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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 8017 |
. 2
| |
| 2 | ax-rnegex 7911 |
. . . 4
   | |
| 3 | 2 | 3ad2ant3 1020 |
. . 3
|
| 4 | simpl3 1002 |
. . . . . . 7
| |
| 5 | simpl1 1000 |
. . . . . . 7
| |
| 6 | 4, 5 | readdcld 7977 |
. . . . . 6
|
| 7 | simpl2 1001 |
. . . . . . 7
| |
| 8 | 4, 7 | readdcld 7977 |
. . . . . 6
|
| 9 | simprl 529 |
. . . . . 6
| |
| 10 | axltadd 8017 |
. . . . . 6
| |
| 11 | 6, 8, 9, 10 | syl3anc 1238 |
. . . . 5
|
| 12 | 9 | recnd 7976 |
. . . . . . 7
 |
| 13 | 4 | recnd 7976 |
. . . . . . 7
|
| 14 | 5 | recnd 7976 |
. . . . . . 7
|
| 15 | 12, 13, 14 | addassd 7970 |
. . . . . 6
|
| 16 | 7 | recnd 7976 |
. . . . . . 7
|
| 17 | 12, 13, 16 | addassd 7970 |
. . . . . 6
|
| 18 | 15, 17 | breq12d 4013 |
. . . . 5
|
| 19 | 11, 18 | sylibrd 169 |
. . . 4
|
| 20 | simprr 531 |
. . . . . . . 8
| |
| 21 | addcom 8084 |
. . . . . . . . . 10
| |
| 22 | 21 | eqeq1d 2186 |
. . . . . . . . 9
|
| 23 | 13, 12, 22 | syl2anc 411 |
. . . . . . . 8
|
| 24 | 20, 23 | mpbid 147 |
. . . . . . 7
|
| 25 | 24 | oveq1d 5884 |
. . . . . 6
|
| 26 | 14 | addid2d 8097 |
. . . . . 6
|
| 27 | 25, 26 | eqtrd 2210 |
. . . . 5
|
| 28 | 24 | oveq1d 5884 |
. . . . . 6
|
| 29 | 16 | addid2d 8097 |
. . . . . 6
|
| 30 | 28, 29 | eqtrd 2210 |
. . . . 5
|
| 31 | 27, 30 | breq12d 4013 |
. . . 4
|
| 32 | 19, 31 | sylibd 149 |
. . 3
|
| 33 | 3, 32 | rexlimddv 2599 |
. 2
|
| 34 | 1, 33 | impbid 129 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 w3a 978 wceq 1353 wcel 2148 wrex 2456 class class class wbr 4000
(class class class)co 5869 cc 7800 cr 7801 cc0 7802 caddc 7805 clt 7982 |
| 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 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-addcom 7902 ax-addass 7904 ax-i2m1 7907 ax-0id 7910 ax-rnegex 7911 ax-pre-ltadd 7918 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-xp 4629 df-iota 5174 df-fv 5220 df-ov 5872 df-pnf 7984 df-mnf 7985 df-ltxr 7987 |
| This theorem is referenced by: ltadd2i 8367 ltadd2d 8368 ltaddneg 8371 ltadd1 8376 ltaddpos 8399 ltsub2 8406 ltaddsublt 8518 avglt1 9146 flqbi2 10277 |
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