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 |
|
elrgspnlem1.1 |
⊢ 𝑆 = ran ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
9 |
1
|
a1i |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑅 ) ) |
10 |
4
|
a1i |
⊢ ( 𝜑 → 𝑁 = ( RingSpan ‘ 𝑅 ) ) |
11 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑁 ‘ 𝐴 ) = ( 𝑁 ‘ 𝐴 ) ) |
12 |
6 9 7 10 11
|
rgspnval |
⊢ ( 𝜑 → ( 𝑁 ‘ 𝐴 ) = ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } ) |
13 |
|
sseq2 |
⊢ ( 𝑡 = 𝑆 → ( 𝐴 ⊆ 𝑡 ↔ 𝐴 ⊆ 𝑆 ) ) |
14 |
1 2 3 4 5 6 7 8
|
elrgspnlem2 |
⊢ ( 𝜑 → 𝑆 ∈ ( SubRing ‘ 𝑅 ) ) |
15 |
1 2 3 4 5 6 7 8
|
elrgspnlem3 |
⊢ ( 𝜑 → 𝐴 ⊆ 𝑆 ) |
16 |
13 14 15
|
elrabd |
⊢ ( 𝜑 → 𝑆 ∈ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } ) |
17 |
|
intss1 |
⊢ ( 𝑆 ∈ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } → ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } ⊆ 𝑆 ) |
18 |
16 17
|
syl |
⊢ ( 𝜑 → ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } ⊆ 𝑆 ) |
19 |
|
simpr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑠 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑠 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → 𝑠 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
20 |
|
eqidd |
⊢ ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) → ( 𝑔 supp 0 ) = ( 𝑔 supp 0 ) ) |
21 |
|
oveq1 |
⊢ ( 𝑓 = 𝑔 → ( 𝑓 supp 0 ) = ( 𝑔 supp 0 ) ) |
22 |
21
|
eqeq1d |
⊢ ( 𝑓 = 𝑔 → ( ( 𝑓 supp 0 ) = ( 𝑔 supp 0 ) ↔ ( 𝑔 supp 0 ) = ( 𝑔 supp 0 ) ) ) |
23 |
|
fveq1 |
⊢ ( 𝑓 = 𝑔 → ( 𝑓 ‘ 𝑤 ) = ( 𝑔 ‘ 𝑤 ) ) |
24 |
23
|
oveq1d |
⊢ ( 𝑓 = 𝑔 → ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
25 |
24
|
mpteq2dv |
⊢ ( 𝑓 = 𝑔 → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
26 |
25
|
oveq2d |
⊢ ( 𝑓 = 𝑔 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
27 |
26
|
eleq1d |
⊢ ( 𝑓 = 𝑔 → ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ↔ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) |
28 |
22 27
|
imbi12d |
⊢ ( 𝑓 = 𝑔 → ( ( ( 𝑓 supp 0 ) = ( 𝑔 supp 0 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ↔ ( ( 𝑔 supp 0 ) = ( 𝑔 supp 0 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ) |
29 |
|
eqeq2 |
⊢ ( 𝑖 = ∅ → ( ( 𝑓 supp 0 ) = 𝑖 ↔ ( 𝑓 supp 0 ) = ∅ ) ) |
30 |
29
|
imbi1d |
⊢ ( 𝑖 = ∅ → ( ( ( 𝑓 supp 0 ) = 𝑖 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ↔ ( ( 𝑓 supp 0 ) = ∅ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ) |
31 |
30
|
ralbidv |
⊢ ( 𝑖 = ∅ → ( ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = 𝑖 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ↔ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ∅ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ) |
32 |
|
eqeq2 |
⊢ ( 𝑖 = ℎ → ( ( 𝑓 supp 0 ) = 𝑖 ↔ ( 𝑓 supp 0 ) = ℎ ) ) |
33 |
32
|
imbi1d |
⊢ ( 𝑖 = ℎ → ( ( ( 𝑓 supp 0 ) = 𝑖 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ↔ ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ) |
34 |
33
|
ralbidv |
⊢ ( 𝑖 = ℎ → ( ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = 𝑖 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ↔ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ) |
35 |
|
eqeq2 |
⊢ ( 𝑖 = ( ℎ ∪ { 𝑥 } ) → ( ( 𝑓 supp 0 ) = 𝑖 ↔ ( 𝑓 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ) |
36 |
35
|
imbi1d |
⊢ ( 𝑖 = ( ℎ ∪ { 𝑥 } ) → ( ( ( 𝑓 supp 0 ) = 𝑖 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ↔ ( ( 𝑓 supp 0 ) = ( ℎ ∪ { 𝑥 } ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ) |
37 |
36
|
ralbidv |
⊢ ( 𝑖 = ( ℎ ∪ { 𝑥 } ) → ( ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = 𝑖 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ↔ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ( ℎ ∪ { 𝑥 } ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ) |
38 |
|
eqeq2 |
⊢ ( 𝑖 = ( 𝑔 supp 0 ) → ( ( 𝑓 supp 0 ) = 𝑖 ↔ ( 𝑓 supp 0 ) = ( 𝑔 supp 0 ) ) ) |
39 |
38
|
imbi1d |
⊢ ( 𝑖 = ( 𝑔 supp 0 ) → ( ( ( 𝑓 supp 0 ) = 𝑖 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ↔ ( ( 𝑓 supp 0 ) = ( 𝑔 supp 0 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ) |
40 |
39
|
ralbidv |
⊢ ( 𝑖 = ( 𝑔 supp 0 ) → ( ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = 𝑖 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ↔ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ( 𝑔 supp 0 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ) |
41 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
42 |
6
|
ringcmnd |
⊢ ( 𝜑 → 𝑅 ∈ CMnd ) |
43 |
42
|
ad5antr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) → 𝑅 ∈ CMnd ) |
44 |
1
|
fvexi |
⊢ 𝐵 ∈ V |
45 |
44
|
a1i |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
46 |
45 7
|
ssexd |
⊢ ( 𝜑 → 𝐴 ∈ V ) |
47 |
|
wrdexg |
⊢ ( 𝐴 ∈ V → Word 𝐴 ∈ V ) |
48 |
46 47
|
syl |
⊢ ( 𝜑 → Word 𝐴 ∈ V ) |
49 |
48
|
ad5antr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) → Word 𝐴 ∈ V ) |
50 |
|
simp-4l |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) → 𝜑 ) |
51 |
5
|
reqabi |
⊢ ( 𝑓 ∈ 𝐹 ↔ ( 𝑓 ∈ ( ℤ ↑m Word 𝐴 ) ∧ 𝑓 finSupp 0 ) ) |
52 |
51
|
simplbi |
⊢ ( 𝑓 ∈ 𝐹 → 𝑓 ∈ ( ℤ ↑m Word 𝐴 ) ) |
53 |
52
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) → 𝑓 ∈ ( ℤ ↑m Word 𝐴 ) ) |
54 |
