| Step |
Hyp |
Ref |
Expression |
| 1 |
|
elrgspn.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
| 2 |
|
elrgspn.m |
⊢ 𝑀 = ( mulGrp ‘ 𝑅 ) |
| 3 |
|
elrgspn.x |
⊢ · = ( .g ‘ 𝑅 ) |
| 4 |
|
elrgspn.n |
⊢ 𝑁 = ( RingSpan ‘ 𝑅 ) |
| 5 |
|
elrgspn.f |
⊢ 𝐹 = { 𝑓 ∈ ( ℤ ↑m Word 𝐴 ) ∣ 𝑓 finSupp 0 } |
| 6 |
|
elrgspn.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
| 7 |
|
elrgspn.a |
⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 ) |
| 8 |
1
|
a1i |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑅 ) ) |
| 9 |
4
|
a1i |
⊢ ( 𝜑 → 𝑁 = ( RingSpan ‘ 𝑅 ) ) |
| 10 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑁 ‘ 𝐴 ) = ( 𝑁 ‘ 𝐴 ) ) |
| 11 |
6 8 7 9 10
|
rgspncl |
⊢ ( 𝜑 → ( 𝑁 ‘ 𝐴 ) ∈ ( SubRing ‘ 𝑅 ) ) |
| 12 |
1
|
subrgss |
⊢ ( ( 𝑁 ‘ 𝐴 ) ∈ ( SubRing ‘ 𝑅 ) → ( 𝑁 ‘ 𝐴 ) ⊆ 𝐵 ) |
| 13 |
11 12
|
syl |
⊢ ( 𝜑 → ( 𝑁 ‘ 𝐴 ) ⊆ 𝐵 ) |
| 14 |
13
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑁 ‘ 𝐴 ) ) → 𝑋 ∈ 𝐵 ) |
| 15 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → 𝑋 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
| 16 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
| 17 |
6
|
ringcmnd |
⊢ ( 𝜑 → 𝑅 ∈ CMnd ) |
| 18 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → 𝑅 ∈ CMnd ) |
| 19 |
1
|
fvexi |
⊢ 𝐵 ∈ V |
| 20 |
19
|
a1i |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
| 21 |
20 7
|
ssexd |
⊢ ( 𝜑 → 𝐴 ∈ V ) |
| 22 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → 𝐴 ∈ V ) |
| 23 |
|
wrdexg |
⊢ ( 𝐴 ∈ V → Word 𝐴 ∈ V ) |
| 24 |
22 23
|
syl |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → Word 𝐴 ∈ V ) |
| 25 |
6
|
ringgrpd |
⊢ ( 𝜑 → 𝑅 ∈ Grp ) |
| 26 |
25
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑅 ∈ Grp ) |
| 27 |
|
zex |
⊢ ℤ ∈ V |
| 28 |
27
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ℤ ∈ V ) |
| 29 |
|
breq1 |
⊢ ( 𝑓 = 𝑔 → ( 𝑓 finSupp 0 ↔ 𝑔 finSupp 0 ) ) |
| 30 |
29 5
|
elrab2 |
⊢ ( 𝑔 ∈ 𝐹 ↔ ( 𝑔 ∈ ( ℤ ↑m Word 𝐴 ) ∧ 𝑔 finSupp 0 ) ) |
| 31 |
30
|
biimpi |
⊢ ( 𝑔 ∈ 𝐹 → ( 𝑔 ∈ ( ℤ ↑m Word 𝐴 ) ∧ 𝑔 finSupp 0 ) ) |
| 32 |
31
|
simpld |
⊢ ( 𝑔 ∈ 𝐹 → 𝑔 ∈ ( ℤ ↑m Word 𝐴 ) ) |
| 33 |
32
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → 𝑔 ∈ ( ℤ ↑m Word 𝐴 ) ) |
| 34 |
24 28 33
|
elmaprd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → 𝑔 : Word 𝐴 ⟶ ℤ ) |
| 35 |
34
|
ffvelcdmda |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑔 ‘ 𝑤 ) ∈ ℤ ) |
| 36 |
2
|
ringmgp |
⊢ ( 𝑅 ∈ Ring → 𝑀 ∈ Mnd ) |
| 37 |
6 36
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ Mnd ) |
| 38 |
37
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑀 ∈ Mnd ) |
| 39 |
|
sswrd |
⊢ ( 𝐴 ⊆ 𝐵 → Word 𝐴 ⊆ Word 𝐵 ) |
| 40 |
7 39
|
syl |
⊢ ( 𝜑 → Word 𝐴 ⊆ Word 𝐵 ) |
| 41 |
40
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → Word 𝐴 ⊆ Word 𝐵 ) |
| 42 |
41
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑤 ∈ Word 𝐵 ) |
| 43 |
2 1
|
mgpbas |
⊢ 𝐵 = ( Base ‘ 𝑀 ) |
| 44 |
43
|
gsumwcl |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑤 ∈ Word 𝐵 ) → ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) |
| 45 |
38 42 44
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) |
| 46 |
1 3 26 35 45
|
mulgcld |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) |
| 47 |
46
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) : Word 𝐴 ⟶ 𝐵 ) |
| 48 |
34
|
feqmptd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → 𝑔 = ( 𝑤 ∈ Word 𝐴 ↦ ( 𝑔 ‘ 𝑤 ) ) ) |
| 49 |
31
|
simprd |
⊢ ( 𝑔 ∈ 𝐹 → 𝑔 finSupp 0 ) |
| 50 |
49
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → 𝑔 finSupp 0 ) |
| 51 |
48 50
|
eqbrtrrd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( 𝑔 ‘ 𝑤 ) ) finSupp 0 ) |
