Step |
Hyp |
Ref |
Expression |
1 |
|
rgspnval.r |
|- ( ph -> R e. Ring ) |
2 |
|
rgspnval.b |
|- ( ph -> B = ( Base ` R ) ) |
3 |
|
rgspnval.ss |
|- ( ph -> A C_ B ) |
4 |
|
rgspnval.n |
|- ( ph -> N = ( RingSpan ` R ) ) |
5 |
|
rgspnval.sp |
|- ( ph -> U = ( N ` A ) ) |
6 |
4
|
fveq1d |
|- ( ph -> ( N ` A ) = ( ( RingSpan ` R ) ` A ) ) |
7 |
|
elex |
|- ( R e. Ring -> R e. _V ) |
8 |
|
fveq2 |
|- ( a = R -> ( Base ` a ) = ( Base ` R ) ) |
9 |
8
|
pweqd |
|- ( a = R -> ~P ( Base ` a ) = ~P ( Base ` R ) ) |
10 |
|
fveq2 |
|- ( a = R -> ( SubRing ` a ) = ( SubRing ` R ) ) |
11 |
|
rabeq |
|- ( ( SubRing ` a ) = ( SubRing ` R ) -> { t e. ( SubRing ` a ) | b C_ t } = { t e. ( SubRing ` R ) | b C_ t } ) |
12 |
10 11
|
syl |
|- ( a = R -> { t e. ( SubRing ` a ) | b C_ t } = { t e. ( SubRing ` R ) | b C_ t } ) |
13 |
12
|
inteqd |
|- ( a = R -> |^| { t e. ( SubRing ` a ) | b C_ t } = |^| { t e. ( SubRing ` R ) | b C_ t } ) |
14 |
9 13
|
mpteq12dv |
|- ( a = R -> ( b e. ~P ( Base ` a ) |-> |^| { t e. ( SubRing ` a ) | b C_ t } ) = ( b e. ~P ( Base ` R ) |-> |^| { t e. ( SubRing ` R ) | b C_ t } ) ) |
15 |
|
df-rgspn |
|- RingSpan = ( a e. _V |-> ( b e. ~P ( Base ` a ) |-> |^| { t e. ( SubRing ` a ) | b C_ t } ) ) |
16 |
|
fvex |
|- ( Base ` R ) e. _V |
17 |
16
|
pwex |
|- ~P ( Base ` R ) e. _V |
18 |
17
|
mptex |
|- ( b e. ~P ( Base ` R ) |-> |^| { t e. ( SubRing ` R ) | b C_ t } ) e. _V |
19 |
14 15 18
|
fvmpt |
|- ( R e. _V -> ( RingSpan ` R ) = ( b e. ~P ( Base ` R ) |-> |^| { t e. ( SubRing ` R ) | b C_ t } ) ) |
20 |
1 7 19
|
3syl |
|- ( ph -> ( RingSpan ` R ) = ( b e. ~P ( Base ` R ) |-> |^| { t e. ( SubRing ` R ) | b C_ t } ) ) |
21 |
20
|
fveq1d |
|- ( ph -> ( ( RingSpan ` R ) ` A ) = ( ( b e. ~P ( Base ` R ) |-> |^| { t e. ( SubRing ` R ) | b C_ t } ) ` A ) ) |
22 |
|
eqid |
|- ( b e. ~P ( Base ` R ) |-> |^| { t e. ( SubRing ` R ) | b C_ t } ) = ( b e. ~P ( Base ` R ) |-> |^| { t e. ( SubRing ` R ) | b C_ t } ) |
23 |
|
sseq1 |
|- ( b = A -> ( b C_ t <-> A C_ t ) ) |
24 |
23
|
rabbidv |
|- ( b = A -> { t e. ( SubRing ` R ) | b C_ t } = { t e. ( SubRing ` R ) | A C_ t } ) |
25 |
24
|
inteqd |
|- ( b = A -> |^| { t e. ( SubRing ` R ) | b C_ t } = |^| { t e. ( SubRing ` R ) | A C_ t } ) |
26 |
3 2
|
sseqtrd |
|- ( ph -> A C_ ( Base ` R ) ) |
27 |
16
|
elpw2 |
|- ( A e. ~P ( Base ` R ) <-> A C_ ( Base ` R ) ) |
28 |
26 27
|
sylibr |
|- ( ph -> A e. ~P ( Base ` R ) ) |
29 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
30 |
29
|
subrgid |
|- ( R e. Ring -> ( Base ` R ) e. ( SubRing ` R ) ) |
31 |
1 30
|
syl |
|- ( ph -> ( Base ` R ) e. ( SubRing ` R ) ) |
32 |
2 31
|
eqeltrd |
|- ( ph -> B e. ( SubRing ` R ) ) |
33 |
|
sseq2 |
|- ( t = B -> ( A C_ t <-> A C_ B ) ) |
34 |
33
|
rspcev |
|- ( ( B e. ( SubRing ` R ) /\ A C_ B ) -> E. t e. ( SubRing ` R ) A C_ t ) |
35 |
32 3 34
|
syl2anc |
|- ( ph -> E. t e. ( SubRing ` R ) A C_ t ) |
36 |
|
intexrab |
|- ( E. t e. ( SubRing ` R ) A C_ t <-> |^| { t e. ( SubRing ` R ) | A C_ t } e. _V ) |
37 |
35 36
|
sylib |
|- ( ph -> |^| { t e. ( SubRing ` R ) | A C_ t } e. _V ) |
38 |
22 25 28 37
|
fvmptd3 |
|- ( ph -> ( ( b e. ~P ( Base ` R ) |-> |^| { t e. ( SubRing ` R ) | b C_ t } ) ` A ) = |^| { t e. ( SubRing ` R ) | A C_ t } ) |
39 |
21 38
|
eqtrd |
|- ( ph -> ( ( RingSpan ` R ) ` A ) = |^| { t e. ( SubRing ` R ) | A C_ t } ) |
40 |
5 6 39
|
3eqtrd |
|- ( ph -> U = |^| { t e. ( SubRing ` R ) | A C_ t } ) |