|
zex |
⊢ ℤ ∈ V |
55 |
54
|
a1i |
⊢ ( 𝜑 → ℤ ∈ V ) |
56 |
55 48
|
elmapd |
⊢ ( 𝜑 → ( 𝑓 ∈ ( ℤ ↑m Word 𝐴 ) ↔ 𝑓 : Word 𝐴 ⟶ ℤ ) ) |
57 |
56
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( ℤ ↑m Word 𝐴 ) ) → 𝑓 : Word 𝐴 ⟶ ℤ ) |
58 |
50 53 57
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) → 𝑓 : Word 𝐴 ⟶ ℤ ) |
59 |
58
|
ffnd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) → 𝑓 Fn Word 𝐴 ) |
60 |
59
|
ad2antrr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ∅ ) ) → 𝑓 Fn Word 𝐴 ) |
61 |
49
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ∅ ) ) → Word 𝐴 ∈ V ) |
62 |
|
0zd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ∅ ) ) → 0 ∈ ℤ ) |
63 |
|
simpr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ∅ ) ) → 𝑤 ∈ ( Word 𝐴 ∖ ∅ ) ) |
64 |
63
|
eldifad |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ∅ ) ) → 𝑤 ∈ Word 𝐴 ) |
65 |
63
|
eldifbd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ∅ ) ) → ¬ 𝑤 ∈ ∅ ) |
66 |
|
simplr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ∅ ) ) → ( 𝑓 supp 0 ) = ∅ ) |
67 |
65 66
|
neleqtrrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ∅ ) ) → ¬ 𝑤 ∈ ( 𝑓 supp 0 ) ) |
68 |
64 67
|
eldifd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ∅ ) ) → 𝑤 ∈ ( Word 𝐴 ∖ ( 𝑓 supp 0 ) ) ) |
69 |
60 61 62 68
|
fvdifsupp |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ∅ ) ) → ( 𝑓 ‘ 𝑤 ) = 0 ) |
70 |
69
|
oveq1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ∅ ) ) → ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( 0 · ( 𝑀 Σg 𝑤 ) ) ) |
71 |
2
|
ringmgp |
⊢ ( 𝑅 ∈ Ring → 𝑀 ∈ Mnd ) |
72 |
6 71
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ Mnd ) |
73 |
72
|
ad6antr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ∅ ) ) → 𝑀 ∈ Mnd ) |
74 |
|
sswrd |
⊢ ( 𝐴 ⊆ 𝐵 → Word 𝐴 ⊆ Word 𝐵 ) |
75 |
7 74
|
syl |
⊢ ( 𝜑 → Word 𝐴 ⊆ Word 𝐵 ) |
76 |
75
|
ad6antr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ∅ ) ) → Word 𝐴 ⊆ Word 𝐵 ) |
77 |
76 64
|
sseldd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ∅ ) ) → 𝑤 ∈ Word 𝐵 ) |
78 |
2 1
|
mgpbas |
⊢ 𝐵 = ( Base ‘ 𝑀 ) |
79 |
78
|
gsumwcl |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑤 ∈ Word 𝐵 ) → ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) |
80 |
73 77 79
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ∅ ) ) → ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) |
81 |
1 41 3
|
mulg0 |
⊢ ( ( 𝑀 Σg 𝑤 ) ∈ 𝐵 → ( 0 · ( 𝑀 Σg 𝑤 ) ) = ( 0g ‘ 𝑅 ) ) |
82 |
80 81
|
syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ∅ ) ) → ( 0 · ( 𝑀 Σg 𝑤 ) ) = ( 0g ‘ 𝑅 ) ) |
83 |
70 82
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ∅ ) ) → ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( 0g ‘ 𝑅 ) ) |
84 |
|
0fi |
⊢ ∅ ∈ Fin |
85 |
84
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) → ∅ ∈ Fin ) |
86 |
6
|
ringgrpd |
⊢ ( 𝜑 → 𝑅 ∈ Grp ) |
87 |
86
|
ad6antr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑅 ∈ Grp ) |
88 |
58
|
ad2antrr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑓 : Word 𝐴 ⟶ ℤ ) |
89 |
|
simpr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑤 ∈ Word 𝐴 ) |
90 |
88 89
|
ffvelcdmd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑓 ‘ 𝑤 ) ∈ ℤ ) |
91 |
72
|
ad6antr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑀 ∈ Mnd ) |
92 |
75
|
ad6antr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ Word 𝐴 ) → Word 𝐴 ⊆ Word 𝐵 ) |
93 |
92 89
|
sseldd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑤 ∈ Word 𝐵 ) |
94 |
91 93 79
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) |
95 |
1 3 87 90 94
|
mulgcld |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) |
96 |
|
0ss |
⊢ ∅ ⊆ Word 𝐴 |
97 |
96
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) → ∅ ⊆ Word 𝐴 ) |
98 |
1 41 43 49 83 85 95 97
|
gsummptres2 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ ∅ ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
99 |
|
mpt0 |
⊢ ( 𝑤 ∈ ∅ ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ∅ |
100 |
99
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) → ( 𝑤 ∈ ∅ ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ∅ ) |
101 |
100
|
oveq2d |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) → ( 𝑅 Σg ( 𝑤 ∈ ∅ ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ∅ ) ) |
102 |
41
|
gsum0 |
⊢ ( 𝑅 Σg ∅ ) = ( 0g ‘ 𝑅 ) |
103 |
102
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) → ( 𝑅 Σg ∅ ) = ( 0g ‘ 𝑅 ) ) |
104 |
98 101 103
|
3eqtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 0g ‘ 𝑅 ) ) |
105 |
|
subrgsubg |
⊢ ( 𝑡 ∈ ( SubRing ‘ 𝑅 ) → 𝑡 ∈ ( SubGrp ‘ 𝑅 ) ) |
106 |
41
|
subg0cl |
⊢ ( 𝑡 ∈ ( SubGrp ‘ 𝑅 ) → ( 0g ‘ 𝑅 ) ∈ 𝑡 ) |
107 |
105 106
|
syl |
⊢ ( 𝑡 ∈ ( SubRing ‘ 𝑅 ) → ( 0g ‘ 𝑅 ) ∈ 𝑡 ) |
108 |
107
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) → ( 0g ‘ 𝑅 ) ∈ 𝑡 ) |
109 |
108
|
ad4antr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) → ( 0g ‘ 𝑅 ) ∈ 𝑡 ) |
110 |
104 109
|
eqeltrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) ∧ ( 𝑓 supp 0 ) = ∅ ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) |
111 |
110
|
