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Mirrors > Home > ILE Home > Th. List > xltnegi | Unicode version |
Description: Forward direction of xltneg 9619. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xltnegi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9563 | . . 3 | |
2 | elxr 9563 | . . . . . 6 | |
3 | ltneg 8224 | . . . . . . . . 9 | |
4 | rexneg 9613 | . . . . . . . . . 10 | |
5 | rexneg 9613 | . . . . . . . . . 10 | |
6 | 4, 5 | breqan12rd 3946 | . . . . . . . . 9 |
7 | 3, 6 | bitr4d 190 | . . . . . . . 8 |
8 | 7 | biimpd 143 | . . . . . . 7 |
9 | xnegeq 9610 | . . . . . . . . . . 11 | |
10 | xnegpnf 9611 | . . . . . . . . . . 11 | |
11 | 9, 10 | syl6eq 2188 | . . . . . . . . . 10 |
12 | 11 | adantl 275 | . . . . . . . . 9 |
13 | renegcl 8023 | . . . . . . . . . . . 12 | |
14 | 5, 13 | eqeltrd 2216 | . . . . . . . . . . 11 |
15 | mnflt 9569 | . . . . . . . . . . 11 | |
16 | 14, 15 | syl 14 | . . . . . . . . . 10 |
17 | 16 | adantr 274 | . . . . . . . . 9 |
18 | 12, 17 | eqbrtrd 3950 | . . . . . . . 8 |
19 | 18 | a1d 22 | . . . . . . 7 |
20 | simpr 109 | . . . . . . . . 9 | |
21 | 20 | breq2d 3941 | . . . . . . . 8 |
22 | rexr 7811 | . . . . . . . . . . 11 | |
23 | nltmnf 9574 | . . . . . . . . . . 11 | |
24 | 22, 23 | syl 14 | . . . . . . . . . 10 |
25 | 24 | adantr 274 | . . . . . . . . 9 |
26 | 25 | pm2.21d 608 | . . . . . . . 8 |
27 | 21, 26 | sylbid 149 | . . . . . . 7 |
28 | 8, 19, 27 | 3jaodan 1284 | . . . . . 6 |
29 | 2, 28 | sylan2b 285 | . . . . 5 |
30 | 29 | expimpd 360 | . . . 4 |
31 | simpl 108 | . . . . . . 7 | |
32 | 31 | breq1d 3939 | . . . . . 6 |
33 | pnfnlt 9573 | . . . . . . . 8 | |
34 | 33 | adantl 275 | . . . . . . 7 |
35 | 34 | pm2.21d 608 | . . . . . 6 |
36 | 32, 35 | sylbid 149 | . . . . 5 |
37 | 36 | expimpd 360 | . . . 4 |
38 | breq1 3932 | . . . . . 6 | |
39 | 38 | anbi2d 459 | . . . . 5 |
40 | renegcl 8023 | . . . . . . . . . . 11 | |
41 | 4, 40 | eqeltrd 2216 | . . . . . . . . . 10 |
42 | 41 | adantr 274 | . . . . . . . . 9 |
43 | ltpnf 9567 | . . . . . . . . 9 | |
44 | 42, 43 | syl 14 | . . . . . . . 8 |
45 | 11 | adantr 274 | . . . . . . . . 9 |
46 | mnfltpnf 9571 | . . . . . . . . 9 | |
47 | 45, 46 | eqbrtrdi 3967 | . . . . . . . 8 |
48 | breq2 3933 | . . . . . . . . . 10 | |
49 | mnfxr 7822 | . . . . . . . . . . . 12 | |
50 | nltmnf 9574 | . . . . . . . . . . . 12 | |
51 | 49, 50 | ax-mp 5 | . . . . . . . . . . 11 |
52 | 51 | pm2.21i 635 | . . . . . . . . . 10 |
53 | 48, 52 | syl6bi 162 | . . . . . . . . 9 |
54 | 53 | imp 123 | . . . . . . . 8 |
55 | 44, 47, 54 | 3jaoian 1283 | . . . . . . 7 |
56 | 2, 55 | sylanb 282 | . . . . . 6 |
57 | xnegeq 9610 | . . . . . . . 8 | |
58 | xnegmnf 9612 | . . . . . . . 8 | |
59 | 57, 58 | syl6eq 2188 | . . . . . . 7 |
60 | 59 | breq2d 3941 | . . . . . 6 |
61 | 56, 60 | syl5ibr 155 | . . . . 5 |
62 | 39, 61 | sylbid 149 | . . . 4 |
63 | 30, 37, 62 | 3jaoi 1281 | . . 3 |
64 | 1, 63 | sylbi 120 | . 2 |
65 | 64 | 3impib 1179 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 w3o 961 w3a 962 wceq 1331 wcel 1480 class class class wbr 3929 cr 7619 cpnf 7797 cmnf 7798 cxr 7799 clt 7800 cneg 7934 cxne 9556 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-sub 7935 df-neg 7936 df-xneg 9559 |
This theorem is referenced by: xltneg 9619 |
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