Metamath Proof Explorer


Theorem tdeglem3OLD

Description: Obsolete version of tdeglem3 as of 7-Aug-2024. (Contributed by Stefan O'Rear, 26-Mar-2015) (Proof shortened by AV, 27-Jul-2019) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Hypotheses tdeglem.a A = m 0 I | m -1 Fin
tdeglem.h H = h A fld h
Assertion tdeglem3OLD I V X A Y A H X + f Y = H X + H Y

Proof

Step Hyp Ref Expression
1 tdeglem.a A = m 0 I | m -1 Fin
2 tdeglem.h H = h A fld h
3 cnfldbas = Base fld
4 cnfld0 0 = 0 fld
5 cnfldadd + = + fld
6 cnring fld Ring
7 ringcmn fld Ring fld CMnd
8 6 7 mp1i I V X A Y A fld CMnd
9 simp1 I V X A Y A I V
10 1 psrbagfOLD I V X A X : I 0
11 nn0sscn 0
12 fss X : I 0 0 X : I
13 10 11 12 sylancl I V X A X : I
14 13 3adant3 I V X A Y A X : I
15 1 psrbagfOLD I V Y A Y : I 0
16 fss Y : I 0 0 Y : I
17 15 11 16 sylancl I V Y A Y : I
18 17 3adant2 I V X A Y A Y : I
19 1 psrbagfsuppOLD X A I V finSupp 0 X
20 19 ancoms I V X A finSupp 0 X
21 20 3adant3 I V X A Y A finSupp 0 X
22 1 psrbagfsuppOLD Y A I V finSupp 0 Y
23 22 ancoms I V Y A finSupp 0 Y
24 23 3adant2 I V X A Y A finSupp 0 Y
25 3 4 5 8 9 14 18 21 24 gsumadd I V X A Y A fld X + f Y = fld X + fld Y
26 1 psrbagaddclOLD I V X A Y A X + f Y A
27 oveq2 h = X + f Y fld h = fld X + f Y
28 ovex fld X + f Y V
29 27 2 28 fvmpt X + f Y A H X + f Y = fld X + f Y
30 26 29 syl I V X A Y A H X + f Y = fld X + f Y
31 oveq2 h = X fld h = fld X
32 ovex fld X V
33 31 2 32 fvmpt X A H X = fld X
34 oveq2 h = Y fld h = fld Y
35 ovex fld Y V
36 34 2 35 fvmpt Y A H Y = fld Y
37 33 36 oveqan12d X A Y A H X + H Y = fld X + fld Y
38 37 3adant1 I V X A Y A H X + H Y = fld X + fld Y
39 25 30 38 3eqtr4d I V X A Y A H X + f Y = H X + H Y