ex |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑓 ∈ 𝐹 ) → ( ( 𝑓 supp 0 ) = ∅ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) |
112 |
111
|
ralrimiva |
⊢ ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) → ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ∅ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) |
113 |
42
|
ad7antr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 𝑅 ∈ CMnd ) |
114 |
48
|
ad7antr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → Word 𝐴 ∈ V ) |
115 |
|
simp-5l |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) → 𝜑 ) |
116 |
|
breq1 |
⊢ ( 𝑓 = 𝑒 → ( 𝑓 finSupp 0 ↔ 𝑒 finSupp 0 ) ) |
117 |
116 5
|
elrab2 |
⊢ ( 𝑒 ∈ 𝐹 ↔ ( 𝑒 ∈ ( ℤ ↑m Word 𝐴 ) ∧ 𝑒 finSupp 0 ) ) |
118 |
117
|
simplbi |
⊢ ( 𝑒 ∈ 𝐹 → 𝑒 ∈ ( ℤ ↑m Word 𝐴 ) ) |
119 |
55 48
|
elmapd |
⊢ ( 𝜑 → ( 𝑒 ∈ ( ℤ ↑m Word 𝐴 ) ↔ 𝑒 : Word 𝐴 ⟶ ℤ ) ) |
120 |
119
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ( ℤ ↑m Word 𝐴 ) ) → 𝑒 : Word 𝐴 ⟶ ℤ ) |
121 |
118 120
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝐹 ) → 𝑒 : Word 𝐴 ⟶ ℤ ) |
122 |
115 121
|
sylancom |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) → 𝑒 : Word 𝐴 ⟶ ℤ ) |
123 |
122
|
adantl3r |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) → 𝑒 : Word 𝐴 ⟶ ℤ ) |
124 |
123
|
ffnd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) → 𝑒 Fn Word 𝐴 ) |
125 |
124
|
ad2antrr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( 𝑒 supp 0 ) ) ) → 𝑒 Fn Word 𝐴 ) |
126 |
114
|
adantr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( 𝑒 supp 0 ) ) ) → Word 𝐴 ∈ V ) |
127 |
|
0zd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( 𝑒 supp 0 ) ) ) → 0 ∈ ℤ ) |
128 |
|
simpr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( 𝑒 supp 0 ) ) ) → 𝑤 ∈ ( Word 𝐴 ∖ ( 𝑒 supp 0 ) ) ) |
129 |
125 126 127 128
|
fvdifsupp |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( 𝑒 supp 0 ) ) ) → ( 𝑒 ‘ 𝑤 ) = 0 ) |
130 |
129
|
oveq1d |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( 𝑒 supp 0 ) ) ) → ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( 0 · ( 𝑀 Σg 𝑤 ) ) ) |
131 |
72
|
ad8antr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( 𝑒 supp 0 ) ) ) → 𝑀 ∈ Mnd ) |
132 |
75
|
ad8antr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( 𝑒 supp 0 ) ) ) → Word 𝐴 ⊆ Word 𝐵 ) |
133 |
128
|
eldifad |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( 𝑒 supp 0 ) ) ) → 𝑤 ∈ Word 𝐴 ) |
134 |
132 133
|
sseldd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( 𝑒 supp 0 ) ) ) → 𝑤 ∈ Word 𝐵 ) |
135 |
131 134 79
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( 𝑒 supp 0 ) ) ) → ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) |
136 |
135 81
|
syl |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( 𝑒 supp 0 ) ) ) → ( 0 · ( 𝑀 Σg 𝑤 ) ) = ( 0g ‘ 𝑅 ) ) |
137 |
130 136
|
eqtrd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( 𝑒 supp 0 ) ) ) → ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( 0g ‘ 𝑅 ) ) |
138 |
117
|
simprbi |
⊢ ( 𝑒 ∈ 𝐹 → 𝑒 finSupp 0 ) |
139 |
138
|
ad2antlr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 𝑒 finSupp 0 ) |
140 |
139
|
fsuppimpd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑒 supp 0 ) ∈ Fin ) |
141 |
86
|
ad8antr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑅 ∈ Grp ) |
142 |
123
|
ad2antrr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑒 : Word 𝐴 ⟶ ℤ ) |
143 |
|
simpr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑤 ∈ Word 𝐴 ) |
144 |
142 143
|
ffvelcdmd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑒 ‘ 𝑤 ) ∈ ℤ ) |
145 |
72
|
ad8antr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑀 ∈ Mnd ) |
146 |
75
|
ad7antr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → Word 𝐴 ⊆ Word 𝐵 ) |
147 |
146
|
sselda |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑤 ∈ Word 𝐵 ) |
148 |
145 147 79
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) |
149 |
1 3 141 144 148
|
mulgcld |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) |
150 |
|
suppssdm |
⊢ ( 𝑒 supp 0 ) ⊆ dom 𝑒 |
151 |
123
|
adantr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 𝑒 : Word 𝐴 ⟶ ℤ ) |
152 |
150 151
|
fssdm |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑒 supp 0 ) ⊆ Word 𝐴 ) |
153 |
1 41 113 114 137 140 149 152
|
gsummptres2 |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ ( 𝑒 supp 0 ) ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
154 |
153
|
adantllr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ ( 𝑒 supp 0 ) ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
155 |
|
simpr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) |
156 |
155
|
mpteq1d |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑤 ∈ ( 𝑒 supp 0 ) ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ ( ℎ ∪ { 𝑥 } ) ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
157 |
156
|
oveq2d |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑅 Σg ( 𝑤 ∈ ( 𝑒 supp 0 ) ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ ( ℎ ∪ { 𝑥 } ) ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
158 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
