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Mirrors > Home > ILE Home > Th. List > xltnegi | Unicode version |
Description: Forward direction of xltneg 9793. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xltnegi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9733 | . . 3 | |
2 | elxr 9733 | . . . . . 6 | |
3 | ltneg 8381 | . . . . . . . . 9 | |
4 | rexneg 9787 | . . . . . . . . . 10 | |
5 | rexneg 9787 | . . . . . . . . . 10 | |
6 | 4, 5 | breqan12rd 4006 | . . . . . . . . 9 |
7 | 3, 6 | bitr4d 190 | . . . . . . . 8 |
8 | 7 | biimpd 143 | . . . . . . 7 |
9 | xnegeq 9784 | . . . . . . . . . . 11 | |
10 | xnegpnf 9785 | . . . . . . . . . . 11 | |
11 | 9, 10 | eqtrdi 2219 | . . . . . . . . . 10 |
12 | 11 | adantl 275 | . . . . . . . . 9 |
13 | renegcl 8180 | . . . . . . . . . . . 12 | |
14 | 5, 13 | eqeltrd 2247 | . . . . . . . . . . 11 |
15 | mnflt 9740 | . . . . . . . . . . 11 | |
16 | 14, 15 | syl 14 | . . . . . . . . . 10 |
17 | 16 | adantr 274 | . . . . . . . . 9 |
18 | 12, 17 | eqbrtrd 4011 | . . . . . . . 8 |
19 | 18 | a1d 22 | . . . . . . 7 |
20 | simpr 109 | . . . . . . . . 9 | |
21 | 20 | breq2d 4001 | . . . . . . . 8 |
22 | rexr 7965 | . . . . . . . . . . 11 | |
23 | nltmnf 9745 | . . . . . . . . . . 11 | |
24 | 22, 23 | syl 14 | . . . . . . . . . 10 |
25 | 24 | adantr 274 | . . . . . . . . 9 |
26 | 25 | pm2.21d 614 | . . . . . . . 8 |
27 | 21, 26 | sylbid 149 | . . . . . . 7 |
28 | 8, 19, 27 | 3jaodan 1301 | . . . . . 6 |
29 | 2, 28 | sylan2b 285 | . . . . 5 |
30 | 29 | expimpd 361 | . . . 4 |
31 | simpl 108 | . . . . . . 7 | |
32 | 31 | breq1d 3999 | . . . . . 6 |
33 | pnfnlt 9744 | . . . . . . . 8 | |
34 | 33 | adantl 275 | . . . . . . 7 |
35 | 34 | pm2.21d 614 | . . . . . 6 |
36 | 32, 35 | sylbid 149 | . . . . 5 |
37 | 36 | expimpd 361 | . . . 4 |
38 | breq1 3992 | . . . . . 6 | |
39 | 38 | anbi2d 461 | . . . . 5 |
40 | renegcl 8180 | . . . . . . . . . . 11 | |
41 | 4, 40 | eqeltrd 2247 | . . . . . . . . . 10 |
42 | 41 | adantr 274 | . . . . . . . . 9 |
43 | ltpnf 9737 | . . . . . . . . 9 | |
44 | 42, 43 | syl 14 | . . . . . . . 8 |
45 | 11 | adantr 274 | . . . . . . . . 9 |
46 | mnfltpnf 9742 | . . . . . . . . 9 | |
47 | 45, 46 | eqbrtrdi 4028 | . . . . . . . 8 |
48 | breq2 3993 | . . . . . . . . . 10 | |
49 | mnfxr 7976 | . . . . . . . . . . . 12 | |
50 | nltmnf 9745 | . . . . . . . . . . . 12 | |
51 | 49, 50 | ax-mp 5 | . . . . . . . . . . 11 |
52 | 51 | pm2.21i 641 | . . . . . . . . . 10 |
53 | 48, 52 | syl6bi 162 | . . . . . . . . 9 |
54 | 53 | imp 123 | . . . . . . . 8 |
55 | 44, 47, 54 | 3jaoian 1300 | . . . . . . 7 |
56 | 2, 55 | sylanb 282 | . . . . . 6 |
57 | xnegeq 9784 | . . . . . . . 8 | |
58 | xnegmnf 9786 | . . . . . . . 8 | |
59 | 57, 58 | eqtrdi 2219 | . . . . . . 7 |
60 | 59 | breq2d 4001 | . . . . . 6 |
61 | 56, 60 | syl5ibr 155 | . . . . 5 |
62 | 39, 61 | sylbid 149 | . . . 4 |
63 | 30, 37, 62 | 3jaoi 1298 | . . 3 |
64 | 1, 63 | sylbi 120 | . 2 |
65 | 64 | 3impib 1196 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 w3o 972 w3a 973 wceq 1348 wcel 2141 class class class wbr 3989 cr 7773 cpnf 7951 cmnf 7952 cxr 7953 clt 7954 cneg 8091 cxne 9726 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-sub 8092 df-neg 8093 df-xneg 9729 |
This theorem is referenced by: xltneg 9793 |
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