Step |
Hyp |
Ref |
Expression |
1 |
|
dmdprdsplitlem.0 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
2 |
|
dmdprdsplitlem.w |
⊢ 𝑊 = { ℎ ∈ X 𝑖 ∈ 𝐼 ( 𝑆 ‘ 𝑖 ) ∣ ℎ finSupp 0 } |
3 |
|
dmdprdsplitlem.1 |
⊢ ( 𝜑 → 𝐺 dom DProd 𝑆 ) |
4 |
|
dmdprdsplitlem.2 |
⊢ ( 𝜑 → dom 𝑆 = 𝐼 ) |
5 |
|
dmdprdsplitlem.3 |
⊢ ( 𝜑 → 𝐴 ⊆ 𝐼 ) |
6 |
|
dmdprdsplitlem.4 |
⊢ ( 𝜑 → 𝐹 ∈ 𝑊 ) |
7 |
|
dmdprdsplitlem.5 |
⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) ∈ ( 𝐺 DProd ( 𝑆 ↾ 𝐴 ) ) ) |
8 |
3 4
|
dprdf2 |
⊢ ( 𝜑 → 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ) |
9 |
8 5
|
fssresd |
⊢ ( 𝜑 → ( 𝑆 ↾ 𝐴 ) : 𝐴 ⟶ ( SubGrp ‘ 𝐺 ) ) |
10 |
|
fdm |
⊢ ( ( 𝑆 ↾ 𝐴 ) : 𝐴 ⟶ ( SubGrp ‘ 𝐺 ) → dom ( 𝑆 ↾ 𝐴 ) = 𝐴 ) |
11 |
|
eqid |
⊢ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } = { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } |
12 |
1 11
|
eldprd |
⊢ ( dom ( 𝑆 ↾ 𝐴 ) = 𝐴 → ( ( 𝐺 Σg 𝐹 ) ∈ ( 𝐺 DProd ( 𝑆 ↾ 𝐴 ) ) ↔ ( 𝐺 dom DProd ( 𝑆 ↾ 𝐴 ) ∧ ∃ 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ) |
13 |
9 10 12
|
3syl |
⊢ ( 𝜑 → ( ( 𝐺 Σg 𝐹 ) ∈ ( 𝐺 DProd ( 𝑆 ↾ 𝐴 ) ) ↔ ( 𝐺 dom DProd ( 𝑆 ↾ 𝐴 ) ∧ ∃ 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ) |
14 |
7 13
|
mpbid |
⊢ ( 𝜑 → ( 𝐺 dom DProd ( 𝑆 ↾ 𝐴 ) ∧ ∃ 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) |
15 |
14
|
simprd |
⊢ ( 𝜑 → ∃ 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) |
16 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) → ∃ 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) |
17 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) |
18 |
14
|
simpld |
⊢ ( 𝜑 → 𝐺 dom DProd ( 𝑆 ↾ 𝐴 ) ) |
19 |
18
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → 𝐺 dom DProd ( 𝑆 ↾ 𝐴 ) ) |
20 |
9 10
|
syl |
⊢ ( 𝜑 → dom ( 𝑆 ↾ 𝐴 ) = 𝐴 ) |
21 |
20
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → dom ( 𝑆 ↾ 𝐴 ) = 𝐴 ) |
22 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ) |
23 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
24 |
11 19 21 22 23
|
dprdff |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → 𝑓 : 𝐴 ⟶ ( Base ‘ 𝐺 ) ) |
25 |
24
|
feqmptd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → 𝑓 = ( 𝑛 ∈ 𝐴 ↦ ( 𝑓 ‘ 𝑛 ) ) ) |
26 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → 𝐴 ⊆ 𝐼 ) |
27 |
26
|
resmptd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → ( ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ↾ 𝐴 ) = ( 𝑛 ∈ 𝐴 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ) |
28 |
|
iftrue |
⊢ ( 𝑛 ∈ 𝐴 → if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) = ( 𝑓 ‘ 𝑛 ) ) |
29 |
28
|
mpteq2ia |
⊢ ( 𝑛 ∈ 𝐴 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) = ( 𝑛 ∈ 𝐴 ↦ ( 𝑓 ‘ 𝑛 ) ) |
30 |
27 29
|
eqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → ( ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ↾ 𝐴 ) = ( 𝑛 ∈ 𝐴 ↦ ( 𝑓 ‘ 𝑛 ) ) ) |