159 |
|
breq1 |
⊢ ( 𝑓 = 𝑔 → ( 𝑓 finSupp 0 ↔ 𝑔 finSupp 0 ) ) |
160 |
159 5
|
elrab2 |
⊢ ( 𝑔 ∈ 𝐹 ↔ ( 𝑔 ∈ ( ℤ ↑m Word 𝐴 ) ∧ 𝑔 finSupp 0 ) ) |
161 |
160
|
simprbi |
⊢ ( 𝑔 ∈ 𝐹 → 𝑔 finSupp 0 ) |
162 |
161
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) → 𝑔 finSupp 0 ) |
163 |
162
|
fsuppimpd |
⊢ ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) → ( 𝑔 supp 0 ) ∈ Fin ) |
164 |
163
|
ad4antr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑔 supp 0 ) ∈ Fin ) |
165 |
|
simp-4r |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ℎ ⊆ ( 𝑔 supp 0 ) ) |
166 |
164 165
|
ssfid |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ℎ ∈ Fin ) |
167 |
86
|
ad8antr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ℎ ) → 𝑅 ∈ Grp ) |
168 |
151
|
adantr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ℎ ) → 𝑒 : Word 𝐴 ⟶ ℤ ) |
169 |
|
suppssdm |
⊢ ( 𝑔 supp 0 ) ⊆ dom 𝑔 |
170 |
|
simp-7l |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 𝜑 ) |
171 |
|
simp-5r |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 𝑔 ∈ 𝐹 ) |
172 |
160
|
simplbi |
⊢ ( 𝑔 ∈ 𝐹 → 𝑔 ∈ ( ℤ ↑m Word 𝐴 ) ) |
173 |
55 48
|
elmapd |
⊢ ( 𝜑 → ( 𝑔 ∈ ( ℤ ↑m Word 𝐴 ) ↔ 𝑔 : Word 𝐴 ⟶ ℤ ) ) |
174 |
173
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( ℤ ↑m Word 𝐴 ) ) → 𝑔 : Word 𝐴 ⟶ ℤ ) |
175 |
172 174
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → 𝑔 : Word 𝐴 ⟶ ℤ ) |
176 |
170 171 175
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 𝑔 : Word 𝐴 ⟶ ℤ ) |
177 |
169 176
|
fssdm |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑔 supp 0 ) ⊆ Word 𝐴 ) |
178 |
165 177
|
sstrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ℎ ⊆ Word 𝐴 ) |
179 |
178
|
sselda |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ℎ ) → 𝑤 ∈ Word 𝐴 ) |
180 |
168 179
|
ffvelcdmd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ℎ ) → ( 𝑒 ‘ 𝑤 ) ∈ ℤ ) |
181 |
179 148
|
syldan |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ℎ ) → ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) |
182 |
1 3 167 180 181
|
mulgcld |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ℎ ) → ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) |
183 |
|
simpllr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) |
184 |
183
|
eldifbd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ¬ 𝑥 ∈ ℎ ) |
185 |
170 86
|
syl |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 𝑅 ∈ Grp ) |
186 |
183
|
eldifad |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 𝑥 ∈ ( 𝑔 supp 0 ) ) |
187 |
177 186
|
sseldd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 𝑥 ∈ Word 𝐴 ) |
188 |
151 187
|
ffvelcdmd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑒 ‘ 𝑥 ) ∈ ℤ ) |
189 |
170 72
|
syl |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 𝑀 ∈ Mnd ) |
190 |
146 187
|
sseldd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 𝑥 ∈ Word 𝐵 ) |
191 |
78
|
gsumwcl |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑥 ∈ Word 𝐵 ) → ( 𝑀 Σg 𝑥 ) ∈ 𝐵 ) |
192 |
189 190 191
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑀 Σg 𝑥 ) ∈ 𝐵 ) |
193 |
1 3 185 188 192
|
mulgcld |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( ( 𝑒 ‘ 𝑥 ) · ( 𝑀 Σg 𝑥 ) ) ∈ 𝐵 ) |
194 |
|
fveq2 |
⊢ ( 𝑤 = 𝑥 → ( 𝑒 ‘ 𝑤 ) = ( 𝑒 ‘ 𝑥 ) ) |
195 |
|
oveq2 |
⊢ ( 𝑤 = 𝑥 → ( 𝑀 Σg 𝑤 ) = ( 𝑀 Σg 𝑥 ) ) |
196 |
194 195
|
oveq12d |
⊢ ( 𝑤 = 𝑥 → ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( ( 𝑒 ‘ 𝑥 ) · ( 𝑀 Σg 𝑥 ) ) ) |
197 |
1 158 113 166 182 183 184 193 196
|
gsumunsn |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑅 Σg ( 𝑤 ∈ ( ℎ ∪ { 𝑥 } ) ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( ( 𝑅 Σg ( 𝑤 ∈ ℎ ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑒 ‘ 𝑥 ) · ( 𝑀 Σg 𝑥 ) ) ) ) |
198 |
197
|
adantllr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑅 Σg ( 𝑤 ∈ ( ℎ ∪ { 𝑥 } ) ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( ( 𝑅 Σg ( 𝑤 ∈ ℎ ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑒 ‘ 𝑥 ) · ( 𝑀 Σg 𝑥 ) ) ) ) |
199 |
154 157 198
|
3eqtrd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( ( 𝑅 Σg ( 𝑤 ∈ ℎ ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑒 ‘ 𝑥 ) · ( 𝑀 Σg 𝑥 ) ) ) ) |
200 |
105
|
ad8antlr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 𝑡 ∈ ( SubGrp ‘ 𝑅 ) ) |
201 |
124
|
adantr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 𝑒 Fn Word 𝐴 ) |
202 |
|
0zd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 0 ∈ ℤ ) |
203 |
201 187 202
|
fmptunsnop |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑦 ∈ Word 𝐴 ↦ if ( 𝑦 = 𝑥 , 0 , ( 𝑒 ‘ 𝑦 ) ) ) = ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ) |
204 |
203
|
adantr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) → ( 𝑦 ∈ Word 𝐴 ↦ if ( 𝑦 = 𝑥 , 0 , ( 𝑒 ‘ 𝑦 ) ) ) = ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ) |
205 |
204
|
fveq1d |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) → ( ( 𝑦 ∈ Word 𝐴 ↦ if ( 𝑦 = 𝑥 , 0 , ( 𝑒 ‘ 𝑦 ) ) ) ‘ 𝑤 ) = ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) ) |
206 |
|
eqid |
⊢ ( 𝑦 ∈ Word 𝐴 ↦ if ( 𝑦 = 𝑥 , 0 , ( 𝑒 ‘ 𝑦 ) ) ) = ( 𝑦 ∈ Word 𝐴 ↦ if ( 𝑦 = 𝑥 , 0 , ( 𝑒 ‘ 𝑦 ) ) ) |
