Metamath Proof Explorer

Theorem sn-inelr

Description: inelr without ax-mulcom . (Contributed by SN, 1-Jun-2024)

Ref Expression
Assertion sn-inelr 
|- -. _i e. RR

Proof

Step Hyp Ref Expression
1 0re 
 |-  0 e. RR
2 lttri4 
 |-  ( ( _i e. RR /\ 0 e. RR ) -> ( _i < 0 \/ _i = 0 \/ 0 < _i ) )
3 1 2 mpan2 
 |-  ( _i e. RR -> ( _i < 0 \/ _i = 0 \/ 0 < _i ) )
4 reneg1lt0 
 |-  ( 0 -R 1 ) < 0
5 1re 
 |-  1 e. RR
6 rernegcl 
 |-  ( 1 e. RR -> ( 0 -R 1 ) e. RR )
7 5 6 ax-mp 
 |-  ( 0 -R 1 ) e. RR
8 7 1 ltnsymi 
 |-  ( ( 0 -R 1 ) < 0 -> -. 0 < ( 0 -R 1 ) )
9 4 8 ax-mp 
 |-  -. 0 < ( 0 -R 1 )
10 relt0neg1 
 |-  ( _i e. RR -> ( _i < 0 <-> 0 < ( 0 -R _i ) ) )
11 rernegcl 
 |-  ( _i e. RR -> ( 0 -R _i ) e. RR )
12 mulgt0 
 |-  ( ( ( ( 0 -R _i ) e. RR /\ 0 < ( 0 -R _i ) ) /\ ( ( 0 -R _i ) e. RR /\ 0 < ( 0 -R _i ) ) ) -> 0 < ( ( 0 -R _i ) x. ( 0 -R _i ) ) )
13 12 anidms 
 |-  ( ( ( 0 -R _i ) e. RR /\ 0 < ( 0 -R _i ) ) -> 0 < ( ( 0 -R _i ) x. ( 0 -R _i ) ) )
14 11 13 sylan 
 |-  ( ( _i e. RR /\ 0 < ( 0 -R _i ) ) -> 0 < ( ( 0 -R _i ) x. ( 0 -R _i ) ) )
15 id 
 |-  ( _i e. RR -> _i e. RR )
16 11 15 remulneg2d 
 |-  ( _i e. RR -> ( ( 0 -R _i ) x. ( 0 -R _i ) ) = ( 0 -R ( ( 0 -R _i ) x. _i ) ) )
17 15 15 remulcld 
 |-  ( _i e. RR -> ( _i x. _i ) e. RR )
18 15 17 remulcld 
 |-  ( _i e. RR -> ( _i x. ( _i x. _i ) ) e. RR )
19 ipiiie0 
 |-  ( _i + ( _i x. ( _i x. _i ) ) ) = 0
20 renegadd 
 |-  ( ( _i e. RR /\ ( _i x. ( _i x. _i ) ) e. RR ) -> ( ( 0 -R _i ) = ( _i x. ( _i x. _i ) ) <-> ( _i + ( _i x. ( _i x. _i ) ) ) = 0 ) )
21 19 20 mpbiri 
 |-  ( ( _i e. RR /\ ( _i x. ( _i x. _i ) ) e. RR ) -> ( 0 -R _i ) = ( _i x. ( _i x. _i ) ) )
22 18 21 mpdan 
 |-  ( _i e. RR -> ( 0 -R _i ) = ( _i x. ( _i x. _i ) ) )
23 22 oveq1d 
 |-  ( _i e. RR -> ( ( 0 -R _i ) x. _i ) = ( ( _i x. ( _i x. _i ) ) x. _i ) )
24 ax-icn 
 |-  _i e. CC
25 24 24 24 mulassi 
 |-  ( ( _i x. _i ) x. _i ) = ( _i x. ( _i x. _i ) )
26 25 oveq1i 
 |-  ( ( ( _i x. _i ) x. _i ) x. _i ) = ( ( _i x. ( _i x. _i ) ) x. _i )
27 24 24 mulcli 
 |-  ( _i x. _i ) e. CC
28 27 24 24 mulassi 
 |-  ( ( ( _i x. _i ) x. _i ) x. _i ) = ( ( _i x. _i ) x. ( _i x. _i ) )
29 26 28 eqtr3i 
 |-  ( ( _i x. ( _i x. _i ) ) x. _i ) = ( ( _i x. _i ) x. ( _i x. _i ) )
