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Mirrors > Home > ILE Home > Th. List > xlesubadd | Unicode version |
Description: Under certain conditions, the conclusion of lesubadd 8164 is true even in the extended reals. (Contributed by Mario Carneiro, 4-Sep-2015.) |
Ref | Expression |
---|---|
xlesubadd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 969 | . . . . . 6 | |
2 | simpl2 970 | . . . . . . 7 | |
3 | 2 | xnegcld 9606 | . . . . . 6 |
4 | xaddcl 9611 | . . . . . 6 | |
5 | 1, 3, 4 | syl2anc 408 | . . . . 5 |
6 | 5 | adantr 274 | . . . 4 |
7 | simpll3 1007 | . . . 4 | |
8 | simpr 109 | . . . 4 | |
9 | xleadd1 9626 | . . . 4 | |
10 | 6, 7, 8, 9 | syl3anc 1201 | . . 3 |
11 | xnpcan 9623 | . . . . 5 | |
12 | 1, 11 | sylan 281 | . . . 4 |
13 | 12 | breq1d 3909 | . . 3 |
14 | 10, 13 | bitrd 187 | . 2 |
15 | simpr3 974 | . . . . . . 7 | |
16 | oveq1 5749 | . . . . . . . . 9 | |
17 | pnfaddmnf 9601 | . . . . . . . . 9 | |
18 | 16, 17 | syl6eq 2166 | . . . . . . . 8 |
19 | 18 | breq1d 3909 | . . . . . . 7 |
20 | 15, 19 | syl5ibrcom 156 | . . . . . 6 |
21 | xaddmnf1 9599 | . . . . . . . . 9 | |
22 | 21 | ex 114 | . . . . . . . 8 |
23 | 1, 22 | syl 14 | . . . . . . 7 |
24 | simpl3 971 | . . . . . . . . 9 | |
25 | mnfle 9546 | . . . . . . . . 9 | |
26 | 24, 25 | syl 14 | . . . . . . . 8 |
27 | breq1 3902 | . . . . . . . 8 | |
28 | 26, 27 | syl5ibrcom 156 | . . . . . . 7 |
29 | 23, 28 | syld 45 | . . . . . 6 |
30 | xrpnfdc 9593 | . . . . . . . 8 DECID | |
31 | dcne 2296 | . . . . . . . 8 DECID | |
32 | 30, 31 | sylib 121 | . . . . . . 7 |
33 | 1, 32 | syl 14 | . . . . . 6 |
34 | 20, 29, 33 | mpjaod 692 | . . . . 5 |
35 | pnfge 9543 | . . . . . . 7 | |
36 | 1, 35 | syl 14 | . . . . . 6 |
37 | ge0nemnf 9575 | . . . . . . . 8 | |
38 | 24, 15, 37 | syl2anc 408 | . . . . . . 7 |
39 | xaddpnf1 9597 | . . . . . . 7 | |
40 | 24, 38, 39 | syl2anc 408 | . . . . . 6 |
41 | 36, 40 | breqtrrd 3926 | . . . . 5 |
42 | 34, 41 | 2thd 174 | . . . 4 |
43 | xnegeq 9578 | . . . . . . . 8 | |
44 | xnegpnf 9579 | . . . . . . . 8 | |
45 | 43, 44 | syl6eq 2166 | . . . . . . 7 |
46 | 45 | oveq2d 5758 | . . . . . 6 |
47 | 46 | breq1d 3909 | . . . . 5 |
48 | oveq2 5750 | . . . . . 6 | |
49 | 48 | breq2d 3911 | . . . . 5 |
50 | 47, 49 | bibi12d 234 | . . . 4 |
51 | 42, 50 | syl5ibrcom 156 | . . 3 |
52 | 51 | imp 123 | . 2 |
53 | simpr2 973 | . . . 4 | |
54 | 2, 53 | jca 304 | . . 3 |
55 | xrnemnf 9532 | . . 3 | |
56 | 54, 55 | sylib 121 | . 2 |
57 | 14, 52, 56 | mpjaodan 772 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 682 DECID wdc 804 w3a 947 wceq 1316 wcel 1465 wne 2285 class class class wbr 3899 (class class class)co 5742 cr 7587 cc0 7588 cpnf 7765 cmnf 7766 cxr 7767 cle 7769 cxne 9524 cxad 9525 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-xneg 9527 df-xadd 9528 |
This theorem is referenced by: xmetrtri 12472 |
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