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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 7889 | . . 3 | |
2 | 1 | adantl 275 | . 2 |
3 | cnre 7889 | . . . . . 6 | |
4 | 3 | ad3antrrr 484 | . . . . 5 |
5 | simpr 109 | . . . . . . . . 9 | |
6 | simpllr 524 | . . . . . . . . 9 | |
7 | 5, 6 | breq12d 3992 | . . . . . . . 8 # # |
8 | simplrl 525 | . . . . . . . . 9 | |
9 | simplrr 526 | . . . . . . . . 9 | |
10 | simprl 521 | . . . . . . . . . 10 | |
11 | 10 | ad3antrrr 484 | . . . . . . . . 9 |
12 | simprr 522 | . . . . . . . . . 10 | |
13 | 12 | ad3antrrr 484 | . . . . . . . . 9 |
14 | apreim 8495 | . . . . . . . . 9 # # # | |
15 | 8, 9, 11, 13, 14 | syl22anc 1228 | . . . . . . . 8 # # # |
16 | 8 | renegcld 8272 | . . . . . . . . . 10 |
17 | 9 | renegcld 8272 | . . . . . . . . . 10 |
18 | 11 | renegcld 8272 | . . . . . . . . . 10 |
19 | 13 | renegcld 8272 | . . . . . . . . . 10 |
20 | apreim 8495 | . . . . . . . . . 10 # # # | |
21 | 16, 17, 18, 19, 20 | syl22anc 1228 | . . . . . . . . 9 # # # |
22 | 8 | recnd 7921 | . . . . . . . . . . . 12 |
23 | ax-icn 7842 | . . . . . . . . . . . . . 14 | |
24 | 23 | a1i 9 | . . . . . . . . . . . . 13 |
25 | 9 | recnd 7921 | . . . . . . . . . . . . 13 |
26 | 24, 25 | mulcld 7913 | . . . . . . . . . . . 12 |
27 | 22, 26 | negdid 8216 | . . . . . . . . . . 11 |
28 | 5 | negeqd 8087 | . . . . . . . . . . 11 |
29 | 24, 25 | mulneg2d 8304 | . . . . . . . . . . . 12 |
30 | 29 | oveq2d 5855 | . . . . . . . . . . 11 |
31 | 27, 28, 30 | 3eqtr4d 2207 | . . . . . . . . . 10 |
32 | 11 | recnd 7921 | . . . . . . . . . . . 12 |
33 | 13 | recnd 7921 | . . . . . . . . . . . . 13 |
34 | 24, 33 | mulcld 7913 | . . . . . . . . . . . 12 |
35 | 32, 34 | negdid 8216 | . . . . . . . . . . 11 |
36 | 6 | negeqd 8087 | . . . . . . . . . . 11 |
37 | 24, 33 | mulneg2d 8304 | . . . . . . . . . . . 12 |
38 | 37 | oveq2d 5855 | . . . . . . . . . . 11 |
39 | 35, 36, 38 | 3eqtr4d 2207 | . . . . . . . . . 10 |
40 | 31, 39 | breq12d 3992 | . . . . . . . . 9 # # |
41 | reapneg 8489 | . . . . . . . . . . 11 # # | |
42 | 8, 11, 41 | syl2anc 409 | . . . . . . . . . 10 # # |
43 | reapneg 8489 | . . . . . . . . . . 11 # # | |
44 | 9, 13, 43 | syl2anc 409 | . . . . . . . . . 10 # # |
45 | 42, 44 | orbi12d 783 | . . . . . . . . 9 # # # # |
46 | 21, 40, 45 | 3bitr4rd 220 | . . . . . . . 8 # # # |
47 | 7, 15, 46 | 3bitrd 213 | . . . . . . 7 # # |
48 | 47 | ex 114 | . . . . . 6 # # |
49 | 48 | rexlimdvva 2589 | . . . . 5 # # |
50 | 4, 49 | mpd 13 | . . . 4 # # |
51 | 50 | ex 114 | . . 3 # # |
52 | 51 | rexlimdvva 2589 | . 2 # # |
53 | 2, 52 | mpd 13 | 1 # # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 698 wceq 1342 wcel 2135 wrex 2443 class class class wbr 3979 (class class class)co 5839 cc 7745 cr 7746 ci 7749 caddc 7750 cmul 7752 cneg 8064 # cap 8473 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4097 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-cnex 7838 ax-resscn 7839 ax-1cn 7840 ax-1re 7841 ax-icn 7842 ax-addcl 7843 ax-addrcl 7844 ax-mulcl 7845 ax-mulrcl 7846 ax-addcom 7847 ax-mulcom 7848 ax-addass 7849 ax-mulass 7850 ax-distr 7851 ax-i2m1 7852 ax-0lt1 7853 ax-1rid 7854 ax-0id 7855 ax-rnegex 7856 ax-precex 7857 ax-cnre 7858 ax-pre-ltirr 7859 ax-pre-lttrn 7861 ax-pre-apti 7862 ax-pre-ltadd 7863 ax-pre-mulgt0 7864 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2726 df-sbc 2950 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-br 3980 df-opab 4041 df-id 4268 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-iota 5150 df-fun 5187 df-fv 5193 df-riota 5795 df-ov 5842 df-oprab 5843 df-mpo 5844 df-pnf 7929 df-mnf 7930 df-ltxr 7932 df-sub 8065 df-neg 8066 df-reap 8467 df-ap 8474 |
This theorem is referenced by: mulext1 8504 negap0 8522 div2subap 8727 cjap 10842 geosergap 11441 |
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