31 |
25 30
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → 𝑓 = ( ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ↾ 𝐴 ) ) |
32 |
31
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → ( 𝐺 Σg 𝑓 ) = ( 𝐺 Σg ( ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ↾ 𝐴 ) ) ) |
33 |
|
eqid |
⊢ ( Cntz ‘ 𝐺 ) = ( Cntz ‘ 𝐺 ) |
34 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → 𝐺 dom DProd 𝑆 ) |
35 |
|
dprdgrp |
⊢ ( 𝐺 dom DProd 𝑆 → 𝐺 ∈ Grp ) |
36 |
|
grpmnd |
⊢ ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd ) |
37 |
34 35 36
|
3syl |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → 𝐺 ∈ Mnd ) |
38 |
3 4
|
dprddomcld |
⊢ ( 𝜑 → 𝐼 ∈ V ) |
39 |
38
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → 𝐼 ∈ V ) |
40 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → dom 𝑆 = 𝐼 ) |
41 |
19
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ∧ 𝑛 ∈ 𝐼 ) → 𝐺 dom DProd ( 𝑆 ↾ 𝐴 ) ) |
42 |
21
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ∧ 𝑛 ∈ 𝐼 ) → dom ( 𝑆 ↾ 𝐴 ) = 𝐴 ) |
43 |
|
simplrl |
⊢ ( ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ∧ 𝑛 ∈ 𝐼 ) → 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ) |
44 |
11 41 42 43
|
dprdfcl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ∧ 𝑛 ∈ 𝐼 ) ∧ 𝑛 ∈ 𝐴 ) → ( 𝑓 ‘ 𝑛 ) ∈ ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑛 ) ) |
45 |
|
fvres |
⊢ ( 𝑛 ∈ 𝐴 → ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑛 ) = ( 𝑆 ‘ 𝑛 ) ) |
46 |
45
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ∧ 𝑛 ∈ 𝐼 ) ∧ 𝑛 ∈ 𝐴 ) → ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑛 ) = ( 𝑆 ‘ 𝑛 ) ) |
47 |
44 46
|
eleqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ∧ 𝑛 ∈ 𝐼 ) ∧ 𝑛 ∈ 𝐴 ) → ( 𝑓 ‘ 𝑛 ) ∈ ( 𝑆 ‘ 𝑛 ) ) |
48 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ) |
49 |
48
|
ffvelrnda |
⊢ ( ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ∧ 𝑛 ∈ 𝐼 ) → ( 𝑆 ‘ 𝑛 ) ∈ ( SubGrp ‘ 𝐺 ) ) |
50 |
1
|
subg0cl |
⊢ ( ( 𝑆 ‘ 𝑛 ) ∈ ( SubGrp ‘ 𝐺 ) → 0 ∈ ( 𝑆 ‘ 𝑛 ) ) |
51 |
49 50
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ∧ 𝑛 ∈ 𝐼 ) → 0 ∈ ( 𝑆 ‘ 𝑛 ) ) |
52 |
51
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ∧ 𝑛 ∈ 𝐼 ) ∧ ¬ 𝑛 ∈ 𝐴 ) → 0 ∈ ( 𝑆 ‘ 𝑛 ) ) |
53 |
47 52
|
ifclda |
⊢ ( ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ∧ 𝑛 ∈ 𝐼 ) → if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ∈ ( 𝑆 ‘ 𝑛 ) ) |
54 |
38
|
mptexd |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ∈ V ) |
55 |
54
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ∈ V ) |
56 |
|
funmpt |
⊢ Fun ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) |
57 |
56
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → Fun ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ) |
58 |
11 19 21 22
|
dprdffsupp |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → 𝑓 finSupp 0 ) |
