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Mirrors > Home > ILE Home > Th. List > xltnegi | Unicode version |
Description: Forward direction of xltneg 9807. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xltnegi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9747 | . . 3 | |
2 | elxr 9747 | . . . . . 6 | |
3 | ltneg 8393 | . . . . . . . . 9 | |
4 | rexneg 9801 | . . . . . . . . . 10 | |
5 | rexneg 9801 | . . . . . . . . . 10 | |
6 | 4, 5 | breqan12rd 4015 | . . . . . . . . 9 |
7 | 3, 6 | bitr4d 191 | . . . . . . . 8 |
8 | 7 | biimpd 144 | . . . . . . 7 |
9 | xnegeq 9798 | . . . . . . . . . . 11 | |
10 | xnegpnf 9799 | . . . . . . . . . . 11 | |
11 | 9, 10 | eqtrdi 2224 | . . . . . . . . . 10 |
12 | 11 | adantl 277 | . . . . . . . . 9 |
13 | renegcl 8192 | . . . . . . . . . . . 12 | |
14 | 5, 13 | eqeltrd 2252 | . . . . . . . . . . 11 |
15 | mnflt 9754 | . . . . . . . . . . 11 | |
16 | 14, 15 | syl 14 | . . . . . . . . . 10 |
17 | 16 | adantr 276 | . . . . . . . . 9 |
18 | 12, 17 | eqbrtrd 4020 | . . . . . . . 8 |
19 | 18 | a1d 22 | . . . . . . 7 |
20 | simpr 110 | . . . . . . . . 9 | |
21 | 20 | breq2d 4010 | . . . . . . . 8 |
22 | rexr 7977 | . . . . . . . . . . 11 | |
23 | nltmnf 9759 | . . . . . . . . . . 11 | |
24 | 22, 23 | syl 14 | . . . . . . . . . 10 |
25 | 24 | adantr 276 | . . . . . . . . 9 |
26 | 25 | pm2.21d 619 | . . . . . . . 8 |
27 | 21, 26 | sylbid 150 | . . . . . . 7 |
28 | 8, 19, 27 | 3jaodan 1306 | . . . . . 6 |
29 | 2, 28 | sylan2b 287 | . . . . 5 |
30 | 29 | expimpd 363 | . . . 4 |
31 | simpl 109 | . . . . . . 7 | |
32 | 31 | breq1d 4008 | . . . . . 6 |
33 | pnfnlt 9758 | . . . . . . . 8 | |
34 | 33 | adantl 277 | . . . . . . 7 |
35 | 34 | pm2.21d 619 | . . . . . 6 |
36 | 32, 35 | sylbid 150 | . . . . 5 |
37 | 36 | expimpd 363 | . . . 4 |
38 | breq1 4001 | . . . . . 6 | |
39 | 38 | anbi2d 464 | . . . . 5 |
40 | renegcl 8192 | . . . . . . . . . . 11 | |
41 | 4, 40 | eqeltrd 2252 | . . . . . . . . . 10 |
42 | 41 | adantr 276 | . . . . . . . . 9 |
43 | ltpnf 9751 | . . . . . . . . 9 | |
44 | 42, 43 | syl 14 | . . . . . . . 8 |
45 | 11 | adantr 276 | . . . . . . . . 9 |
46 | mnfltpnf 9756 | . . . . . . . . 9 | |
47 | 45, 46 | eqbrtrdi 4037 | . . . . . . . 8 |
48 | breq2 4002 | . . . . . . . . . 10 | |
49 | mnfxr 7988 | . . . . . . . . . . . 12 | |
50 | nltmnf 9759 | . . . . . . . . . . . 12 | |
51 | 49, 50 | ax-mp 5 | . . . . . . . . . . 11 |
52 | 51 | pm2.21i 646 | . . . . . . . . . 10 |
53 | 48, 52 | syl6bi 163 | . . . . . . . . 9 |
54 | 53 | imp 124 | . . . . . . . 8 |
55 | 44, 47, 54 | 3jaoian 1305 | . . . . . . 7 |
56 | 2, 55 | sylanb 284 | . . . . . 6 |
57 | xnegeq 9798 | . . . . . . . 8 | |
58 | xnegmnf 9800 | . . . . . . . 8 | |
59 | 57, 58 | eqtrdi 2224 | . . . . . . 7 |
60 | 59 | breq2d 4010 | . . . . . 6 |
61 | 56, 60 | syl5ibr 156 | . . . . 5 |
62 | 39, 61 | sylbid 150 | . . . 4 |
63 | 30, 37, 62 | 3jaoi 1303 | . . 3 |
64 | 1, 63 | sylbi 121 | . 2 |
65 | 64 | 3impib 1201 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 104 w3o 977 w3a 978 wceq 1353 wcel 2146 class class class wbr 3998 cr 7785 cpnf 7963 cmnf 7964 cxr 7965 clt 7966 cneg 8103 cxne 9740 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-addcom 7886 ax-addass 7888 ax-distr 7890 ax-i2m1 7891 ax-0id 7894 ax-rnegex 7895 ax-cnre 7897 ax-pre-ltadd 7902 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-iota 5170 df-fun 5210 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-sub 8104 df-neg 8105 df-xneg 9743 |
This theorem is referenced by: xltneg 9807 |
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