207 |
|
eqidd |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) ∧ 𝑦 = 𝑤 ) ∧ 𝑦 = 𝑥 ) → 0 = 0 ) |
208 |
201
|
ad3antrrr |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) ∧ 𝑦 = 𝑤 ) ∧ ¬ 𝑦 = 𝑥 ) → 𝑒 Fn Word 𝐴 ) |
209 |
114
|
ad3antrrr |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) ∧ 𝑦 = 𝑤 ) ∧ ¬ 𝑦 = 𝑥 ) → Word 𝐴 ∈ V ) |
210 |
|
0zd |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) ∧ 𝑦 = 𝑤 ) ∧ ¬ 𝑦 = 𝑥 ) → 0 ∈ ℤ ) |
211 |
|
simplr |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) ∧ 𝑦 = 𝑤 ) ∧ ¬ 𝑦 = 𝑥 ) → 𝑦 = 𝑤 ) |
212 |
|
simpllr |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) ∧ 𝑦 = 𝑤 ) ∧ ¬ 𝑦 = 𝑥 ) → 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) |
213 |
212
|
eldifad |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) ∧ 𝑦 = 𝑤 ) ∧ ¬ 𝑦 = 𝑥 ) → 𝑤 ∈ Word 𝐴 ) |
214 |
211 213
|
eqeltrd |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) ∧ 𝑦 = 𝑤 ) ∧ ¬ 𝑦 = 𝑥 ) → 𝑦 ∈ Word 𝐴 ) |
215 |
|
simp-4r |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) ∧ 𝑦 = 𝑤 ) ∧ ¬ 𝑦 = 𝑥 ) → ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) |
216 |
212
|
eldifbd |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) ∧ 𝑦 = 𝑤 ) ∧ ¬ 𝑦 = 𝑥 ) → ¬ 𝑤 ∈ ℎ ) |
217 |
211 216
|
eqneltrd |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) ∧ 𝑦 = 𝑤 ) ∧ ¬ 𝑦 = 𝑥 ) → ¬ 𝑦 ∈ ℎ ) |
218 |
|
simpr |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) ∧ 𝑦 = 𝑤 ) ∧ ¬ 𝑦 = 𝑥 ) → ¬ 𝑦 = 𝑥 ) |
219 |
218
|
neqned |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) ∧ 𝑦 = 𝑤 ) ∧ ¬ 𝑦 = 𝑥 ) → 𝑦 ≠ 𝑥 ) |
220 |
|
nelsn |
⊢ ( 𝑦 ≠ 𝑥 → ¬ 𝑦 ∈ { 𝑥 } ) |
221 |
219 220
|
syl |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) ∧ 𝑦 = 𝑤 ) ∧ ¬ 𝑦 = 𝑥 ) → ¬ 𝑦 ∈ { 𝑥 } ) |
222 |
|
nelun |
⊢ ( ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) → ( ¬ 𝑦 ∈ ( 𝑒 supp 0 ) ↔ ( ¬ 𝑦 ∈ ℎ ∧ ¬ 𝑦 ∈ { 𝑥 } ) ) ) |
223 |
222
|
biimpar |
⊢ ( ( ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ∧ ( ¬ 𝑦 ∈ ℎ ∧ ¬ 𝑦 ∈ { 𝑥 } ) ) → ¬ 𝑦 ∈ ( 𝑒 supp 0 ) ) |
224 |
215 217 221 223
|
syl12anc |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) ∧ 𝑦 = 𝑤 ) ∧ ¬ 𝑦 = 𝑥 ) → ¬ 𝑦 ∈ ( 𝑒 supp 0 ) ) |
225 |
214 224
|
eldifd |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) ∧ 𝑦 = 𝑤 ) ∧ ¬ 𝑦 = 𝑥 ) → 𝑦 ∈ ( Word 𝐴 ∖ ( 𝑒 supp 0 ) ) ) |
226 |
208 209 210 225
|
fvdifsupp |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) ∧ 𝑦 = 𝑤 ) ∧ ¬ 𝑦 = 𝑥 ) → ( 𝑒 ‘ 𝑦 ) = 0 ) |
227 |
207 226
|
ifeqda |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) ∧ 𝑦 = 𝑤 ) → if ( 𝑦 = 𝑥 , 0 , ( 𝑒 ‘ 𝑦 ) ) = 0 ) |
228 |
|
simpr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) → 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) |
229 |
228
|
eldifad |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) → 𝑤 ∈ Word 𝐴 ) |
230 |
|
0zd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) → 0 ∈ ℤ ) |
231 |
206 227 229 230
|
fvmptd2 |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) → ( ( 𝑦 ∈ Word 𝐴 ↦ if ( 𝑦 = 𝑥 , 0 , ( 𝑒 ‘ 𝑦 ) ) ) ‘ 𝑤 ) = 0 ) |
232 |
205 231
|
eqtr3d |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) → ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) = 0 ) |
233 |
232
|
oveq1d |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) → ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( 0 · ( 𝑀 Σg 𝑤 ) ) ) |
234 |
229 148
|
syldan |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) → ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) |
235 |
234 81
|
syl |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) → ( 0 · ( 𝑀 Σg 𝑤 ) ) = ( 0g ‘ 𝑅 ) ) |
236 |
233 235
|
eqtrd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ℎ ) ) → ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( 0g ‘ 𝑅 ) ) |
237 |
203
|
adantr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑦 ∈ Word 𝐴 ↦ if ( 𝑦 = 𝑥 , 0 , ( 𝑒 ‘ 𝑦 ) ) ) = ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ) |
238 |
237
|
fveq1d |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( 𝑦 ∈ Word 𝐴 ↦ if ( 𝑦 = 𝑥 , 0 , ( 𝑒 ‘ 𝑦 ) ) ) ‘ 𝑤 ) = ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) ) |
239 |
|
0zd |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑦 ∈ Word 𝐴 ) ∧ 𝑦 = 𝑥 ) → 0 ∈ ℤ ) |
240 |
151
|
ad2antrr |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑦 ∈ Word 𝐴 ) ∧ ¬ 𝑦 = 𝑥 ) → 𝑒 : Word 𝐴 ⟶ ℤ ) |
241 |
|
simplr |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑦 ∈ Word 𝐴 ) ∧ ¬ 𝑦 = 𝑥 ) → 𝑦 ∈ Word 𝐴 ) |
242 |
240 241
|
ffvelcdmd |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑦 ∈ Word 𝐴 ) ∧ ¬ 𝑦 = 𝑥 ) → ( 𝑒 ‘ 𝑦 ) ∈ ℤ ) |
243 |
239 242
|
ifclda |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑦 ∈ Word 𝐴 ) → if ( 𝑦 = 𝑥 , 0 , ( 𝑒 ‘ 𝑦 ) ) ∈ ℤ ) |
244 |
243
|
fmpttd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑦 ∈ Word 𝐴 ↦ if ( 𝑦 = 𝑥 , 0 , ( 𝑒 ‘ 𝑦 ) ) ) : Word 𝐴 ⟶ ℤ ) |
245 |
244
|
ffvelcdmda |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( 𝑦 ∈ Word 𝐴 ↦ if ( 𝑦 = 𝑥 , 0 , ( 𝑒 ‘ 𝑦 ) ) ) ‘ 𝑤 ) ∈ ℤ ) |
246 |
238 245
|
eqeltrrd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) ∈ ℤ ) |
247 |
1 3 141 246 148
|
mulgcld |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) |
