Description: Write a group sum over a cartesian product as a double sum in two ways. This corresponds to the first equation in Lang p. 6. (Contributed by AV, 27-Dec-2023)
Ref | Expression | ||
---|---|---|---|
Hypotheses | gsumxp2.b | |
|
gsumxp2.z | |
||
gsumxp2.g | |
||
gsumxp2.a | |
||
gsumxp2.r | |
||
gsumxp2.f | |
||
gsumxp2.w | |
||
Assertion | gsumxp2 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumxp2.b | |
|
2 | gsumxp2.z | |
|
3 | gsumxp2.g | |
|
4 | gsumxp2.a | |
|
5 | gsumxp2.r | |
|
6 | gsumxp2.f | |
|
7 | gsumxp2.w | |
|
8 | 6 | fovrnda | |
9 | 7 | fsuppimpd | |
10 | simpl | |
|
11 | opelxpi | |
|
12 | 11 | ad2antlr | |
13 | simpr | |
|
14 | 12 13 | eldifd | |
15 | ssidd | |
|
16 | 4 5 | xpexd | |
17 | 2 | fvexi | |
18 | 17 | a1i | |
19 | 6 15 16 18 | suppssr | |
20 | 10 14 19 | syl2an2r | |
21 | 20 | ex | |
22 | df-br | |
|
23 | 22 | notbii | |
24 | df-ov | |
|
25 | 24 | eqeq1i | |
26 | 21 23 25 | 3imtr4g | |
27 | 26 | impr | |
28 | 1 2 3 4 5 8 9 27 | gsumcom3 | |
29 | 28 | eqcomd | |