Step |
Hyp |
Ref |
Expression |
1 |
|
mplmon.s |
⊢ 𝑃 = ( 𝐼 mPoly 𝑅 ) |
2 |
|
mplmon.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
3 |
|
mplmon.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
4 |
|
mplmon.o |
⊢ 1 = ( 1r ‘ 𝑅 ) |
5 |
|
mplmon.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
6 |
|
mplmon.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
7 |
|
mplmon.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
8 |
|
mplmon.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐷 ) |
9 |
|
mplmonmul.t |
⊢ · = ( .r ‘ 𝑃 ) |
10 |
|
mplmonmul.x |
⊢ ( 𝜑 → 𝑌 ∈ 𝐷 ) |
11 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
12 |
1 2 3 4 5 6 7 8
|
mplmon |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ 𝐵 ) |
13 |
1 2 3 4 5 6 7 10
|
mplmon |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ∈ 𝐵 ) |
14 |
1 2 11 9 5 12 13
|
mplmul |
⊢ ( 𝜑 → ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) · ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ) = ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ) ) ) |
15 |
|
eqeq1 |
⊢ ( 𝑦 = 𝑘 → ( 𝑦 = ( 𝑋 ∘f + 𝑌 ) ↔ 𝑘 = ( 𝑋 ∘f + 𝑌 ) ) ) |
16 |
15
|
ifbid |
⊢ ( 𝑦 = 𝑘 → if ( 𝑦 = ( 𝑋 ∘f + 𝑌 ) , 1 , 0 ) = if ( 𝑘 = ( 𝑋 ∘f + 𝑌 ) , 1 , 0 ) ) |
17 |
16
|
cbvmptv |
⊢ ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = ( 𝑋 ∘f + 𝑌 ) , 1 , 0 ) ) = ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = ( 𝑋 ∘f + 𝑌 ) , 1 , 0 ) ) |
18 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) |
19 |
18
|
snssd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → { 𝑋 } ⊆ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) |
20 |
19
|
resmptd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ↾ { 𝑋 } ) = ( 𝑗 ∈ { 𝑋 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ) |
21 |
20
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( 𝑅 Σg ( ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ↾ { 𝑋 } ) ) = ( 𝑅 Σg ( 𝑗 ∈ { 𝑋 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ) ) |
22 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → 𝑅 ∈ Ring ) |
23 |
|
ringmnd |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Mnd ) |
24 |
22 23
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → 𝑅 ∈ Mnd ) |
25 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → 𝑋 ∈ 𝐷 ) |
26 |
|
iftrue |
⊢ ( 𝑦 = 𝑋 → if ( 𝑦 = 𝑋 , 1 , 0 ) = 1 ) |
27 |
|
eqid |
⊢ ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) = ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) |
28 |
4
|
fvexi |
⊢ 1 ∈ V |
29 |
26 27 28
|
fvmpt |
⊢ ( 𝑋 ∈ 𝐷 → ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑋 ) = 1 ) |
30 |
25 29
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑋 ) = 1 ) |
31 |
|
ssrab2 |
⊢ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ⊆ 𝐷 |
32 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → 𝐼 ∈ 𝑊 ) |
33 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → 𝑘 ∈ 𝐷 ) |
34 |
|
eqid |
⊢ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } = { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } |
35 |
5 34
|
psrbagconcl |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷 ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( 𝑘 ∘f − 𝑋 ) ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) |
36 |
32 33 18 35
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( 𝑘 ∘f − 𝑋 ) ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) |
37 |
31 36
|
sseldi |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( 𝑘 ∘f − 𝑋 ) ∈ 𝐷 ) |
