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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 8053. (Contributed by Eric Schmidt, 22-May-2007.) |
Ref | Expression |
---|---|
cnegexlem2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 7870 | . 2 | |
2 | cnre 7874 | . 2 | |
3 | ax-rnegex 7841 | . . . . . 6 | |
4 | 3 | adantr 274 | . . . . 5 |
5 | recn 7865 | . . . . . . . . . . 11 | |
6 | ax-icn 7827 | . . . . . . . . . . . 12 | |
7 | recn 7865 | . . . . . . . . . . . 12 | |
8 | mulcl 7859 | . . . . . . . . . . . 12 | |
9 | 6, 7, 8 | sylancr 411 | . . . . . . . . . . 11 |
10 | recn 7865 | . . . . . . . . . . 11 | |
11 | addid2 8014 | . . . . . . . . . . . . . . 15 | |
12 | 11 | 3ad2ant3 1005 | . . . . . . . . . . . . . 14 |
13 | 12 | adantr 274 | . . . . . . . . . . . . 13 |
14 | oveq1 5831 | . . . . . . . . . . . . . . 15 | |
15 | 14 | ad2antrl 482 | . . . . . . . . . . . . . 14 |
16 | add32 8034 | . . . . . . . . . . . . . . . . 17 | |
17 | 16 | 3com23 1191 | . . . . . . . . . . . . . . . 16 |
18 | oveq1 5831 | . . . . . . . . . . . . . . . . 17 | |
19 | 18 | eqcomd 2163 | . . . . . . . . . . . . . . . 16 |
20 | 17, 19 | sylan9eq 2210 | . . . . . . . . . . . . . . 15 |
21 | 20 | adantrl 470 | . . . . . . . . . . . . . 14 |
22 | addid2 8014 | . . . . . . . . . . . . . . . 16 | |
23 | 22 | 3ad2ant2 1004 | . . . . . . . . . . . . . . 15 |
24 | 23 | adantr 274 | . . . . . . . . . . . . . 14 |
25 | 15, 21, 24 | 3eqtr3d 2198 | . . . . . . . . . . . . 13 |
26 | 13, 25 | eqtr3d 2192 | . . . . . . . . . . . 12 |
27 | 26 | ex 114 | . . . . . . . . . . 11 |
28 | 5, 9, 10, 27 | syl3an 1262 | . . . . . . . . . 10 |
29 | 28 | 3expa 1185 | . . . . . . . . 9 |
30 | 29 | imp 123 | . . . . . . . 8 |
31 | simplr 520 | . . . . . . . 8 | |
32 | 30, 31 | eqeltrrd 2235 | . . . . . . 7 |
33 | 32 | exp32 363 | . . . . . 6 |
34 | 33 | rexlimdva 2574 | . . . . 5 |
35 | 4, 34 | mpd 13 | . . . 4 |
36 | 35 | reximdva 2559 | . . 3 |
37 | 36 | rexlimiv 2568 | . 2 |
38 | 1, 2, 37 | mp2b 8 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 963 wceq 1335 wcel 2128 wrex 2436 (class class class)co 5824 cc 7730 cr 7731 cc0 7732 ci 7734 caddc 7735 cmul 7737 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-resscn 7824 ax-1cn 7825 ax-icn 7827 ax-addcl 7828 ax-mulcl 7830 ax-addcom 7832 ax-addass 7834 ax-i2m1 7837 ax-0id 7840 ax-rnegex 7841 ax-cnre 7843 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-iota 5135 df-fv 5178 df-ov 5827 |
This theorem is referenced by: cnegex 8053 |
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