Nearby theorems
|
|
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
| 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 8216 |
. 2
| |
| 2 | ax-rnegex 8108 |
. . . 4
   | |
| 3 | 2 | 3ad2ant3 1044 |
. . 3
|
| 4 | simpl3 1026 |
. . . . . . 7
| |
| 5 | simpl1 1024 |
. . . . . . 7
| |
| 6 | 4, 5 | readdcld 8176 |
. . . . . 6
|
| 7 | simpl2 1025 |
. . . . . . 7
| |
| 8 | 4, 7 | readdcld 8176 |
. . . . . 6
|
| 9 | simprl 529 |
. . . . . 6
| |
| 10 | axltadd 8216 |
. . . . . 6
| |
| 11 | 6, 8, 9, 10 | syl3anc 1271 |
. . . . 5
|
| 12 | 9 | recnd 8175 |
. . . . . . 7
 |
| 13 | 4 | recnd 8175 |
. . . . . . 7
|
| 14 | 5 | recnd 8175 |
. . . . . . 7
|
| 15 | 12, 13, 14 | addassd 8169 |
. . . . . 6
|
| 16 | 7 | recnd 8175 |
. . . . . . 7
|
| 17 | 12, 13, 16 | addassd 8169 |
. . . . . 6
|
| 18 | 15, 17 | breq12d 4096 |
. . . . 5
|
| 19 | 11, 18 | sylibrd 169 |
. . . 4
|
| 20 | simprr 531 |
. . . . . . . 8
| |
| 21 | addcom 8283 |
. . . . . . . . . 10
| |
| 22 | 21 | eqeq1d 2238 |
. . . . . . . . 9
|
| 23 | 13, 12, 22 | syl2anc 411 |
. . . . . . . 8
|
| 24 | 20, 23 | mpbid 147 |
. . . . . . 7
|
| 25 | 24 | oveq1d 6016 |
. . . . . 6
|
| 26 | 14 | addlidd 8296 |
. . . . . 6
|
| 27 | 25, 26 | eqtrd 2262 |
. . . . 5
|
| 28 | 24 | oveq1d 6016 |
. . . . . 6
|
| 29 | 16 | addlidd 8296 |
. . . . . 6
|
| 30 | 28, 29 | eqtrd 2262 |
. . . . 5
|
| 31 | 27, 30 | breq12d 4096 |
. . . 4
|
| 32 | 19, 31 | sylibd 149 |
. . 3
|
| 33 | 3, 32 | rexlimddv 2653 |
. 2
|
| 34 | 1, 33 | impbid 129 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 w3a 1002 wceq 1395 wcel 2200 wrex 2509 class class class wbr 4083
(class class class)co 6001 cc 7997 cr 7998 cc0 7999 caddc 8002 clt 8181 |
| 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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-addcom 8099 ax-addass 8101 ax-i2m1 8104 ax-0id 8107 ax-rnegex 8108 ax-pre-ltadd 8115 |
| This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-rab 2517 df-v 2801 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-br 4084 df-opab 4146 df-xp 4725 df-iota 5278 df-fv 5326 df-ov 6004 df-pnf 8183 df-mnf 8184 df-ltxr 8186 |
| This theorem is referenced by: ltadd2i 8567 ltadd2d 8568 ltaddneg 8571 ltadd1 8576 ltaddpos 8599 ltsub2 8606 ltaddsublt 8718 avglt1 9350 flqbi2 10511 |
| Copyright terms: Public domain | W3C validator |