38 |
|
eqeq1 |
⊢ ( 𝑦 = ( 𝑘 ∘f − 𝑋 ) → ( 𝑦 = 𝑌 ↔ ( 𝑘 ∘f − 𝑋 ) = 𝑌 ) ) |
39 |
38
|
ifbid |
⊢ ( 𝑦 = ( 𝑘 ∘f − 𝑋 ) → if ( 𝑦 = 𝑌 , 1 , 0 ) = if ( ( 𝑘 ∘f − 𝑋 ) = 𝑌 , 1 , 0 ) ) |
40 |
|
eqid |
⊢ ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) = ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) |
41 |
3
|
fvexi |
⊢ 0 ∈ V |
42 |
28 41
|
ifex |
⊢ if ( ( 𝑘 ∘f − 𝑋 ) = 𝑌 , 1 , 0 ) ∈ V |
43 |
39 40 42
|
fvmpt |
⊢ ( ( 𝑘 ∘f − 𝑋 ) ∈ 𝐷 → ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑋 ) ) = if ( ( 𝑘 ∘f − 𝑋 ) = 𝑌 , 1 , 0 ) ) |
44 |
37 43
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑋 ) ) = if ( ( 𝑘 ∘f − 𝑋 ) = 𝑌 , 1 , 0 ) ) |
45 |
30 44
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑋 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑋 ) ) ) = ( 1 ( .r ‘ 𝑅 ) if ( ( 𝑘 ∘f − 𝑋 ) = 𝑌 , 1 , 0 ) ) ) |
46 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
47 |
46 4
|
ringidcl |
⊢ ( 𝑅 ∈ Ring → 1 ∈ ( Base ‘ 𝑅 ) ) |
48 |
46 3
|
ring0cl |
⊢ ( 𝑅 ∈ Ring → 0 ∈ ( Base ‘ 𝑅 ) ) |
49 |
47 48
|
ifcld |
⊢ ( 𝑅 ∈ Ring → if ( ( 𝑘 ∘f − 𝑋 ) = 𝑌 , 1 , 0 ) ∈ ( Base ‘ 𝑅 ) ) |
50 |
22 49
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → if ( ( 𝑘 ∘f − 𝑋 ) = 𝑌 , 1 , 0 ) ∈ ( Base ‘ 𝑅 ) ) |
51 |
46 11 4
|
ringlidm |
⊢ ( ( 𝑅 ∈ Ring ∧ if ( ( 𝑘 ∘f − 𝑋 ) = 𝑌 , 1 , 0 ) ∈ ( Base ‘ 𝑅 ) ) → ( 1 ( .r ‘ 𝑅 ) if ( ( 𝑘 ∘f − 𝑋 ) = 𝑌 , 1 , 0 ) ) = if ( ( 𝑘 ∘f − 𝑋 ) = 𝑌 , 1 , 0 ) ) |
52 |
22 50 51
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( 1 ( .r ‘ 𝑅 ) if ( ( 𝑘 ∘f − 𝑋 ) = 𝑌 , 1 , 0 ) ) = if ( ( 𝑘 ∘f − 𝑋 ) = 𝑌 , 1 , 0 ) ) |
53 |
5
|
psrbagf |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷 ) → 𝑘 : 𝐼 ⟶ ℕ0 ) |
54 |
32 33 53
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → 𝑘 : 𝐼 ⟶ ℕ0 ) |
55 |
54
|
ffvelrnda |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) ∧ 𝑧 ∈ 𝐼 ) → ( 𝑘 ‘ 𝑧 ) ∈ ℕ0 ) |
56 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → 𝐼 ∈ 𝑊 ) |
57 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → 𝑋 ∈ 𝐷 ) |
58 |
5
|
psrbagf |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐷 ) → 𝑋 : 𝐼 ⟶ ℕ0 ) |
59 |
56 57 58
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → 𝑋 : 𝐼 ⟶ ℕ0 ) |
60 |
59
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑧 ∈ 𝐼 ) → ( 𝑋 ‘ 𝑧 ) ∈ ℕ0 ) |
61 |
60
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) ∧ 𝑧 ∈ 𝐼 ) → ( 𝑋 ‘ 𝑧 ) ∈ ℕ0 ) |
62 |
5
|
psrbagf |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑌 ∈ 𝐷 ) → 𝑌 : 𝐼 ⟶ ℕ0 ) |
63 |
6 10 62
|
syl2anc |
⊢ ( 𝜑 → 𝑌 : 𝐼 ⟶ ℕ0 ) |
64 |
63
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → 𝑌 : 𝐼 ⟶ ℕ0 ) |
65 |
64
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑧 ∈ 𝐼 ) → ( 𝑌 ‘ 𝑧 ) ∈ ℕ0 ) |
66 |
65
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) ∧ 𝑧 ∈ 𝐼 ) → ( 𝑌 ‘ 𝑧 ) ∈ ℕ0 ) |
67 |
|
nn0cn |
⊢ ( ( 𝑘 ‘ 𝑧 ) ∈ ℕ0 → ( 𝑘 ‘ 𝑧 ) ∈ ℂ ) |
68 |
|
nn0cn |
⊢ ( ( 𝑋 ‘ 𝑧 ) ∈ ℕ0 → ( 𝑋 ‘ 𝑧 ) ∈ ℂ ) |
69 |
|
nn0cn |
⊢ ( ( 𝑌 ‘ 𝑧 ) ∈ ℕ0 → ( 𝑌 ‘ 𝑧 ) ∈ ℂ ) |
70 |
|
subadd |
⊢ ( ( ( 𝑘 ‘ 𝑧 ) ∈ ℂ ∧ ( 𝑋 ‘ 𝑧 ) ∈ ℂ ∧ ( 𝑌 ‘ 𝑧 ) ∈ ℂ ) → ( ( ( 𝑘 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) = ( 𝑌 ‘ 𝑧 ) ↔ ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) = ( 𝑘 ‘ 𝑧 ) ) ) |
71 |
67 68 69 70
|
syl3an |
⊢ ( ( ( 𝑘 ‘ 𝑧 ) ∈ ℕ0 ∧ ( 𝑋 ‘ 𝑧 ) ∈ ℕ0 ∧ ( 𝑌 ‘ 𝑧 ) ∈ ℕ0 ) → ( ( ( 𝑘 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) = ( 𝑌 ‘ 𝑧 ) ↔ ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) = ( 𝑘 ‘ 𝑧 ) ) ) |
72 |