59 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ∧ 𝑛 ∈ ( 𝐼 ∖ ( 𝑓 supp 0 ) ) ) ∧ 𝑛 ∈ 𝐴 ) → 𝑛 ∈ 𝐴 ) |
60 |
|
eldifn |
⊢ ( 𝑛 ∈ ( 𝐼 ∖ ( 𝑓 supp 0 ) ) → ¬ 𝑛 ∈ ( 𝑓 supp 0 ) ) |
61 |
60
|
ad2antlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ∧ 𝑛 ∈ ( 𝐼 ∖ ( 𝑓 supp 0 ) ) ) ∧ 𝑛 ∈ 𝐴 ) → ¬ 𝑛 ∈ ( 𝑓 supp 0 ) ) |
62 |
59 61
|
eldifd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ∧ 𝑛 ∈ ( 𝐼 ∖ ( 𝑓 supp 0 ) ) ) ∧ 𝑛 ∈ 𝐴 ) → 𝑛 ∈ ( 𝐴 ∖ ( 𝑓 supp 0 ) ) ) |
63 |
|
ssidd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → ( 𝑓 supp 0 ) ⊆ ( 𝑓 supp 0 ) ) |
64 |
38 5
|
ssexd |
⊢ ( 𝜑 → 𝐴 ∈ V ) |
65 |
64
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → 𝐴 ∈ V ) |
66 |
1
|
fvexi |
⊢ 0 ∈ V |
67 |
66
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → 0 ∈ V ) |
68 |
24 63 65 67
|
suppssr |
⊢ ( ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∖ ( 𝑓 supp 0 ) ) ) → ( 𝑓 ‘ 𝑛 ) = 0 ) |
69 |
68
|
adantlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ∧ 𝑛 ∈ ( 𝐼 ∖ ( 𝑓 supp 0 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∖ ( 𝑓 supp 0 ) ) ) → ( 𝑓 ‘ 𝑛 ) = 0 ) |
70 |
62 69
|
syldan |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ∧ 𝑛 ∈ ( 𝐼 ∖ ( 𝑓 supp 0 ) ) ) ∧ 𝑛 ∈ 𝐴 ) → ( 𝑓 ‘ 𝑛 ) = 0 ) |
71 |
70
|
ifeq1da |
⊢ ( ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ∧ 𝑛 ∈ ( 𝐼 ∖ ( 𝑓 supp 0 ) ) ) → if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) = if ( 𝑛 ∈ 𝐴 , 0 , 0 ) ) |
72 |
|
ifid |
⊢ if ( 𝑛 ∈ 𝐴 , 0 , 0 ) = 0 |
73 |
71 72
|
eqtrdi |
⊢ ( ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ∧ 𝑛 ∈ ( 𝐼 ∖ ( 𝑓 supp 0 ) ) ) → if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) = 0 ) |
74 |
73 39
|
suppss2 |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → ( ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) supp 0 ) ⊆ ( 𝑓 supp 0 ) ) |
75 |
|
fsuppsssupp |
⊢ ( ( ( ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ∈ V ∧ Fun ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ) ∧ ( 𝑓 finSupp 0 ∧ ( ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) supp 0 ) ⊆ ( 𝑓 supp 0 ) ) ) → ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) finSupp 0 ) |
76 |
55 57 58 74 75
|
syl22anc |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) finSupp 0 ) |
77 |
2 34 40 53 76
|
dprdwd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ∈ 𝑊 ) |
78 |
2 34 40 77 23
|
dprdff |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) : 𝐼 ⟶ ( Base ‘ 𝐺 ) ) |
79 |
2 34 40 77 33
|
dprdfcntz |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → ran ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ran ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ) ) |
80 |
|
eldifn |
⊢ ( 𝑛 ∈ ( 𝐼 ∖ 𝐴 ) → ¬ 𝑛 ∈ 𝐴 ) |
81 |
80
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ∧ 𝑛 ∈ ( 𝐼 ∖ 𝐴 ) ) → ¬ 𝑛 ∈ 𝐴 ) |
