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Mirrors > Home > ILE Home > Th. List > xltnegi | Unicode version |
Description: Forward direction of xltneg 8749. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xltnegi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 8696 | . . 3 | |
2 | elxr 8696 | . . . . . 6 | |
3 | ltneg 7457 | . . . . . . . . 9 | |
4 | rexneg 8743 | . . . . . . . . . 10 | |
5 | rexneg 8743 | . . . . . . . . . 10 | |
6 | 4, 5 | breqan12rd 3780 | . . . . . . . . 9 |
7 | 3, 6 | bitr4d 180 | . . . . . . . 8 |
8 | 7 | biimpd 132 | . . . . . . 7 |
9 | xnegeq 8740 | . . . . . . . . . . 11 | |
10 | xnegpnf 8741 | . . . . . . . . . . 11 | |
11 | 9, 10 | syl6eq 2088 | . . . . . . . . . 10 |
12 | 11 | adantl 262 | . . . . . . . . 9 |
13 | renegcl 7272 | . . . . . . . . . . . 12 | |
14 | 5, 13 | eqeltrd 2114 | . . . . . . . . . . 11 |
15 | mnflt 8704 | . . . . . . . . . . 11 | |
16 | 14, 15 | syl 14 | . . . . . . . . . 10 |
17 | 16 | adantr 261 | . . . . . . . . 9 |
18 | 12, 17 | eqbrtrd 3784 | . . . . . . . 8 |
19 | 18 | a1d 22 | . . . . . . 7 |
20 | simpr 103 | . . . . . . . . 9 | |
21 | 20 | breq2d 3776 | . . . . . . . 8 |
22 | rexr 7071 | . . . . . . . . . . 11 | |
23 | nltmnf 8709 | . . . . . . . . . . 11 | |
24 | 22, 23 | syl 14 | . . . . . . . . . 10 |
25 | 24 | adantr 261 | . . . . . . . . 9 |
26 | 25 | pm2.21d 549 | . . . . . . . 8 |
27 | 21, 26 | sylbid 139 | . . . . . . 7 |
28 | 8, 19, 27 | 3jaodan 1201 | . . . . . 6 |
29 | 2, 28 | sylan2b 271 | . . . . 5 |
30 | 29 | expimpd 345 | . . . 4 |
31 | simpl 102 | . . . . . . 7 | |
32 | 31 | breq1d 3774 | . . . . . 6 |
33 | pnfnlt 8708 | . . . . . . . 8 | |
34 | 33 | adantl 262 | . . . . . . 7 |
35 | 34 | pm2.21d 549 | . . . . . 6 |
36 | 32, 35 | sylbid 139 | . . . . 5 |
37 | 36 | expimpd 345 | . . . 4 |
38 | breq1 3767 | . . . . . 6 | |
39 | 38 | anbi2d 437 | . . . . 5 |
40 | renegcl 7272 | . . . . . . . . . . 11 | |
41 | 4, 40 | eqeltrd 2114 | . . . . . . . . . 10 |
42 | 41 | adantr 261 | . . . . . . . . 9 |
43 | ltpnf 8702 | . . . . . . . . 9 | |
44 | 42, 43 | syl 14 | . . . . . . . 8 |
45 | 11 | adantr 261 | . . . . . . . . 9 |
46 | mnfltpnf 8706 | . . . . . . . . 9 | |
47 | 45, 46 | syl6eqbr 3801 | . . . . . . . 8 |
48 | breq2 3768 | . . . . . . . . . 10 | |
49 | mnfxr 8694 | . . . . . . . . . . . 12 | |
50 | nltmnf 8709 | . . . . . . . . . . . 12 | |
51 | 49, 50 | ax-mp 7 | . . . . . . . . . . 11 |
52 | 51 | pm2.21i 575 | . . . . . . . . . 10 |
53 | 48, 52 | syl6bi 152 | . . . . . . . . 9 |
54 | 53 | imp 115 | . . . . . . . 8 |
55 | 44, 47, 54 | 3jaoian 1200 | . . . . . . 7 |
56 | 2, 55 | sylanb 268 | . . . . . 6 |
57 | xnegeq 8740 | . . . . . . . 8 | |
58 | xnegmnf 8742 | . . . . . . . 8 | |
59 | 57, 58 | syl6eq 2088 | . . . . . . 7 |
60 | 59 | breq2d 3776 | . . . . . 6 |
61 | 56, 60 | syl5ibr 145 | . . . . 5 |
62 | 39, 61 | sylbid 139 | . . . 4 |
63 | 30, 37, 62 | 3jaoi 1198 | . . 3 |
64 | 1, 63 | sylbi 114 | . 2 |
65 | 64 | 3impib 1102 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 w3o 884 w3a 885 wceq 1243 wcel 1393 class class class wbr 3764 cr 6888 cpnf 7057 cmnf 7058 cxr 7059 clt 7060 cneg 7183 cxne 8686 |
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-in1 544 ax-in2 545 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-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 ax-pre-ltadd 7000 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-if 3332 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-sub 7184 df-neg 7185 df-xneg 8689 |
This theorem is referenced by: xltneg 8749 |
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