55 61 66 71
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) ∧ 𝑧 ∈ 𝐼 ) → ( ( ( 𝑘 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) = ( 𝑌 ‘ 𝑧 ) ↔ ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) = ( 𝑘 ‘ 𝑧 ) ) ) |
73 |
|
eqcom |
⊢ ( ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) = ( 𝑘 ‘ 𝑧 ) ↔ ( 𝑘 ‘ 𝑧 ) = ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) ) |
74 |
72 73
|
syl6bb |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) ∧ 𝑧 ∈ 𝐼 ) → ( ( ( 𝑘 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) = ( 𝑌 ‘ 𝑧 ) ↔ ( 𝑘 ‘ 𝑧 ) = ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) ) ) |
75 |
74
|
ralbidva |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( ∀ 𝑧 ∈ 𝐼 ( ( 𝑘 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) = ( 𝑌 ‘ 𝑧 ) ↔ ∀ 𝑧 ∈ 𝐼 ( 𝑘 ‘ 𝑧 ) = ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) ) ) |
76 |
|
mpteqb |
⊢ ( ∀ 𝑧 ∈ 𝐼 ( ( 𝑘 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) ∈ V → ( ( 𝑧 ∈ 𝐼 ↦ ( ( 𝑘 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) ) = ( 𝑧 ∈ 𝐼 ↦ ( 𝑌 ‘ 𝑧 ) ) ↔ ∀ 𝑧 ∈ 𝐼 ( ( 𝑘 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) = ( 𝑌 ‘ 𝑧 ) ) ) |
77 |
|
ovexd |
⊢ ( 𝑧 ∈ 𝐼 → ( ( 𝑘 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) ∈ V ) |
78 |
76 77
|
mprg |
⊢ ( ( 𝑧 ∈ 𝐼 ↦ ( ( 𝑘 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) ) = ( 𝑧 ∈ 𝐼 ↦ ( 𝑌 ‘ 𝑧 ) ) ↔ ∀ 𝑧 ∈ 𝐼 ( ( 𝑘 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) = ( 𝑌 ‘ 𝑧 ) ) |
79 |
|
mpteqb |
⊢ ( ∀ 𝑧 ∈ 𝐼 ( 𝑘 ‘ 𝑧 ) ∈ V → ( ( 𝑧 ∈ 𝐼 ↦ ( 𝑘 ‘ 𝑧 ) ) = ( 𝑧 ∈ 𝐼 ↦ ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) ) ↔ ∀ 𝑧 ∈ 𝐼 ( 𝑘 ‘ 𝑧 ) = ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) ) ) |
80 |
|
fvexd |
⊢ ( 𝑧 ∈ 𝐼 → ( 𝑘 ‘ 𝑧 ) ∈ V ) |
81 |
79 80
|
mprg |
⊢ ( ( 𝑧 ∈ 𝐼 ↦ ( 𝑘 ‘ 𝑧 ) ) = ( 𝑧 ∈ 𝐼 ↦ ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) ) ↔ ∀ 𝑧 ∈ 𝐼 ( 𝑘 ‘ 𝑧 ) = ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) ) |
82 |
75 78 81
|
3bitr4g |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( ( 𝑧 ∈ 𝐼 ↦ ( ( 𝑘 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) ) = ( 𝑧 ∈ 𝐼 ↦ ( 𝑌 ‘ 𝑧 ) ) ↔ ( 𝑧 ∈ 𝐼 ↦ ( 𝑘 ‘ 𝑧 ) ) = ( 𝑧 ∈ 𝐼 ↦ ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) ) ) ) |
83 |
54
|
feqmptd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → 𝑘 = ( 𝑧 ∈ 𝐼 ↦ ( 𝑘 ‘ 𝑧 ) ) ) |
84 |
59
|
feqmptd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → 𝑋 = ( 𝑧 ∈ 𝐼 ↦ ( 𝑋 ‘ 𝑧 ) ) ) |
85 |
84
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → 𝑋 = ( 𝑧 ∈ 𝐼 ↦ ( 𝑋 ‘ 𝑧 ) ) ) |
86 |
32 55 61 83 85
|
offval2 |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( 𝑘 ∘f − 𝑋 ) = ( 𝑧 ∈ 𝐼 ↦ ( ( 𝑘 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) ) ) |
87 |
64
|
feqmptd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → 𝑌 = ( 𝑧 ∈ 𝐼 ↦ ( 𝑌 ‘ 𝑧 ) ) ) |
88 |
87
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → 𝑌 = ( 𝑧 ∈ 𝐼 ↦ ( 𝑌 ‘ 𝑧 ) ) ) |
89 |
86 88
|
eqeq12d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( ( 𝑘 ∘f − 𝑋 ) = 𝑌 ↔ ( 𝑧 ∈ 𝐼 ↦ ( ( 𝑘 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) ) = ( 𝑧 ∈ 𝐼 ↦ ( 𝑌 ‘ 𝑧 ) ) ) ) |
90 |
56 60 65 84 87
|
offval2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( 𝑋 ∘f + 𝑌 ) = ( 𝑧 ∈ 𝐼 ↦ ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) ) ) |
91 |
90
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( 𝑋 ∘f + 𝑌 ) = ( 𝑧 ∈ 𝐼 ↦ ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) ) ) |
92 |
83 91
|
