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Mirrors > Home > ILE Home > Th. List > cnegexlem3 | Unicode version |
Description: Existence of real number difference. Lemma for cnegex 7189. (Contributed by Eric Schmidt, 22-May-2007.) |
Ref | Expression |
---|---|
cnegexlem3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | readdcl 7007 | . . . . . 6 | |
2 | ax-rnegex 6993 | . . . . . 6 | |
3 | 1, 2 | syl 14 | . . . . 5 |
4 | 3 | adantlr 446 | . . . 4 |
5 | 4 | adantr 261 | . . 3 |
6 | recn 7014 | . . . . . . . 8 | |
7 | recn 7014 | . . . . . . . 8 | |
8 | 6, 7 | anim12i 321 | . . . . . . 7 |
9 | 8 | anim1i 323 | . . . . . 6 |
10 | 9 | anim1i 323 | . . . . 5 |
11 | recn 7014 | . . . . 5 | |
12 | recn 7014 | . . . . . . . . . 10 | |
13 | add32 7170 | . . . . . . . . . . . 12 | |
14 | 13 | 3expa 1104 | . . . . . . . . . . 11 |
15 | addcl 7006 | . . . . . . . . . . . . 13 | |
16 | addcom 7150 | . . . . . . . . . . . . 13 | |
17 | 15, 16 | sylan 267 | . . . . . . . . . . . 12 |
18 | 17 | an32s 502 | . . . . . . . . . . 11 |
19 | 14, 18 | eqtr2d 2073 | . . . . . . . . . 10 |
20 | 12, 19 | sylanl2 383 | . . . . . . . . 9 |
21 | 20 | adantllr 450 | . . . . . . . 8 |
22 | 21 | adantlr 446 | . . . . . . 7 |
23 | addcom 7150 | . . . . . . . . . . . 12 | |
24 | 23 | ancoms 255 | . . . . . . . . . . 11 |
25 | 12, 24 | sylan2 270 | . . . . . . . . . 10 |
26 | id 19 | . . . . . . . . . 10 | |
27 | 25, 26 | sylan9eq 2092 | . . . . . . . . 9 |
28 | 27 | adantlll 449 | . . . . . . . 8 |
29 | 28 | adantr 261 | . . . . . . 7 |
30 | 22, 29 | eqeq12d 2054 | . . . . . 6 |
31 | simplr 482 | . . . . . . . 8 | |
32 | 15 | adantlr 446 | . . . . . . . . 9 |
33 | 32 | adantlr 446 | . . . . . . . 8 |
34 | simpllr 486 | . . . . . . . 8 | |
35 | cnegexlem1 7186 | . . . . . . . 8 | |
36 | 31, 33, 34, 35 | syl3anc 1135 | . . . . . . 7 |
37 | 36 | adantlr 446 | . . . . . 6 |
38 | 30, 37 | bitr3d 179 | . . . . 5 |
39 | 10, 11, 38 | syl2an 273 | . . . 4 |
40 | 39 | rexbidva 2323 | . . 3 |
41 | 5, 40 | mpbid 135 | . 2 |
42 | ax-rnegex 6993 | . . 3 | |
43 | 42 | adantl 262 | . 2 |
44 | 41, 43 | r19.29a 2454 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wrex 2307 (class class class)co 5512 cc 6887 cr 6888 cc0 6889 caddc 6892 |
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-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 |
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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