| 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 ) |