eqeq12d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( 𝑘 = ( 𝑋 ∘f + 𝑌 ) ↔ ( 𝑧 ∈ 𝐼 ↦ ( 𝑘 ‘ 𝑧 ) ) = ( 𝑧 ∈ 𝐼 ↦ ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) ) ) ) |
93 |
82 89 92
|
3bitr4d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( ( 𝑘 ∘f − 𝑋 ) = 𝑌 ↔ 𝑘 = ( 𝑋 ∘f + 𝑌 ) ) ) |
94 |
93
|
ifbid |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → if ( ( 𝑘 ∘f − 𝑋 ) = 𝑌 , 1 , 0 ) = if ( 𝑘 = ( 𝑋 ∘f + 𝑌 ) , 1 , 0 ) ) |
95 |
45 52 94
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑋 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑋 ) ) ) = if ( 𝑘 = ( 𝑋 ∘f + 𝑌 ) , 1 , 0 ) ) |
96 |
94 50
|
eqeltrrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → if ( 𝑘 = ( 𝑋 ∘f + 𝑌 ) , 1 , 0 ) ∈ ( Base ‘ 𝑅 ) ) |
97 |
95 96
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑋 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑋 ) ) ) ∈ ( Base ‘ 𝑅 ) ) |
98 |
|
fveq2 |
⊢ ( 𝑗 = 𝑋 → ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) = ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑋 ) ) |
99 |
|
oveq2 |
⊢ ( 𝑗 = 𝑋 → ( 𝑘 ∘f − 𝑗 ) = ( 𝑘 ∘f − 𝑋 ) ) |
100 |
99
|
fveq2d |
⊢ ( 𝑗 = 𝑋 → ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) = ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑋 ) ) ) |
101 |
98 100
|
oveq12d |
⊢ ( 𝑗 = 𝑋 → ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) = ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑋 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑋 ) ) ) ) |
102 |
46 101
|
gsumsn |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝑋 ∈ 𝐷 ∧ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑋 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑋 ) ) ) ∈ ( Base ‘ 𝑅 ) ) → ( 𝑅 Σg ( 𝑗 ∈ { 𝑋 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ) = ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑋 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑋 ) ) ) ) |
103 |
24 25 97 102
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( 𝑅 Σg ( 𝑗 ∈ { 𝑋 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ) = ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑋 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑋 ) ) ) ) |
104 |
21 103 95
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( 𝑅 Σg ( ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ↾ { 𝑋 } ) ) = if ( 𝑘 = ( 𝑋 ∘f + 𝑌 ) , 1 , 0 ) ) |
105 |
3
|
gsum0 |
⊢ ( 𝑅 Σg ∅ ) = 0 |
106 |
|
disjsn |
⊢ ( ( { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ∩ { 𝑋 } ) = ∅ ↔ ¬ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) |
107 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → 𝑅 ∈ Ring ) |
108 |
1 46 2 5 12
|
mplelf |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
109 |
108
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
110 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) |
111 |
31 110
|
sseldi |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → 𝑗 ∈ 𝐷 ) |
112 |
109 111
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ∈ ( Base ‘ 𝑅 ) ) |
113 |
1 46 2 5 13
|
mplelf |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
114 |
113
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
115 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → 𝐼 ∈ 𝑊 ) |
116 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → 𝑘 ∈ 𝐷 ) |
117 |
5 34
|
psrbagconcl |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷 ∧ 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( 𝑘 ∘f − 𝑗 ) ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) |
