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Mirrors > Home > ILE Home > Th. List > apneg | Unicode version |
Description: Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.) |
Ref | Expression |
---|---|
apneg | # # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnre 7730 | . . 3 | |
2 | 1 | adantl 275 | . 2 |
3 | cnre 7730 | . . . . . 6 | |
4 | 3 | ad3antrrr 483 | . . . . 5 |
5 | simpr 109 | . . . . . . . . 9 | |
6 | simpllr 508 | . . . . . . . . 9 | |
7 | 5, 6 | breq12d 3912 | . . . . . . . 8 # # |
8 | simplrl 509 | . . . . . . . . 9 | |
9 | simplrr 510 | . . . . . . . . 9 | |
10 | simprl 505 | . . . . . . . . . 10 | |
11 | 10 | ad3antrrr 483 | . . . . . . . . 9 |
12 | simprr 506 | . . . . . . . . . 10 | |
13 | 12 | ad3antrrr 483 | . . . . . . . . 9 |
14 | apreim 8332 | . . . . . . . . 9 # # # | |
15 | 8, 9, 11, 13, 14 | syl22anc 1202 | . . . . . . . 8 # # # |
16 | 8 | renegcld 8110 | . . . . . . . . . 10 |
17 | 9 | renegcld 8110 | . . . . . . . . . 10 |
18 | 11 | renegcld 8110 | . . . . . . . . . 10 |
19 | 13 | renegcld 8110 | . . . . . . . . . 10 |
20 | apreim 8332 | . . . . . . . . . 10 # # # | |
21 | 16, 17, 18, 19, 20 | syl22anc 1202 | . . . . . . . . 9 # # # |
22 | 8 | recnd 7762 | . . . . . . . . . . . 12 |
23 | ax-icn 7683 | . . . . . . . . . . . . . 14 | |
24 | 23 | a1i 9 | . . . . . . . . . . . . 13 |
25 | 9 | recnd 7762 | . . . . . . . . . . . . 13 |
26 | 24, 25 | mulcld 7754 | . . . . . . . . . . . 12 |
27 | 22, 26 | negdid 8054 | . . . . . . . . . . 11 |
28 | 5 | negeqd 7925 | . . . . . . . . . . 11 |
29 | 24, 25 | mulneg2d 8142 | . . . . . . . . . . . 12 |
30 | 29 | oveq2d 5758 | . . . . . . . . . . 11 |
31 | 27, 28, 30 | 3eqtr4d 2160 | . . . . . . . . . 10 |
32 | 11 | recnd 7762 | . . . . . . . . . . . 12 |
33 | 13 | recnd 7762 | . . . . . . . . . . . . 13 |
34 | 24, 33 | mulcld 7754 | . . . . . . . . . . . 12 |
35 | 32, 34 | negdid 8054 | . . . . . . . . . . 11 |
36 | 6 | negeqd 7925 | . . . . . . . . . . 11 |
37 | 24, 33 | mulneg2d 8142 | . . . . . . . . . . . 12 |
38 | 37 | oveq2d 5758 | . . . . . . . . . . 11 |
39 | 35, 36, 38 | 3eqtr4d 2160 | . . . . . . . . . 10 |
40 | 31, 39 | breq12d 3912 | . . . . . . . . 9 # # |
41 | reapneg 8326 | . . . . . . . . . . 11 # # | |
42 | 8, 11, 41 | syl2anc 408 | . . . . . . . . . 10 # # |
43 | reapneg 8326 | . . . . . . . . . . 11 # # | |
44 | 9, 13, 43 | syl2anc 408 | . . . . . . . . . 10 # # |
45 | 42, 44 | orbi12d 767 | . . . . . . . . 9 # # # # |
46 | 21, 40, 45 | 3bitr4rd 220 | . . . . . . . 8 # # # |
47 | 7, 15, 46 | 3bitrd 213 | . . . . . . 7 # # |
48 | 47 | ex 114 | . . . . . 6 # # |
49 | 48 | rexlimdvva 2534 | . . . . 5 # # |
50 | 4, 49 | mpd 13 | . . . 4 # # |
51 | 50 | ex 114 | . . 3 # # |
52 | 51 | rexlimdvva 2534 | . 2 # # |
53 | 2, 52 | mpd 13 | 1 # # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 682 wceq 1316 wcel 1465 wrex 2394 class class class wbr 3899 (class class class)co 5742 cc 7586 cr 7587 ci 7590 caddc 7591 cmul 7593 cneg 7902 # cap 8310 |
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-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-ltxr 7773 df-sub 7903 df-neg 7904 df-reap 8304 df-ap 8311 |
This theorem is referenced by: mulext1 8341 negap0 8359 div2subap 8563 cjap 10633 geosergap 11230 |
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