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Mirrors > Home > ILE Home > Th. List > xlesubadd | Unicode version |
Description: Under certain conditions, the conclusion of lesubadd 8323 is true even in the extended reals. (Contributed by Mario Carneiro, 4-Sep-2015.) |
Ref | Expression |
---|---|
xlesubadd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 989 | . . . . . 6 | |
2 | simpl2 990 | . . . . . . 7 | |
3 | 2 | xnegcld 9782 | . . . . . 6 |
4 | xaddcl 9787 | . . . . . 6 | |
5 | 1, 3, 4 | syl2anc 409 | . . . . 5 |
6 | 5 | adantr 274 | . . . 4 |
7 | simpll3 1027 | . . . 4 | |
8 | simpr 109 | . . . 4 | |
9 | xleadd1 9802 | . . . 4 | |
10 | 6, 7, 8, 9 | syl3anc 1227 | . . 3 |
11 | xnpcan 9799 | . . . . 5 | |
12 | 1, 11 | sylan 281 | . . . 4 |
13 | 12 | breq1d 3986 | . . 3 |
14 | 10, 13 | bitrd 187 | . 2 |
15 | simpr3 994 | . . . . . . 7 | |
16 | oveq1 5843 | . . . . . . . . 9 | |
17 | pnfaddmnf 9777 | . . . . . . . . 9 | |
18 | 16, 17 | eqtrdi 2213 | . . . . . . . 8 |
19 | 18 | breq1d 3986 | . . . . . . 7 |
20 | 15, 19 | syl5ibrcom 156 | . . . . . 6 |
21 | xaddmnf1 9775 | . . . . . . . . 9 | |
22 | 21 | ex 114 | . . . . . . . 8 |
23 | 1, 22 | syl 14 | . . . . . . 7 |
24 | simpl3 991 | . . . . . . . . 9 | |
25 | mnfle 9719 | . . . . . . . . 9 | |
26 | 24, 25 | syl 14 | . . . . . . . 8 |
27 | breq1 3979 | . . . . . . . 8 | |
28 | 26, 27 | syl5ibrcom 156 | . . . . . . 7 |
29 | 23, 28 | syld 45 | . . . . . 6 |
30 | xrpnfdc 9769 | . . . . . . . 8 DECID | |
31 | dcne 2345 | . . . . . . . 8 DECID | |
32 | 30, 31 | sylib 121 | . . . . . . 7 |
33 | 1, 32 | syl 14 | . . . . . 6 |
34 | 20, 29, 33 | mpjaod 708 | . . . . 5 |
35 | pnfge 9716 | . . . . . . 7 | |
36 | 1, 35 | syl 14 | . . . . . 6 |
37 | ge0nemnf 9751 | . . . . . . . 8 | |
38 | 24, 15, 37 | syl2anc 409 | . . . . . . 7 |
39 | xaddpnf1 9773 | . . . . . . 7 | |
40 | 24, 38, 39 | syl2anc 409 | . . . . . 6 |
41 | 36, 40 | breqtrrd 4004 | . . . . 5 |
42 | 34, 41 | 2thd 174 | . . . 4 |
43 | xnegeq 9754 | . . . . . . . 8 | |
44 | xnegpnf 9755 | . . . . . . . 8 | |
45 | 43, 44 | eqtrdi 2213 | . . . . . . 7 |
46 | 45 | oveq2d 5852 | . . . . . 6 |
47 | 46 | breq1d 3986 | . . . . 5 |
48 | oveq2 5844 | . . . . . 6 | |
49 | 48 | breq2d 3988 | . . . . 5 |
50 | 47, 49 | bibi12d 234 | . . . 4 |
51 | 42, 50 | syl5ibrcom 156 | . . 3 |
52 | 51 | imp 123 | . 2 |
53 | simpr2 993 | . . . 4 | |
54 | 2, 53 | jca 304 | . . 3 |
55 | xrnemnf 9704 | . . 3 | |
56 | 54, 55 | sylib 121 | . 2 |
57 | 14, 52, 56 | mpjaodan 788 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 698 DECID wdc 824 w3a 967 wceq 1342 wcel 2135 wne 2334 class class class wbr 3976 (class class class)co 5836 cr 7743 cc0 7744 cpnf 7921 cmnf 7922 cxr 7923 cle 7925 cxne 9696 cxad 9697 |
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 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 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 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-xneg 9699 df-xadd 9700 |
This theorem is referenced by: xmetrtri 12923 |
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