118 |
115 116 110 117
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( 𝑘 ∘f − 𝑗 ) ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) |
119 |
31 118
|
sseldi |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( 𝑘 ∘f − 𝑗 ) ∈ 𝐷 ) |
120 |
114 119
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ∈ ( Base ‘ 𝑅 ) ) |
121 |
46 11
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ∈ ( Base ‘ 𝑅 ) ∧ ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ∈ ( Base ‘ 𝑅 ) ) → ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ∈ ( Base ‘ 𝑅 ) ) |
122 |
107 112 120 121
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ∈ ( Base ‘ 𝑅 ) ) |
123 |
122
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) : { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ⟶ ( Base ‘ 𝑅 ) ) |
124 |
|
ffn |
⊢ ( ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) : { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ⟶ ( Base ‘ 𝑅 ) → ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) Fn { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) |
125 |
|
fnresdisj |
⊢ ( ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) Fn { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } → ( ( { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ∩ { 𝑋 } ) = ∅ ↔ ( ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ↾ { 𝑋 } ) = ∅ ) ) |
126 |
123 124 125
|
3syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( ( { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ∩ { 𝑋 } ) = ∅ ↔ ( ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ↾ { 𝑋 } ) = ∅ ) ) |
127 |
126
|
biimpa |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ ( { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ∩ { 𝑋 } ) = ∅ ) → ( ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ↾ { 𝑋 } ) = ∅ ) |
128 |
106 127
|
sylan2br |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ ¬ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ↾ { 𝑋 } ) = ∅ ) |
129 |
128
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ ¬ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( 𝑅 Σg ( ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ↾ { 𝑋 } ) ) = ( 𝑅 Σg ∅ ) ) |
130 |
|
breq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 ∘r ≤ ( 𝑋 ∘f + 𝑌 ) ↔ 𝑋 ∘r ≤ ( 𝑋 ∘f + 𝑌 ) ) ) |
131 |
60
|
nn0red |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑧 ∈ 𝐼 ) → ( 𝑋 ‘ 𝑧 ) ∈ ℝ ) |
132 |
|
nn0addge1 |
⊢ ( ( ( 𝑋 ‘ 𝑧 ) ∈ ℝ ∧ ( 𝑌 ‘ 𝑧 ) ∈ ℕ0 ) → ( 𝑋 ‘ 𝑧 ) ≤ ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) ) |
133 |
131 65 132
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑧 ∈ 𝐼 ) → ( 𝑋 ‘ 𝑧 ) ≤ ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) ) |
134 |
133
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ∀ 𝑧 ∈ 𝐼 ( 𝑋 ‘ 𝑧 ) ≤ ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) ) |
135 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑧 ∈ 𝐼 ) → ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) ∈ V ) |
136 |
56 60 135 84 90
|
ofrfval2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( 𝑋 ∘r ≤ ( 𝑋 ∘f + 𝑌 ) ↔ ∀ 𝑧 ∈ 𝐼 ( 𝑋 ‘ 𝑧 ) ≤ ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) ) ) |
137 |
134 136
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → 𝑋 ∘r ≤ ( 𝑋 ∘f + 𝑌 ) ) |
138 |
130 57 137
|
elrabd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝑋 ∘f + 𝑌 ) } ) |
139 |
|
breq2 |
⊢ ( 𝑘 = ( 𝑋 ∘f + 𝑌 ) → ( 𝑥 ∘r ≤ 𝑘 ↔ 𝑥 ∘r ≤ ( 𝑋 ∘f + 𝑌 ) ) ) |
