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 7968 |
. 2
| |
| 2 | ax-rnegex 7862 |
. . . 4
   | |
| 3 | 2 | 3ad2ant3 1010 |
. . 3
|
| 4 | simpl3 992 |
. . . . . . 7
| |
| 5 | simpl1 990 |
. . . . . . 7
| |
| 6 | 4, 5 | readdcld 7928 |
. . . . . 6
|
| 7 | simpl2 991 |
. . . . . . 7
| |
| 8 | 4, 7 | readdcld 7928 |
. . . . . 6
|
| 9 | simprl 521 |
. . . . . 6
| |
| 10 | axltadd 7968 |
. . . . . 6
| |
| 11 | 6, 8, 9, 10 | syl3anc 1228 |
. . . . 5
|
| 12 | 9 | recnd 7927 |
. . . . . . 7
 |
| 13 | 4 | recnd 7927 |
. . . . . . 7
|
| 14 | 5 | recnd 7927 |
. . . . . . 7
|
| 15 | 12, 13, 14 | addassd 7921 |
. . . . . 6
|
| 16 | 7 | recnd 7927 |
. . . . . . 7
|
| 17 | 12, 13, 16 | addassd 7921 |
. . . . . 6
|
| 18 | 15, 17 | breq12d 3995 |
. . . . 5
|
| 19 | 11, 18 | sylibrd 168 |
. . . 4
|
| 20 | simprr 522 |
. . . . . . . 8
| |
| 21 | addcom 8035 |
. . . . . . . . . 10
| |
| 22 | 21 | eqeq1d 2174 |
. . . . . . . . 9
|
| 23 | 13, 12, 22 | syl2anc 409 |
. . . . . . . 8
|
| 24 | 20, 23 | mpbid 146 |
. . . . . . 7
|
| 25 | 24 | oveq1d 5857 |
. . . . . 6
|
| 26 | 14 | addid2d 8048 |
. . . . . 6
|
| 27 | 25, 26 | eqtrd 2198 |
. . . . 5
|
| 28 | 24 | oveq1d 5857 |
. . . . . 6
|
| 29 | 16 | addid2d 8048 |
. . . . . 6
|
| 30 | 28, 29 | eqtrd 2198 |
. . . . 5
|
| 31 | 27, 30 | breq12d 3995 |
. . . 4
|
| 32 | 19, 31 | sylibd 148 |
. . 3
|
| 33 | 3, 32 | rexlimddv 2588 |
. 2
|
| 34 | 1, 33 | impbid 128 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 w3a 968 wceq 1343 wcel 2136 wrex 2445 class class class wbr 3982
(class class class)co 5842 cc 7751 cr 7752 cc0 7753 caddc 7756 clt 7933 |
| 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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-addcom 7853 ax-addass 7855 ax-i2m1 7858 ax-0id 7861 ax-rnegex 7862 ax-pre-ltadd 7869 |
| This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-rab 2453 df-v 2728 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-xp 4610 df-iota 5153 df-fv 5196 df-ov 5845 df-pnf 7935 df-mnf 7936 df-ltxr 7938 |
| This theorem is referenced by: ltadd2i 8318 ltadd2d 8319 ltaddneg 8322 ltadd1 8327 ltaddpos 8350 ltsub2 8357 ltaddsublt 8469 avglt1 9095 flqbi2 10226 |
| Copyright terms: Public domain | W3C validator |