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| Mirrors > Home > ILE Home > Th. List > ltadd2 | Unicode version | ||
| Description: Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
| Ref | Expression |
|---|---|
| ltadd2 |
     ![/\](wedge.gif "/\"){kind=small} 
    
|
is distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | axltadd 7858 |
. 2
| |
| 2 | ax-rnegex 7753 |
. . . 4
   | |
| 3 | 2 | 3ad2ant3 1005 |
. . 3
|
| 4 | simpl3 987 |
. . . . . . 7
| |
| 5 | simpl1 985 |
. . . . . . 7
| |
| 6 | 4, 5 | readdcld 7819 |
. . . . . 6
|
| 7 | simpl2 986 |
. . . . . . 7
| |
| 8 | 4, 7 | readdcld 7819 |
. . . . . 6
|
| 9 | simprl 521 |
. . . . . 6
| |
| 10 | axltadd 7858 |
. . . . . 6
| |
| 11 | 6, 8, 9, 10 | syl3anc 1217 |
. . . . 5
|
| 12 | 9 | recnd 7818 |
. . . . . . 7
 |
| 13 | 4 | recnd 7818 |
. . . . . . 7
|
| 14 | 5 | recnd 7818 |
. . . . . . 7
|
| 15 | 12, 13, 14 | addassd 7812 |
. . . . . 6
|
| 16 | 7 | recnd 7818 |
. . . . . . 7
|
| 17 | 12, 13, 16 | addassd 7812 |
. . . . . 6
|
| 18 | 15, 17 | breq12d 3950 |
. . . . 5
|
| 19 | 11, 18 | sylibrd 168 |
. . . 4
|
| 20 | simprr 522 |
. . . . . . . 8
| |
| 21 | addcom 7923 |
. . . . . . . . . 10
| |
| 22 | 21 | eqeq1d 2149 |
. . . . . . . . 9
|
| 23 | 13, 12, 22 | syl2anc 409 |
. . . . . . . 8
|
| 24 | 20, 23 | mpbid 146 |
. . . . . . 7
|
| 25 | 24 | oveq1d 5797 |
. . . . . 6
|
| 26 | 14 | addid2d 7936 |
. . . . . 6
|
| 27 | 25, 26 | eqtrd 2173 |
. . . . 5
|
| 28 | 24 | oveq1d 5797 |
. . . . . 6
|
| 29 | 16 | addid2d 7936 |
. . . . . 6
|
| 30 | 28, 29 | eqtrd 2173 |
. . . . 5
|
| 31 | 27, 30 | breq12d 3950 |
. . . 4
|
| 32 | 19, 31 | sylibd 148 |
. . 3
|
| 33 | 3, 32 | rexlimddv 2557 |
. 2
|
| 34 | 1, 33 | impbid 128 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 w3a 963 wceq 1332 wcel 1481 wrex 2418 class class class wbr 3937
(class class class)co 5782 cc 7642 cr 7643 cc0 7644 caddc 7647 clt 7824 |
| 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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-i2m1 7749 ax-0id 7752 ax-rnegex 7753 ax-pre-ltadd 7760 |
| This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-iota 5096 df-fv 5139 df-ov 5785 df-pnf 7826 df-mnf 7827 df-ltxr 7829 |
| This theorem is referenced by: ltadd2i 8206 ltadd2d 8207 ltaddneg 8210 ltadd1 8215 ltaddpos 8238 ltsub2 8245 ltaddsublt 8357 avglt1 8982 flqbi2 10095 |
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