30 29 a1i 
 |-  ( _i e. RR -> ( ( _i x. ( _i x. _i ) ) x. _i ) = ( ( _i x. _i ) x. ( _i x. _i ) ) )
31 rei4 
 |-  ( ( _i x. _i ) x. ( _i x. _i ) ) = 1
32 31 a1i 
 |-  ( _i e. RR -> ( ( _i x. _i ) x. ( _i x. _i ) ) = 1 )
33 23 30 32 3eqtrd 
 |-  ( _i e. RR -> ( ( 0 -R _i ) x. _i ) = 1 )
34 33 oveq2d 
 |-  ( _i e. RR -> ( 0 -R ( ( 0 -R _i ) x. _i ) ) = ( 0 -R 1 ) )
35 16 34 eqtrd 
 |-  ( _i e. RR -> ( ( 0 -R _i ) x. ( 0 -R _i ) ) = ( 0 -R 1 ) )
36 35 breq2d 
 |-  ( _i e. RR -> ( 0 < ( ( 0 -R _i ) x. ( 0 -R _i ) ) <-> 0 < ( 0 -R 1 ) ) )
37 36 adantr 
 |-  ( ( _i e. RR /\ 0 < ( 0 -R _i ) ) -> ( 0 < ( ( 0 -R _i ) x. ( 0 -R _i ) ) <-> 0 < ( 0 -R 1 ) ) )
38 14 37 mpbid 
 |-  ( ( _i e. RR /\ 0 < ( 0 -R _i ) ) -> 0 < ( 0 -R 1 ) )
39 38 ex 
 |-  ( _i e. RR -> ( 0 < ( 0 -R _i ) -> 0 < ( 0 -R 1 ) ) )
40 10 39 sylbid 
 |-  ( _i e. RR -> ( _i < 0 -> 0 < ( 0 -R 1 ) ) )
41 9 40 mtoi 
 |-  ( _i e. RR -> -. _i < 0 )
42 0ne1 
 |-  0 =/= 1
43 42 neii 
 |-  -. 0 = 1
44 oveq12 
 |-  ( ( _i = 0 /\ _i = 0 ) -> ( _i x. _i ) = ( 0 x. 0 ) )
45 44 anidms 
 |-  ( _i = 0 -> ( _i x. _i ) = ( 0 x. 0 ) )
46 45 oveq1d 
 |-  ( _i = 0 -> ( ( _i x. _i ) + 1 ) = ( ( 0 x. 0 ) + 1 ) )
47 ax-i2m1 
 |-  ( ( _i x. _i ) + 1 ) = 0
48 remul02 
 |-  ( 0 e. RR -> ( 0 x. 0 ) = 0 )
49 1 48 ax-mp 
 |-  ( 0 x. 0 ) = 0
50 49 oveq1i 
 |-  ( ( 0 x. 0 ) + 1 ) = ( 0 + 1 )
51 readdlid 
 |-  ( 1 e. RR -> ( 0 + 1 ) = 1 )
52 5 51 ax-mp 
 |-  ( 0 + 1 ) = 1
53 50 52 eqtri 
 |-  ( ( 0 x. 0 ) + 1 ) = 1
54 46 47 53 3eqtr3g 
 |-  ( _i = 0 -> 0 = 1 )
55 43 54 mto 
 |-  -. _i = 0
56 55 a1i 
 |-  ( _i e. RR -> -. _i = 0 )
57 mulgt0 
 |-  ( ( ( _i e. RR /\ 0 < _i ) /\ ( _i e. RR /\ 0 < _i ) ) -> 0 < ( _i x. _i ) )
58 57 anidms 
 |-  ( ( _i e. RR /\ 0 < _i ) -> 0 < ( _i x. _i ) )
59 reixi 
 |-  ( _i x. _i ) = ( 0 -R 1 )
60 58 59 breqtrdi 
 |-  ( ( _i e. RR /\ 0 < _i ) -> 0 < ( 0 -R 1 ) )
61 60 ex 
 |-  ( _i e. RR -> ( 0 < _i -> 0 < ( 0 -R 1 ) ) )
62 9 61 mtoi 
 |-  ( _i e. RR -> -. 0 < _i )
63 3ioran 
 |-  ( -. ( _i < 0 \/ _i = 0 \/ 0 < _i ) <-> ( -. _i < 0 /\ -. _i = 0 /\ -. 0 < _i ) )
64 41 56 62 63 syl3anbrc 
 |-  ( _i e. RR -> -. ( _i < 0 \/ _i = 0 \/ 0 < _i ) )
65 3 64 pm2.65i 
 |-  -. _i e. RR