248 |
1 41 113 114 236 166 247 178
|
gsummptres2 |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ ℎ ↦ ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
249 |
248
|
adantllr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ ℎ ↦ ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
250 |
203
|
adantr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ℎ ) → ( 𝑦 ∈ Word 𝐴 ↦ if ( 𝑦 = 𝑥 , 0 , ( 𝑒 ‘ 𝑦 ) ) ) = ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ) |
251 |
250
|
fveq1d |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ℎ ) → ( ( 𝑦 ∈ Word 𝐴 ↦ if ( 𝑦 = 𝑥 , 0 , ( 𝑒 ‘ 𝑦 ) ) ) ‘ 𝑤 ) = ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) ) |
252 |
|
simpr |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ℎ ) ∧ 𝑦 = 𝑤 ) → 𝑦 = 𝑤 ) |
253 |
|
simplr |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ℎ ) ∧ 𝑦 = 𝑤 ) → 𝑤 ∈ ℎ ) |
254 |
252 253
|
eqeltrd |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ℎ ) ∧ 𝑦 = 𝑤 ) → 𝑦 ∈ ℎ ) |
255 |
184
|
ad2antrr |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ℎ ) ∧ 𝑦 = 𝑤 ) → ¬ 𝑥 ∈ ℎ ) |
256 |
|
nelneq |
⊢ ( ( 𝑦 ∈ ℎ ∧ ¬ 𝑥 ∈ ℎ ) → ¬ 𝑦 = 𝑥 ) |
257 |
254 255 256
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ℎ ) ∧ 𝑦 = 𝑤 ) → ¬ 𝑦 = 𝑥 ) |
258 |
257
|
iffalsed |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ℎ ) ∧ 𝑦 = 𝑤 ) → if ( 𝑦 = 𝑥 , 0 , ( 𝑒 ‘ 𝑦 ) ) = ( 𝑒 ‘ 𝑦 ) ) |
259 |
252
|
fveq2d |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ℎ ) ∧ 𝑦 = 𝑤 ) → ( 𝑒 ‘ 𝑦 ) = ( 𝑒 ‘ 𝑤 ) ) |
260 |
258 259
|
eqtrd |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ℎ ) ∧ 𝑦 = 𝑤 ) → if ( 𝑦 = 𝑥 , 0 , ( 𝑒 ‘ 𝑦 ) ) = ( 𝑒 ‘ 𝑤 ) ) |
261 |
206 260 179 180
|
fvmptd2 |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ℎ ) → ( ( 𝑦 ∈ Word 𝐴 ↦ if ( 𝑦 = 𝑥 , 0 , ( 𝑒 ‘ 𝑦 ) ) ) ‘ 𝑤 ) = ( 𝑒 ‘ 𝑤 ) ) |
262 |
251 261
|
eqtr3d |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ℎ ) → ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) = ( 𝑒 ‘ 𝑤 ) ) |
263 |
262
|
oveq1d |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑤 ∈ ℎ ) → ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
264 |
263
|
mpteq2dva |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑤 ∈ ℎ ↦ ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ ℎ ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
265 |
264
|
adantllr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑤 ∈ ℎ ↦ ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ ℎ ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
266 |
265
|
oveq2d |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑅 Σg ( 𝑤 ∈ ℎ ↦ ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ ℎ ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
267 |
249 266
|
eqtrd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ ℎ ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
268 |
|
simplr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 𝑒 ∈ 𝐹 ) |
269 |
268
|
resexd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∈ V ) |
270 |
|
snex |
⊢ { 〈 𝑥 , 0 〉 } ∈ V |
271 |
270
|
a1i |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → { 〈 𝑥 , 0 〉 } ∈ V ) |
272 |
269 271 202
|
suppun2 |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) supp 0 ) = ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) supp 0 ) ∪ ( { 〈 𝑥 , 0 〉 } supp 0 ) ) ) |
273 |
114 202 201
|
fdifsupp |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) supp 0 ) = ( ( 𝑒 supp 0 ) ∖ { 𝑥 } ) ) |
274 |
|
simpr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) |
275 |
274
|
difeq1d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( ( 𝑒 supp 0 ) ∖ { 𝑥 } ) = ( ( ℎ ∪ { 𝑥 } ) ∖ { 𝑥 } ) ) |
276 |
|
disjsn |
⊢ ( ( ℎ ∩ { 𝑥 } ) = ∅ ↔ ¬ 𝑥 ∈ ℎ ) |
277 |
|
undif5 |
⊢ ( ( ℎ ∩ { 𝑥 } ) = ∅ → ( ( ℎ ∪ { 𝑥 } ) ∖ { 𝑥 } ) = ℎ ) |
278 |
276 277
|
sylbir |
⊢ ( ¬ 𝑥 ∈ ℎ → ( ( ℎ ∪ { 𝑥 } ) ∖ { 𝑥 } ) = ℎ ) |
279 |
184 278
|
syl |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( ( ℎ ∪ { 𝑥 } ) ∖ { 𝑥 } ) = ℎ ) |
280 |
273 275 279
|
3eqtrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) supp 0 ) = ℎ ) |
281 |
|
vex |
⊢ 𝑥 ∈ V |
282 |
|
c0ex |
⊢ 0 ∈ V |
283 |
281 282
|
xpsn |
⊢ ( { 𝑥 } × { 0 } ) = { 〈 𝑥 , 0 〉 } |
284 |
283
|
oveq1i |
⊢ ( ( { 𝑥 } × { 0 } ) supp 0 ) = ( { 〈 𝑥 , 0 〉 } supp 0 ) |
285 |
|
fczsupp0 |
⊢ ( ( { 𝑥 } × { 0 } ) supp 0 ) = ∅ |
286 |
284 285
|
eqtr3i |
⊢ ( { 〈 𝑥 , 0 〉 } supp 0 ) = ∅ |
287 |
286
|
a1i |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( { 〈 𝑥 , 0 〉 } supp 0 ) = ∅ ) |
288 |
280 287
|
uneq12d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) supp 0 ) ∪ ( { 〈 𝑥 , 0 〉 } supp 0 ) ) = ( ℎ ∪ ∅ ) ) |
289 |
|
un0 |
⊢ ( ℎ ∪ ∅ ) = ℎ |
290 |
289
|
a1i |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( ℎ ∪ ∅ ) = ℎ ) |
291 |
272 288 290
|
3eqtrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) supp 0 ) = ℎ ) |
292 |
291
|
adantllr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) supp 0 ) = ℎ ) |
293 |
|
oveq1 |
⊢ ( 𝑓 = ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) → ( 𝑓 supp 0 ) = ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) supp 0 ) ) |
