Step |
Hyp |
Ref |
Expression |
1 |
|
mulgnndir.b |
|- B = ( Base ` G ) |
2 |
|
mulgnndir.t |
|- .x. = ( .g ` G ) |
3 |
|
mulgnndir.p |
|- .+ = ( +g ` G ) |
4 |
|
mndsgrp |
|- ( G e. Mnd -> G e. Smgrp ) |
5 |
4
|
adantr |
|- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> G e. Smgrp ) |
6 |
5
|
ad2antrr |
|- ( ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M e. NN ) /\ N e. NN ) -> G e. Smgrp ) |
7 |
|
simplr |
|- ( ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M e. NN ) /\ N e. NN ) -> M e. NN ) |
8 |
|
simpr |
|- ( ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M e. NN ) /\ N e. NN ) -> N e. NN ) |
9 |
|
simpr3 |
|- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> X e. B ) |
10 |
9
|
ad2antrr |
|- ( ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M e. NN ) /\ N e. NN ) -> X e. B ) |
11 |
1 2 3
|
mulgnndir |
|- ( ( G e. Smgrp /\ ( M e. NN /\ N e. NN /\ X e. B ) ) -> ( ( M + N ) .x. X ) = ( ( M .x. X ) .+ ( N .x. X ) ) ) |
12 |
6 7 8 10 11
|
syl13anc |
|- ( ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M e. NN ) /\ N e. NN ) -> ( ( M + N ) .x. X ) = ( ( M .x. X ) .+ ( N .x. X ) ) ) |
13 |
|
simpll |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ N = 0 ) -> G e. Mnd ) |
14 |
|
simpr1 |
|- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> M e. NN0 ) |
15 |
14
|
adantr |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ N = 0 ) -> M e. NN0 ) |
16 |
|
simplr3 |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ N = 0 ) -> X e. B ) |
17 |
1 2 13 15 16
|
mulgnn0cld |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ N = 0 ) -> ( M .x. X ) e. B ) |
18 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
19 |
1 3 18
|
mndrid |
|- ( ( G e. Mnd /\ ( M .x. X ) e. B ) -> ( ( M .x. X ) .+ ( 0g ` G ) ) = ( M .x. X ) ) |
20 |
13 17 19
|
syl2anc |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ N = 0 ) -> ( ( M .x. X ) .+ ( 0g ` G ) ) = ( M .x. X ) ) |
21 |
|
simpr |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ N = 0 ) -> N = 0 ) |
22 |
21
|
oveq1d |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ N = 0 ) -> ( N .x. X ) = ( 0 .x. X ) ) |
23 |
1 18 2
|
mulg0 |
|- ( X e. B -> ( 0 .x. X ) = ( 0g ` G ) ) |
24 |
16 23
|
syl |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ N = 0 ) -> ( 0 .x. X ) = ( 0g ` G ) ) |
25 |
22 24
|
eqtrd |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ N = 0 ) -> ( N .x. X ) = ( 0g ` G ) ) |
26 |
25
|
oveq2d |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ N = 0 ) -> ( ( M .x. X ) .+ ( N .x. X ) ) = ( ( M .x. X ) .+ ( 0g ` G ) ) ) |
27 |
21
|
oveq2d |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ N = 0 ) -> ( M + N ) = ( M + 0 ) ) |
28 |
15
|
nn0cnd |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ N = 0 ) -> M e. CC ) |
29 |
28
|
addridd |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ N = 0 ) -> ( M + 0 ) = M ) |
30 |
27 29
|
eqtrd |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ N = 0 ) -> ( M + N ) = M ) |
31 |
30
|
oveq1d |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ N = 0 ) -> ( ( M + N ) .x. X ) = ( M .x. X ) ) |
32 |
20 26 31
|
3eqtr4rd |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ N = 0 ) -> ( ( M + N ) .x. X ) = ( ( M .x. X ) .+ ( N .x. X ) ) ) |
33 |
32
|
adantlr |
|- ( ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M e. NN ) /\ N = 0 ) -> ( ( M + N ) .x. X ) = ( ( M .x. X ) .+ ( N .x. X ) ) ) |
34 |
|
simpr2 |
|- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> N e. NN0 ) |
35 |
|
elnn0 |
|- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) ) |
36 |
34 35
|
sylib |
|- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( N e. NN \/ N = 0 ) ) |
37 |
36
|
adantr |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M e. NN ) -> ( N e. NN \/ N = 0 ) ) |
38 |
12 33 37
|
mpjaodan |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M e. NN ) -> ( ( M + N ) .x. X ) = ( ( M .x. X ) .+ ( N .x. X ) ) ) |
39 |
|
simpll |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M = 0 ) -> G e. Mnd ) |
40 |
|
simplr2 |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M = 0 ) -> N e. NN0 ) |
41 |
|
simplr3 |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M = 0 ) -> X e. B ) |
42 |
1 2 39 40 41
|
mulgnn0cld |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M = 0 ) -> ( N .x. X ) e. B ) |
43 |
1 3 18
|
mndlid |
|- ( ( G e. Mnd /\ ( N .x. X ) e. B ) -> ( ( 0g ` G ) .+ ( N .x. X ) ) = ( N .x. X ) ) |
44 |
39 42 43
|
syl2anc |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M = 0 ) -> ( ( 0g ` G ) .+ ( N .x. X ) ) = ( N .x. X ) ) |
45 |
|
simpr |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M = 0 ) -> M = 0 ) |
46 |
45
|
oveq1d |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M = 0 ) -> ( M .x. X ) = ( 0 .x. X ) ) |
47 |
41 23
|
syl |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M = 0 ) -> ( 0 .x. X ) = ( 0g ` G ) ) |
48 |
46 47
|
eqtrd |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M = 0 ) -> ( M .x. X ) = ( 0g ` G ) ) |
49 |
48
|
oveq1d |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M = 0 ) -> ( ( M .x. X ) .+ ( N .x. X ) ) = ( ( 0g ` G ) .+ ( N .x. X ) ) ) |
50 |
45
|
oveq1d |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M = 0 ) -> ( M + N ) = ( 0 + N ) ) |
51 |
40
|
nn0cnd |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M = 0 ) -> N e. CC ) |
52 |
51
|
addlidd |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M = 0 ) -> ( 0 + N ) = N ) |
53 |
50 52
|
eqtrd |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M = 0 ) -> ( M + N ) = N ) |
54 |
53
|
oveq1d |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M = 0 ) -> ( ( M + N ) .x. X ) = ( N .x. X ) ) |
55 |
44 49 54
|
3eqtr4rd |
|- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M = 0 ) -> ( ( M + N ) .x. X ) = ( ( M .x. X ) .+ ( N .x. X ) ) ) |
56 |
|
elnn0 |
|- ( M e. NN0 <-> ( M e. NN \/ M = 0 ) ) |
57 |
14 56
|
sylib |
|- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( M e. NN \/ M = 0 ) ) |
58 |
38 55 57
|
mpjaodan |
|- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( ( M + N ) .x. X ) = ( ( M .x. X ) .+ ( N .x. X ) ) ) |