Description: A subring of a unital algebra is a subspace and thus a subalgebra iff it contains all scalar multiples of the identity. (Contributed by Mario Carneiro, 9-Mar-2015)
Ref | Expression | ||
---|---|---|---|
Hypotheses | issubassa2.a | |
|
issubassa2.l | |
||
Assertion | issubassa2 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issubassa2.a | |
|
2 | issubassa2.l | |
|
3 | eqid | |
|
4 | eqid | |
|
5 | 1 3 4 | rnascl | |
6 | 5 | ad2antrr | |
7 | assalmod | |
|
8 | 7 | ad2antrr | |
9 | simpr | |
|
10 | 3 | subrg1cl | |
11 | 10 | ad2antlr | |
12 | 2 4 8 9 11 | lspsnel5a | |
13 | 6 12 | eqsstrd | |
14 | subrgsubg | |
|
15 | 14 | ad2antlr | |
16 | simplll | |
|
17 | simprl | |
|
18 | eqid | |
|
19 | 18 | subrgss | |
20 | 19 | ad2antlr | |
21 | 20 | sselda | |
22 | 21 | adantrl | |
23 | eqid | |
|
24 | eqid | |
|
25 | eqid | |
|
26 | eqid | |
|
27 | 1 23 24 18 25 26 | asclmul1 | |
28 | 16 17 22 27 | syl3anc | |
29 | simpllr | |
|
30 | simplr | |
|
31 | 1 23 24 | asclfn | |
32 | 31 | a1i | |
33 | fnfvelrn | |
|
34 | 32 33 | sylan | |
35 | 30 34 | sseldd | |
36 | 35 | adantrr | |
37 | simprr | |
|
38 | 25 | subrgmcl | |
39 | 29 36 37 38 | syl3anc | |
40 | 28 39 | eqeltrrd | |
41 | 40 | ralrimivva | |
42 | 23 24 18 26 2 | islss4 | |
43 | 7 42 | syl | |
44 | 43 | ad2antrr | |
45 | 15 41 44 | mpbir2and | |
46 | 13 45 | impbida | |