |
12
|
sseld |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑆 → 𝑘 ∈ 𝐼 ) ) |
97 |
96
|
imdistani |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑆 ) → ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) ) |
98 |
97 58
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑆 ) → ( 𝑌 ‘ 𝑘 ) ∈ ℕ0 ) |
99 |
56 35
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑆 ) → ( 𝑉 ‘ 𝑘 ) ∈ ( Base ‘ 𝐺 ) ) |
100 |
94 7 95 98 99
|
mulgnn0cld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑆 ) → ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ∈ ( Base ‘ 𝐺 ) ) |
101 |
100
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) : 𝑆 ⟶ ( Base ‘ 𝐺 ) ) |
102 |
101
|
frnd |
⊢ ( 𝜑 → ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ⊆ ( Base ‘ 𝐺 ) ) |
103 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
104 |
|
eqid |
⊢ ( Cntz ‘ 𝐺 ) = ( Cntz ‘ 𝐺 ) |
105 |
94 103 104
|
sscntz |
⊢ ( ( ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ⊆ ( Base ‘ 𝐺 ) ∧ ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ⊆ ( Base ‘ 𝐺 ) ) → ( ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ) ↔ ∀ 𝑥 ∈ ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ∀ 𝑦 ∈ ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) ) |
106 |
102 102 105
|
syl2anc |
⊢ ( 𝜑 → ( ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ) ↔ ∀ 𝑥 ∈ ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ∀ 𝑦 ∈ ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) ) |
107 |
93 106
|
mpbird |
⊢ ( 𝜑 → ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ran ( 𝑘 ∈ 𝑆 ↦ ( ( 𝑌 ‘ 𝑘 ) ↑ ( 𝑉 ‘ 𝑘 ) ) ) ) ) |