294 |
293
|
eqeq1d |
⊢ ( 𝑓 = ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) → ( ( 𝑓 supp 0 ) = ℎ ↔ ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) supp 0 ) = ℎ ) ) |
295 |
|
fveq1 |
⊢ ( 𝑓 = ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) → ( 𝑓 ‘ 𝑤 ) = ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) ) |
296 |
295
|
oveq1d |
⊢ ( 𝑓 = ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) → ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
297 |
296
|
mpteq2dv |
⊢ ( 𝑓 = ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
298 |
297
|
oveq2d |
⊢ ( 𝑓 = ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
299 |
298
|
eleq1d |
⊢ ( 𝑓 = ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) → ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ↔ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) |
300 |
294 299
|
imbi12d |
⊢ ( 𝑓 = ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) → ( ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ↔ ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ) |
301 |
|
simpllr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) |
302 |
|
breq1 |
⊢ ( 𝑓 = ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) → ( 𝑓 finSupp 0 ↔ ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) finSupp 0 ) ) |
303 |
54
|
a1i |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ℤ ∈ V ) |
304 |
114
|
adantllr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → Word 𝐴 ∈ V ) |
305 |
203
|
adantllr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑦 ∈ Word 𝐴 ↦ if ( 𝑦 = 𝑥 , 0 , ( 𝑒 ‘ 𝑦 ) ) ) = ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ) |
306 |
|
0zd |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑦 ∈ Word 𝐴 ) ∧ 𝑦 = 𝑥 ) → 0 ∈ ℤ ) |
307 |
|
simp-10l |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑦 ∈ Word 𝐴 ) ∧ ¬ 𝑦 = 𝑥 ) → 𝜑 ) |
308 |
|
simp-4r |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑦 ∈ Word 𝐴 ) ∧ ¬ 𝑦 = 𝑥 ) → 𝑒 ∈ 𝐹 ) |
309 |
307 308 121
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑦 ∈ Word 𝐴 ) ∧ ¬ 𝑦 = 𝑥 ) → 𝑒 : Word 𝐴 ⟶ ℤ ) |
310 |
|
simplr |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑦 ∈ Word 𝐴 ) ∧ ¬ 𝑦 = 𝑥 ) → 𝑦 ∈ Word 𝐴 ) |
311 |
309 310
|
ffvelcdmd |
⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑦 ∈ Word 𝐴 ) ∧ ¬ 𝑦 = 𝑥 ) → ( 𝑒 ‘ 𝑦 ) ∈ ℤ ) |
312 |
306 311
|
ifclda |
⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ∧ 𝑦 ∈ Word 𝐴 ) → if ( 𝑦 = 𝑥 , 0 , ( 𝑒 ‘ 𝑦 ) ) ∈ ℤ ) |
313 |
312
|
fmpttd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑦 ∈ Word 𝐴 ↦ if ( 𝑦 = 𝑥 , 0 , ( 𝑒 ‘ 𝑦 ) ) ) : Word 𝐴 ⟶ ℤ ) |
314 |
305 313
|
feq1dd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) : Word 𝐴 ⟶ ℤ ) |
315 |
303 304 314
|
elmapdd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ∈ ( ℤ ↑m Word 𝐴 ) ) |
316 |
|
0zd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 0 ∈ ℤ ) |
317 |
314
|
ffund |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → Fun ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ) |
318 |
166
|
adantllr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ℎ ∈ Fin ) |
319 |
292 318
|
eqeltrd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) supp 0 ) ∈ Fin ) |
320 |
315 316 317 319
|
isfsuppd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) finSupp 0 ) |
321 |
302 315 320
|
elrabd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ∈ { 𝑓 ∈ ( ℤ ↑m Word 𝐴 ) ∣ 𝑓 finSupp 0 } ) |
322 |
321 5
|
eleqtrrdi |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ∈ 𝐹 ) |
323 |
300 301 322
|
rspcdva |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) |
324 |
292 323
|
mpd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( ( 𝑒 ↾ ( Word 𝐴 ∖ { 𝑥 } ) ) ∪ { 〈 𝑥 , 0 〉 } ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) |
325 |
267 324
|
eqeltrrd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑅 Σg ( 𝑤 ∈ ℎ ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) |
326 |
86
|
ad8antr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 𝑅 ∈ Grp ) |
327 |
2
|
subrgsubm |
⊢ ( 𝑡 ∈ ( SubRing ‘ 𝑅 ) → 𝑡 ∈ ( SubMnd ‘ 𝑀 ) ) |
328 |
327
|
ad8antlr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 𝑡 ∈ ( SubMnd ‘ 𝑀 ) ) |
329 |
|
sswrd |
⊢ ( 𝐴 ⊆ 𝑡 → Word 𝐴 ⊆ Word 𝑡 ) |
330 |
329
|
ad7antlr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → Word 𝐴 ⊆ Word 𝑡 ) |
331 |
187
|
adantllr |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 𝑥 ∈ Word 𝐴 ) |
332 |
330 331
|
sseldd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 𝑥 ∈ Word 𝑡 ) |
333 |
|
gsumwsubmcl |
⊢ ( ( 𝑡 ∈ ( SubMnd ‘ 𝑀 ) ∧ 𝑥 ∈ Word 𝑡 ) → ( 𝑀 Σg 𝑥 ) ∈ 𝑡 ) |
334 |
328 332 333
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑀 Σg 𝑥 ) ∈ 𝑡 ) |
335 |
123
|
ad4ant13 |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → 𝑒 : Word 𝐴 ⟶ ℤ ) |
336 |
335 331
|
ffvelcdmd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑒 ‘ 𝑥 ) ∈ ℤ ) |
337 |
1 3 326 334 200 336
|
subgmulgcld |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( ( 𝑒 ‘ 𝑥 ) · ( 𝑀 Σg 𝑥 ) ) ∈ 𝑡 ) |
338 |
158 200 325 337
|