| 52 |
1 16 3
|
mulg0 |
⊢ ( 𝑦 ∈ 𝐵 → ( 0 · 𝑦 ) = ( 0g ‘ 𝑅 ) ) |
| 53 |
52
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → ( 0 · 𝑦 ) = ( 0g ‘ 𝑅 ) ) |
| 54 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 0g ‘ 𝑅 ) ∈ V ) |
| 55 |
51 53 35 45 54
|
fsuppssov1 |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
| 56 |
1 16 18 24 47 55
|
gsumcl |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝐵 ) |
| 57 |
56
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝐵 ) |
| 58 |
15 57
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → 𝑋 ∈ 𝐵 ) |
| 59 |
58
|
r19.29an |
⊢ ( ( 𝜑 ∧ ∃ 𝑔 ∈ 𝐹 𝑋 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → 𝑋 ∈ 𝐵 ) |
| 60 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → 𝑅 ∈ Ring ) |
| 61 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → 𝐴 ⊆ 𝐵 ) |
| 62 |
|
fveq1 |
⊢ ( ℎ = 𝑖 → ( ℎ ‘ 𝑤 ) = ( 𝑖 ‘ 𝑤 ) ) |
| 63 |
62
|
oveq1d |
⊢ ( ℎ = 𝑖 → ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
| 64 |
63
|
mpteq2dv |
⊢ ( ℎ = 𝑖 → ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
| 65 |
|
fveq2 |
⊢ ( 𝑤 = 𝑣 → ( 𝑖 ‘ 𝑤 ) = ( 𝑖 ‘ 𝑣 ) ) |
| 66 |
|
oveq2 |
⊢ ( 𝑤 = 𝑣 → ( 𝑀 Σg 𝑤 ) = ( 𝑀 Σg 𝑣 ) ) |
| 67 |
65 66
|
oveq12d |
⊢ ( 𝑤 = 𝑣 → ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) |
| 68 |
67
|
cbvmptv |
⊢ ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑣 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) |
| 69 |
64 68
|
eqtrdi |
⊢ ( ℎ = 𝑖 → ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑣 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) |
| 70 |
69
|
oveq2d |
⊢ ( ℎ = 𝑖 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑣 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) ) |
| 71 |
70
|
cbvmptv |
⊢ ( ℎ ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) = ( 𝑖 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑣 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) ) |
| 72 |
71
|
rneqi |
⊢ ran ( ℎ ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) = ran ( 𝑖 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑣 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) ) |
| 73 |
1 2 3 4 5 60 61 72
|
elrgspnlem4 |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 ‘ 𝐴 ) = ran ( ℎ ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
| 74 |
73
|
eleq2d |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ ( 𝑁 ‘ 𝐴 ) ↔ 𝑋 ∈ ran ( ℎ ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) ) |
| 75 |
|
fveq1 |
⊢ ( ℎ = 𝑔 → ( ℎ ‘ 𝑤 ) = ( 𝑔 ‘ 𝑤 ) ) |
| 76 |
75
|
oveq1d |
⊢ ( ℎ = 𝑔 → ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
| 77 |
76
|
mpteq2dv |
⊢ ( ℎ = 𝑔 → ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
| 78 |
77
|
oveq2d |
⊢ ( ℎ = 𝑔 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
| 79 |
78
|
cbvmptv |
⊢ ( ℎ ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) = ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
| 80 |
79
|
elrnmpt |
⊢ ( 𝑋 ∈ 𝐵 → ( 𝑋 ∈ ran ( ℎ ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ↔ ∃ 𝑔 ∈ 𝐹 𝑋 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
| 81 |
80
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ ran ( ℎ ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ↔ ∃ 𝑔 ∈ 𝐹 𝑋 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
| 82 |
74 81
|
bitrd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ ( 𝑁 ‘ 𝐴 ) ↔ ∃ 𝑔 ∈ 𝐹 𝑋 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
| 83 |
14 59 82
|
bibiad |
⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑁 ‘ 𝐴 ) ↔ ∃ 𝑔 ∈ 𝐹 𝑋 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |