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Mirrors > Home > ILE Home > Th. List > cnegexlem2 | Unicode version |
Description: Existence of a real number which produces a real number when multiplied by . (Hint: zero is such a number, although we don't need to prove that yet). Lemma for cnegex 7189. (Contributed by Eric Schmidt, 22-May-2007.) |
Ref | Expression |
---|---|
cnegexlem2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 7019 | . 2 | |
2 | cnre 7023 | . 2 | |
3 | ax-rnegex 6993 | . . . . . 6 | |
4 | 3 | adantr 261 | . . . . 5 |
5 | recn 7014 | . . . . . . . . . . 11 | |
6 | ax-icn 6979 | . . . . . . . . . . . 12 | |
7 | recn 7014 | . . . . . . . . . . . 12 | |
8 | mulcl 7008 | . . . . . . . . . . . 12 | |
9 | 6, 7, 8 | sylancr 393 | . . . . . . . . . . 11 |
10 | recn 7014 | . . . . . . . . . . 11 | |
11 | addid2 7152 | . . . . . . . . . . . . . . 15 | |
12 | 11 | 3ad2ant3 927 | . . . . . . . . . . . . . 14 |
13 | 12 | adantr 261 | . . . . . . . . . . . . 13 |
14 | oveq1 5519 | . . . . . . . . . . . . . . 15 | |
15 | 14 | ad2antrl 459 | . . . . . . . . . . . . . 14 |
16 | add32 7170 | . . . . . . . . . . . . . . . . 17 | |
17 | 16 | 3com23 1110 | . . . . . . . . . . . . . . . 16 |
18 | oveq1 5519 | . . . . . . . . . . . . . . . . 17 | |
19 | 18 | eqcomd 2045 | . . . . . . . . . . . . . . . 16 |
20 | 17, 19 | sylan9eq 2092 | . . . . . . . . . . . . . . 15 |
21 | 20 | adantrl 447 | . . . . . . . . . . . . . 14 |
22 | addid2 7152 | . . . . . . . . . . . . . . . 16 | |
23 | 22 | 3ad2ant2 926 | . . . . . . . . . . . . . . 15 |
24 | 23 | adantr 261 | . . . . . . . . . . . . . 14 |
25 | 15, 21, 24 | 3eqtr3d 2080 | . . . . . . . . . . . . 13 |
26 | 13, 25 | eqtr3d 2074 | . . . . . . . . . . . 12 |
27 | 26 | ex 108 | . . . . . . . . . . 11 |
28 | 5, 9, 10, 27 | syl3an 1177 | . . . . . . . . . 10 |
29 | 28 | 3expa 1104 | . . . . . . . . 9 |
30 | 29 | imp 115 | . . . . . . . 8 |
31 | simplr 482 | . . . . . . . 8 | |
32 | 30, 31 | eqeltrrd 2115 | . . . . . . 7 |
33 | 32 | exp32 347 | . . . . . 6 |
34 | 33 | rexlimdva 2433 | . . . . 5 |
35 | 4, 34 | mpd 13 | . . . 4 |
36 | 35 | reximdva 2421 | . . 3 |
37 | 36 | rexlimiv 2427 | . 2 |
38 | 1, 2, 37 | mp2b 8 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wceq 1243 wcel 1393 wrex 2307 (class class class)co 5512 cc 6887 cr 6888 cc0 6889 ci 6891 caddc 6892 cmul 6894 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-resscn 6976 ax-1cn 6977 ax-icn 6979 ax-addcl 6980 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 |
This theorem is referenced by: cnegex 7189 |
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