140 |
139
|
rabbidv |
⊢ ( 𝑘 = ( 𝑋 ∘f + 𝑌 ) → { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } = { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝑋 ∘f + 𝑌 ) } ) |
141 |
140
|
eleq2d |
⊢ ( 𝑘 = ( 𝑋 ∘f + 𝑌 ) → ( 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↔ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝑋 ∘f + 𝑌 ) } ) ) |
142 |
138 141
|
syl5ibrcom |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( 𝑘 = ( 𝑋 ∘f + 𝑌 ) → 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) ) |
143 |
142
|
con3dimp |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ ¬ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ¬ 𝑘 = ( 𝑋 ∘f + 𝑌 ) ) |
144 |
143
|
iffalsed |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ ¬ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → if ( 𝑘 = ( 𝑋 ∘f + 𝑌 ) , 1 , 0 ) = 0 ) |
145 |
105 129 144
|
3eqtr4a |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ ¬ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( 𝑅 Σg ( ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ↾ { 𝑋 } ) ) = if ( 𝑘 = ( 𝑋 ∘f + 𝑌 ) , 1 , 0 ) ) |
146 |
104 145
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( 𝑅 Σg ( ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ↾ { 𝑋 } ) ) = if ( 𝑘 = ( 𝑋 ∘f + 𝑌 ) , 1 , 0 ) ) |
147 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → 𝑅 ∈ Ring ) |
148 |
|
ringcmn |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ CMnd ) |
149 |
147 148
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → 𝑅 ∈ CMnd ) |
150 |
5
|
psrbaglefi |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷 ) → { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ∈ Fin ) |
151 |
6 150
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ∈ Fin ) |
152 |
|
ssdif |
⊢ ( { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ⊆ 𝐷 → ( { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ∖ { 𝑋 } ) ⊆ ( 𝐷 ∖ { 𝑋 } ) ) |
153 |
31 152
|
ax-mp |
⊢ ( { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ∖ { 𝑋 } ) ⊆ ( 𝐷 ∖ { 𝑋 } ) |
154 |
153
|
sseli |
⊢ ( 𝑗 ∈ ( { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ∖ { 𝑋 } ) → 𝑗 ∈ ( 𝐷 ∖ { 𝑋 } ) ) |
155 |
108
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
156 |
|
eldifsni |
⊢ ( 𝑦 ∈ ( 𝐷 ∖ { 𝑋 } ) → 𝑦 ≠ 𝑋 ) |
157 |
156
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐷 ∖ { 𝑋 } ) ) → 𝑦 ≠ 𝑋 ) |
158 |
157
|
neneqd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐷 ∖ { 𝑋 } ) ) → ¬ 𝑦 = 𝑋 ) |
159 |
158
|
iffalsed |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐷 ∖ { 𝑋 } ) ) → if ( 𝑦 = 𝑋 , 1 , 0 ) = 0 ) |
160 |
|
ovex |
⊢ ( ℕ0 ↑m 𝐼 ) ∈ V |
161 |
5 160
|
rabex2 |
⊢ 𝐷 ∈ V |
162 |
161
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → 𝐷 ∈ V ) |
163 |
159 162
|
suppss2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) supp 0 ) ⊆ { 𝑋 } ) |
164 |
41
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → 0 ∈ V ) |
165 |
155 163 162 164
|
suppssr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑗 ∈ ( 𝐷 ∖ { 𝑋 } ) ) → ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) = 0 ) |
166 |
154 165
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑗 ∈ ( { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ∖ { 𝑋 } ) ) → ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) = 0 ) |
167 |
166
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑗 ∈ ( { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ∖ { 𝑋 } ) ) → ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) = ( 0 ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) |
