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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 8097. (Contributed by Eric Schmidt, 22-May-2007.) |
Ref | Expression |
---|---|
cnegexlem2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 7912 | . 2 | |
2 | cnre 7916 | . 2 | |
3 | ax-rnegex 7883 | . . . . . 6 | |
4 | 3 | adantr 274 | . . . . 5 |
5 | recn 7907 | . . . . . . . . . . 11 | |
6 | ax-icn 7869 | . . . . . . . . . . . 12 | |
7 | recn 7907 | . . . . . . . . . . . 12 | |
8 | mulcl 7901 | . . . . . . . . . . . 12 | |
9 | 6, 7, 8 | sylancr 412 | . . . . . . . . . . 11 |
10 | recn 7907 | . . . . . . . . . . 11 | |
11 | addid2 8058 | . . . . . . . . . . . . . . 15 | |
12 | 11 | 3ad2ant3 1015 | . . . . . . . . . . . . . 14 |
13 | 12 | adantr 274 | . . . . . . . . . . . . 13 |
14 | oveq1 5860 | . . . . . . . . . . . . . . 15 | |
15 | 14 | ad2antrl 487 | . . . . . . . . . . . . . 14 |
16 | add32 8078 | . . . . . . . . . . . . . . . . 17 | |
17 | 16 | 3com23 1204 | . . . . . . . . . . . . . . . 16 |
18 | oveq1 5860 | . . . . . . . . . . . . . . . . 17 | |
19 | 18 | eqcomd 2176 | . . . . . . . . . . . . . . . 16 |
20 | 17, 19 | sylan9eq 2223 | . . . . . . . . . . . . . . 15 |
21 | 20 | adantrl 475 | . . . . . . . . . . . . . 14 |
22 | addid2 8058 | . . . . . . . . . . . . . . . 16 | |
23 | 22 | 3ad2ant2 1014 | . . . . . . . . . . . . . . 15 |
24 | 23 | adantr 274 | . . . . . . . . . . . . . 14 |
25 | 15, 21, 24 | 3eqtr3d 2211 | . . . . . . . . . . . . 13 |
26 | 13, 25 | eqtr3d 2205 | . . . . . . . . . . . 12 |
27 | 26 | ex 114 | . . . . . . . . . . 11 |
28 | 5, 9, 10, 27 | syl3an 1275 | . . . . . . . . . 10 |
29 | 28 | 3expa 1198 | . . . . . . . . 9 |
30 | 29 | imp 123 | . . . . . . . 8 |
31 | simplr 525 | . . . . . . . 8 | |
32 | 30, 31 | eqeltrrd 2248 | . . . . . . 7 |
33 | 32 | exp32 363 | . . . . . 6 |
34 | 33 | rexlimdva 2587 | . . . . 5 |
35 | 4, 34 | mpd 13 | . . . 4 |
36 | 35 | reximdva 2572 | . . 3 |
37 | 36 | rexlimiv 2581 | . 2 |
38 | 1, 2, 37 | mp2b 8 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 973 wceq 1348 wcel 2141 wrex 2449 (class class class)co 5853 cc 7772 cr 7773 cc0 7774 ci 7776 caddc 7777 cmul 7779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-resscn 7866 ax-1cn 7867 ax-icn 7869 ax-addcl 7870 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-i2m1 7879 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-iota 5160 df-fv 5206 df-ov 5856 |
This theorem is referenced by: cnegex 8097 |
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