82 |
81
|
iffalsed |
⊢ ( ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) ∧ 𝑛 ∈ ( 𝐼 ∖ 𝐴 ) ) → if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) = 0 ) |
83 |
82 39
|
suppss2 |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → ( ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) supp 0 ) ⊆ 𝐴 ) |
84 |
23 1 33 37 39 78 79 83 76
|
gsumzres |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → ( 𝐺 Σg ( ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ↾ 𝐴 ) ) = ( 𝐺 Σg ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ) ) |
85 |
17 32 84
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ) ) |
86 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → 𝐹 ∈ 𝑊 ) |
87 |
1 2 34 40 86 77
|
dprdf11 |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → ( ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ) ↔ 𝐹 = ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ) ) |
88 |
85 87
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → 𝐹 = ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ) |
89 |
88
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → ( 𝐹 ‘ 𝑋 ) = ( ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ‘ 𝑋 ) ) |
90 |
|
eldifi |
⊢ ( 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) → 𝑋 ∈ 𝐼 ) |
91 |
90
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → 𝑋 ∈ 𝐼 ) |
92 |
|
eleq1 |
⊢ ( 𝑛 = 𝑋 → ( 𝑛 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴 ) ) |
93 |
|
fveq2 |
⊢ ( 𝑛 = 𝑋 → ( 𝑓 ‘ 𝑛 ) = ( 𝑓 ‘ 𝑋 ) ) |
94 |
92 93
|
ifbieq1d |
⊢ ( 𝑛 = 𝑋 → if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) = if ( 𝑋 ∈ 𝐴 , ( 𝑓 ‘ 𝑋 ) , 0 ) ) |
95 |
|
eqid |
⊢ ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) = ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) |
96 |
|
fvex |
⊢ ( 𝑓 ‘ 𝑛 ) ∈ V |
97 |
96 66
|
ifex |
⊢ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ∈ V |
98 |
94 95 97
|
fvmpt3i |
⊢ ( 𝑋 ∈ 𝐼 → ( ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ‘ 𝑋 ) = if ( 𝑋 ∈ 𝐴 , ( 𝑓 ‘ 𝑋 ) , 0 ) ) |
99 |
91 98
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → ( ( 𝑛 ∈ 𝐼 ↦ if ( 𝑛 ∈ 𝐴 , ( 𝑓 ‘ 𝑛 ) , 0 ) ) ‘ 𝑋 ) = if ( 𝑋 ∈ 𝐴 , ( 𝑓 ‘ 𝑋 ) , 0 ) ) |
100 |
|
eldifn |
⊢ ( 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) → ¬ 𝑋 ∈ 𝐴 ) |
101 |
100
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → ¬ 𝑋 ∈ 𝐴 ) |
102 |
101
|
iffalsed |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → if ( 𝑋 ∈ 𝐴 , ( 𝑓 ‘ 𝑋 ) , 0 ) = 0 ) |
103 |
89 99 102
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) ∧ ( 𝑓 ∈ { ℎ ∈ X 𝑖 ∈ 𝐴 ( ( 𝑆 ↾ 𝐴 ) ‘ 𝑖 ) ∣ ℎ finSupp 0 } ∧ ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg 𝑓 ) ) ) → ( 𝐹 ‘ 𝑋 ) = 0 ) |
104 |
16 103
|
rexlimddv |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝐼 ∖ 𝐴 ) ) → ( 𝐹 ‘ 𝑋 ) = 0 ) |