168 |
|
eldifi |
⊢ ( 𝑗 ∈ ( { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ∖ { 𝑋 } ) → 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) |
169 |
46 11 3
|
ringlz |
⊢ ( ( 𝑅 ∈ Ring ∧ ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ∈ ( Base ‘ 𝑅 ) ) → ( 0 ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) = 0 ) |
170 |
107 120 169
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ) → ( 0 ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) = 0 ) |
171 |
168 170
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑗 ∈ ( { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ∖ { 𝑋 } ) ) → ( 0 ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) = 0 ) |
172 |
167 171
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) ∧ 𝑗 ∈ ( { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ∖ { 𝑋 } ) ) → ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) = 0 ) |
173 |
161
|
rabex |
⊢ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ∈ V |
174 |
173
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ∈ V ) |
175 |
172 174
|
suppss2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) supp 0 ) ⊆ { 𝑋 } ) |
176 |
161
|
mptrabex |
⊢ ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ∈ V |
177 |
|
funmpt |
⊢ Fun ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) |
178 |
176 177 41
|
3pm3.2i |
⊢ ( ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ∈ V ∧ Fun ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ∧ 0 ∈ V ) |
179 |
178
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ∈ V ∧ Fun ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ∧ 0 ∈ V ) ) |
180 |
|
snfi |
⊢ { 𝑋 } ∈ Fin |
181 |
180
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → { 𝑋 } ∈ Fin ) |
182 |
|
suppssfifsupp |
⊢ ( ( ( ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ∈ V ∧ Fun ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ∧ 0 ∈ V ) ∧ ( { 𝑋 } ∈ Fin ∧ ( ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) supp 0 ) ⊆ { 𝑋 } ) ) → ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) finSupp 0 ) |
183 |
179 181 175 182
|
syl12anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) finSupp 0 ) |
184 |
46 3 149 151 123 175 183
|
gsumres |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( 𝑅 Σg ( ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ↾ { 𝑋 } ) ) = ( 𝑅 Σg ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ) ) |
185 |
146 184
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → if ( 𝑘 = ( 𝑋 ∘f + 𝑌 ) , 1 , 0 ) = ( 𝑅 Σg ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ) ) |
186 |
185
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐷 ↦ if ( 𝑘 = ( 𝑋 ∘f + 𝑌 ) , 1 , 0 ) ) = ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ) ) ) |
187 |
17 186
|
syl5eq |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = ( 𝑋 ∘f + 𝑌 ) , 1 , 0 ) ) = ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘 } ↦ ( ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑗 ) ( .r ‘ 𝑅 ) ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ‘ ( 𝑘 ∘f − 𝑗 ) ) ) ) ) ) ) |
188 |
14 187
|
eqtr4d |
⊢ ( 𝜑 → ( ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) · ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑌 , 1 , 0 ) ) ) = ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = ( 𝑋 ∘f + 𝑌 ) , 1 , 0 ) ) ) |