subgcld |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( ( 𝑅 Σg ( 𝑤 ∈ ℎ ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑒 ‘ 𝑥 ) · ( 𝑀 Σg 𝑥 ) ) ) ∈ 𝑡 ) |
339 |
199 338
|
eqeltrd |
⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) ∧ ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) |
340 |
339
|
ex |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ∧ 𝑒 ∈ 𝐹 ) → ( ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) |
341 |
340
|
ralrimiva |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ∧ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) → ∀ 𝑒 ∈ 𝐹 ( ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) |
342 |
341
|
ex |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ℎ ⊆ ( 𝑔 supp 0 ) ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) → ( ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) → ∀ 𝑒 ∈ 𝐹 ( ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ) |
343 |
342
|
anasss |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ( ℎ ⊆ ( 𝑔 supp 0 ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ) → ( ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) → ∀ 𝑒 ∈ 𝐹 ( ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ) |
344 |
|
oveq1 |
⊢ ( 𝑒 = 𝑓 → ( 𝑒 supp 0 ) = ( 𝑓 supp 0 ) ) |
345 |
344
|
eqeq1d |
⊢ ( 𝑒 = 𝑓 → ( ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ↔ ( 𝑓 supp 0 ) = ( ℎ ∪ { 𝑥 } ) ) ) |
346 |
|
fveq1 |
⊢ ( 𝑒 = 𝑓 → ( 𝑒 ‘ 𝑤 ) = ( 𝑓 ‘ 𝑤 ) ) |
347 |
346
|
oveq1d |
⊢ ( 𝑒 = 𝑓 → ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
348 |
347
|
mpteq2dv |
⊢ ( 𝑒 = 𝑓 → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
349 |
348
|
oveq2d |
⊢ ( 𝑒 = 𝑓 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
350 |
349
|
eleq1d |
⊢ ( 𝑒 = 𝑓 → ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ↔ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) |
351 |
345 350
|
imbi12d |
⊢ ( 𝑒 = 𝑓 → ( ( ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ↔ ( ( 𝑓 supp 0 ) = ( ℎ ∪ { 𝑥 } ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ) |
352 |
351
|
cbvralvw |
⊢ ( ∀ 𝑒 ∈ 𝐹 ( ( 𝑒 supp 0 ) = ( ℎ ∪ { 𝑥 } ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑒 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ↔ ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ( ℎ ∪ { 𝑥 } ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) |
353 |
343 352
|
imbitrdi |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) ∧ ( ℎ ⊆ ( 𝑔 supp 0 ) ∧ 𝑥 ∈ ( ( 𝑔 supp 0 ) ∖ ℎ ) ) ) → ( ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) → ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ( ℎ ∪ { 𝑥 } ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) ) |
354 |
31 34 37 40 112 353 163
|
findcard2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) → ∀ 𝑓 ∈ 𝐹 ( ( 𝑓 supp 0 ) = ( 𝑔 supp 0 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑓 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) |
355 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) → 𝑔 ∈ 𝐹 ) |
356 |
28 354 355
|
rspcdva |
⊢ ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) → ( ( 𝑔 supp 0 ) = ( 𝑔 supp 0 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) ) |
357 |
20 356
|
mpd |
⊢ ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑔 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) |
358 |
357
|
ad4ant13 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑠 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑠 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑡 ) |
359 |
19 358
|
eqeltrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑠 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑠 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → 𝑠 ∈ 𝑡 ) |
360 |
|
eqid |
⊢ ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) = ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
361 |
8
|
eleq2i |
⊢ ( 𝑠 ∈ 𝑆 ↔ 𝑠 ∈ ran ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
362 |
361
|
biimpi |
⊢ ( 𝑠 ∈ 𝑆 → 𝑠 ∈ ran ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
363 |
362
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑠 ∈ 𝑆 ) → 𝑠 ∈ ran ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
364 |
360 363
|
elrnmpt2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑠 ∈ 𝑆 ) → ∃ 𝑔 ∈ 𝐹 𝑠 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
365 |
359 364
|
r19.29a |
⊢ ( ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) ∧ 𝑠 ∈ 𝑆 ) → 𝑠 ∈ 𝑡 ) |
366 |
365
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) → ( 𝑠 ∈ 𝑆 → 𝑠 ∈ 𝑡 ) ) |
367 |
366
|
ssrdv |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) ∧ 𝐴 ⊆ 𝑡 ) → 𝑆 ⊆ 𝑡 ) |
368 |
367
|
ex |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ) → ( 𝐴 ⊆ 𝑡 → 𝑆 ⊆ 𝑡 ) ) |
369 |
368
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ( 𝐴 ⊆ 𝑡 → 𝑆 ⊆ 𝑡 ) ) |
370 |
|
ssintrab |
⊢ ( 𝑆 ⊆ ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } ↔ ∀ 𝑡 ∈ ( SubRing ‘ 𝑅 ) ( 𝐴 ⊆ 𝑡 → 𝑆 ⊆ 𝑡 ) ) |
371 |
369 370
|
sylibr |
⊢ ( 𝜑 → 𝑆 ⊆ ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } ) |
372 |
18 371
|
eqssd |
⊢ ( 𝜑 → ∩ { 𝑡 ∈ ( SubRing ‘ 𝑅 ) ∣ 𝐴 ⊆ 𝑡 } = 𝑆 ) |
373 |
12 372
|
eqtrd |
⊢ ( 𝜑 → ( 𝑁 ‘ 𝐴 ) = 𝑆 ) |