Step |
Hyp |
Ref |
Expression |
1 |
|
omndmul.0 |
|- B = ( Base ` M ) |
2 |
|
omndmul.1 |
|- .<_ = ( le ` M ) |
3 |
|
omndmul3.m |
|- .x. = ( .g ` M ) |
4 |
|
omndmul3.0 |
|- .0. = ( 0g ` M ) |
5 |
|
omndmul3.o |
|- ( ph -> M e. oMnd ) |
6 |
|
omndmul3.1 |
|- ( ph -> N e. NN0 ) |
7 |
|
omndmul3.2 |
|- ( ph -> P e. NN0 ) |
8 |
|
omndmul3.3 |
|- ( ph -> N <_ P ) |
9 |
|
omndmul3.4 |
|- ( ph -> X e. B ) |
10 |
|
omndmul3.5 |
|- ( ph -> .0. .<_ X ) |
11 |
|
omndmnd |
|- ( M e. oMnd -> M e. Mnd ) |
12 |
5 11
|
syl |
|- ( ph -> M e. Mnd ) |
13 |
1 4
|
mndidcl |
|- ( M e. Mnd -> .0. e. B ) |
14 |
12 13
|
syl |
|- ( ph -> .0. e. B ) |
15 |
|
nn0sub |
|- ( ( N e. NN0 /\ P e. NN0 ) -> ( N <_ P <-> ( P - N ) e. NN0 ) ) |
16 |
15
|
biimpa |
|- ( ( ( N e. NN0 /\ P e. NN0 ) /\ N <_ P ) -> ( P - N ) e. NN0 ) |
17 |
6 7 8 16
|
syl21anc |
|- ( ph -> ( P - N ) e. NN0 ) |
18 |
1 3
|
mulgnn0cl |
|- ( ( M e. Mnd /\ ( P - N ) e. NN0 /\ X e. B ) -> ( ( P - N ) .x. X ) e. B ) |
19 |
12 17 9 18
|
syl3anc |
|- ( ph -> ( ( P - N ) .x. X ) e. B ) |
20 |
1 3
|
mulgnn0cl |
|- ( ( M e. Mnd /\ N e. NN0 /\ X e. B ) -> ( N .x. X ) e. B ) |
21 |
12 6 9 20
|
syl3anc |
|- ( ph -> ( N .x. X ) e. B ) |
22 |
1 2 3 4
|
omndmul2 |
|- ( ( M e. oMnd /\ ( X e. B /\ ( P - N ) e. NN0 ) /\ .0. .<_ X ) -> .0. .<_ ( ( P - N ) .x. X ) ) |
23 |
5 9 17 10 22
|
syl121anc |
|- ( ph -> .0. .<_ ( ( P - N ) .x. X ) ) |
24 |
|
eqid |
|- ( +g ` M ) = ( +g ` M ) |
25 |
1 2 24
|
omndadd |
|- ( ( M e. oMnd /\ ( .0. e. B /\ ( ( P - N ) .x. X ) e. B /\ ( N .x. X ) e. B ) /\ .0. .<_ ( ( P - N ) .x. X ) ) -> ( .0. ( +g ` M ) ( N .x. X ) ) .<_ ( ( ( P - N ) .x. X ) ( +g ` M ) ( N .x. X ) ) ) |
26 |
5 14 19 21 23 25
|
syl131anc |
|- ( ph -> ( .0. ( +g ` M ) ( N .x. X ) ) .<_ ( ( ( P - N ) .x. X ) ( +g ` M ) ( N .x. X ) ) ) |
27 |
1 24 4
|
mndlid |
|- ( ( M e. Mnd /\ ( N .x. X ) e. B ) -> ( .0. ( +g ` M ) ( N .x. X ) ) = ( N .x. X ) ) |
28 |
12 21 27
|
syl2anc |
|- ( ph -> ( .0. ( +g ` M ) ( N .x. X ) ) = ( N .x. X ) ) |
29 |
1 3 24
|
mulgnn0dir |
|- ( ( M e. Mnd /\ ( ( P - N ) e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( ( ( P - N ) + N ) .x. X ) = ( ( ( P - N ) .x. X ) ( +g ` M ) ( N .x. X ) ) ) |
30 |
12 17 6 9 29
|
syl13anc |
|- ( ph -> ( ( ( P - N ) + N ) .x. X ) = ( ( ( P - N ) .x. X ) ( +g ` M ) ( N .x. X ) ) ) |
31 |
7
|
nn0cnd |
|- ( ph -> P e. CC ) |
32 |
6
|
nn0cnd |
|- ( ph -> N e. CC ) |
33 |
31 32
|
npcand |
|- ( ph -> ( ( P - N ) + N ) = P ) |
34 |
33
|
oveq1d |
|- ( ph -> ( ( ( P - N ) + N ) .x. X ) = ( P .x. X ) ) |
35 |
30 34
|
eqtr3d |
|- ( ph -> ( ( ( P - N ) .x. X ) ( +g ` M ) ( N .x. X ) ) = ( P .x. X ) ) |
36 |
26 28 35
|
3brtr3d |
|- ( ph -> ( N .x. X ) .<_ ( P .x. X ) ) |