Step |
Hyp |
Ref |
Expression |
1 |
|
simplr |
|- ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) -> ( A *e B ) e. NN0 ) |
2 |
|
simpr |
|- ( ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) /\ A = +oo ) -> A = +oo ) |
3 |
2
|
oveq1d |
|- ( ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) /\ A = +oo ) -> ( A *e B ) = ( +oo *e B ) ) |
4 |
|
xnn0xr |
|- ( B e. NN0* -> B e. RR* ) |
5 |
4
|
ad5antlr |
|- ( ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) /\ A = +oo ) -> B e. RR* ) |
6 |
|
simp-5r |
|- ( ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) /\ A = +oo ) -> B e. NN0* ) |
7 |
|
simprr |
|- ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> B =/= 0 ) |
8 |
7
|
ad3antrrr |
|- ( ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) /\ A = +oo ) -> B =/= 0 ) |
9 |
|
xnn0gt0 |
|- ( ( B e. NN0* /\ B =/= 0 ) -> 0 < B ) |
10 |
6 8 9
|
syl2anc |
|- ( ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) /\ A = +oo ) -> 0 < B ) |
11 |
|
xmulpnf2 |
|- ( ( B e. RR* /\ 0 < B ) -> ( +oo *e B ) = +oo ) |
12 |
5 10 11
|
syl2anc |
|- ( ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) /\ A = +oo ) -> ( +oo *e B ) = +oo ) |
13 |
|
pnfnre2 |
|- -. +oo e. RR |
14 |
|
nn0re |
|- ( +oo e. NN0 -> +oo e. RR ) |
15 |
13 14
|
mto |
|- -. +oo e. NN0 |
16 |
15
|
a1i |
|- ( ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) /\ A = +oo ) -> -. +oo e. NN0 ) |
17 |
12 16
|
eqneltrd |
|- ( ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) /\ A = +oo ) -> -. ( +oo *e B ) e. NN0 ) |
18 |
3 17
|
eqneltrd |
|- ( ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) /\ A = +oo ) -> -. ( A *e B ) e. NN0 ) |
19 |
|
simpr |
|- ( ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) /\ B = +oo ) -> B = +oo ) |
20 |
19
|
oveq2d |
|- ( ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) /\ B = +oo ) -> ( A *e B ) = ( A *e +oo ) ) |
21 |
|
xnn0xr |
|- ( A e. NN0* -> A e. RR* ) |
22 |
21
|
ad5antr |
|- ( ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) /\ B = +oo ) -> A e. RR* ) |
23 |
|
simp-5l |
|- ( ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) /\ B = +oo ) -> A e. NN0* ) |
24 |
|
simprl |
|- ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> A =/= 0 ) |
25 |
24
|
ad3antrrr |
|- ( ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) /\ B = +oo ) -> A =/= 0 ) |
26 |
|
xnn0gt0 |
|- ( ( A e. NN0* /\ A =/= 0 ) -> 0 < A ) |
27 |
23 25 26
|
syl2anc |
|- ( ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) /\ B = +oo ) -> 0 < A ) |
28 |
|
xmulpnf1 |
|- ( ( A e. RR* /\ 0 < A ) -> ( A *e +oo ) = +oo ) |
29 |
22 27 28
|
syl2anc |
|- ( ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) /\ B = +oo ) -> ( A *e +oo ) = +oo ) |
30 |
15
|
a1i |
|- ( ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) /\ B = +oo ) -> -. +oo e. NN0 ) |
31 |
29 30
|
eqneltrd |
|- ( ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) /\ B = +oo ) -> -. ( A *e +oo ) e. NN0 ) |
32 |
20 31
|
eqneltrd |
|- ( ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) /\ B = +oo ) -> -. ( A *e B ) e. NN0 ) |
33 |
|
xnn0nnn0pnf |
|- ( ( A e. NN0* /\ -. A e. NN0 ) -> A = +oo ) |
34 |
33
|
ad5ant15 |
|- ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. A e. NN0 ) -> A = +oo ) |
35 |
34
|
ex |
|- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) -> ( -. A e. NN0 -> A = +oo ) ) |
36 |
|
xnn0nnn0pnf |
|- ( ( B e. NN0* /\ -. B e. NN0 ) -> B = +oo ) |
37 |
36
|
ad5ant25 |
|- ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. B e. NN0 ) -> B = +oo ) |
38 |
37
|
ex |
|- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) -> ( -. B e. NN0 -> B = +oo ) ) |
39 |
35 38
|
orim12d |
|- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) -> ( ( -. A e. NN0 \/ -. B e. NN0 ) -> ( A = +oo \/ B = +oo ) ) ) |
40 |
|
pm3.13 |
|- ( -. ( A e. NN0 /\ B e. NN0 ) -> ( -. A e. NN0 \/ -. B e. NN0 ) ) |
41 |
39 40
|
impel |
|- ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) -> ( A = +oo \/ B = +oo ) ) |
42 |
18 32 41
|
mpjaodan |
|- ( ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) /\ -. ( A e. NN0 /\ B e. NN0 ) ) -> -. ( A *e B ) e. NN0 ) |
43 |
1 42
|
condan |
|- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A *e B ) e. NN0 ) -> ( A e. NN0 /\ B e. NN0 ) ) |
44 |
|
nn0re |
|- ( A e. NN0 -> A e. RR ) |
45 |
44
|
ad2antrl |
|- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A e. NN0 /\ B e. NN0 ) ) -> A e. RR ) |
46 |
|
nn0re |
|- ( B e. NN0 -> B e. RR ) |
47 |
46
|
ad2antll |
|- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A e. NN0 /\ B e. NN0 ) ) -> B e. RR ) |
48 |
|
rexmul |
|- ( ( A e. RR /\ B e. RR ) -> ( A *e B ) = ( A x. B ) ) |
49 |
45 47 48
|
syl2anc |
|- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A e. NN0 /\ B e. NN0 ) ) -> ( A *e B ) = ( A x. B ) ) |
50 |
|
nn0mulcl |
|- ( ( A e. NN0 /\ B e. NN0 ) -> ( A x. B ) e. NN0 ) |
51 |
50
|
adantl |
|- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A e. NN0 /\ B e. NN0 ) ) -> ( A x. B ) e. NN0 ) |
52 |
49 51
|
eqeltrd |
|- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ ( A e. NN0 /\ B e. NN0 ) ) -> ( A *e B ) e. NN0 ) |
53 |
43 52
|
impbida |
|- ( ( ( A e. NN0* /\ B e. NN0* ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( ( A *e B ) e. NN0 <-> ( A e. NN0 /\ B e. NN0 ) ) ) |