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Mirrors > Home > ILE Home > Th. List > leadd2 | Unicode version |
Description: Addition to both sides of 'less than or equal to'. (Contributed by NM, 26-Oct-1999.) |
Ref | Expression |
---|---|
leadd2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leadd1 8328 | . 2 | |
2 | simp1 987 | . . . . 5 | |
3 | 2 | recnd 7927 | . . . 4 |
4 | simp3 989 | . . . . 5 | |
5 | 4 | recnd 7927 | . . . 4 |
6 | 3, 5 | addcomd 8049 | . . 3 |
7 | simp2 988 | . . . . 5 | |
8 | 7 | recnd 7927 | . . . 4 |
9 | 8, 5 | addcomd 8049 | . . 3 |
10 | 6, 9 | breq12d 3995 | . 2 |
11 | 1, 10 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 w3a 968 wcel 2136 class class class wbr 3982 (class class class)co 5842 cr 7752 caddc 7756 cle 7934 |
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-cnv 4612 df-iota 5153 df-fv 5196 df-ov 5845 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 |
This theorem is referenced by: le2add 8342 ltleadd 8344 lesub2 8355 addge01 8370 leadd2i 8402 leadd2d 8438 expubnd 10512 bernneq 10575 faclbnd6 10657 |
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