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Mirrors > Home > ILE Home > Th. List > leadd1 | Unicode version |
Description: Addition to both sides of 'less than or equal to'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 18-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
leadd1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltadd1 8184 | . . . 4 | |
2 | 1 | 3com12 1185 | . . 3 |
3 | 2 | notbid 656 | . 2 |
4 | simp1 981 | . . 3 | |
5 | simp2 982 | . . 3 | |
6 | 4, 5 | lenltd 7873 | . 2 |
7 | simp3 983 | . . . 4 | |
8 | 4, 7 | readdcld 7788 | . . 3 |
9 | 5, 7 | readdcld 7788 | . . 3 |
10 | 8, 9 | lenltd 7873 | . 2 |
11 | 3, 6, 10 | 3bitr4d 219 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 104 w3a 962 wcel 1480 class class class wbr 3924 (class class class)co 5767 cr 7612 caddc 7616 clt 7793 cle 7794 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-cnv 4542 df-iota 5083 df-fv 5126 df-ov 5770 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 |
This theorem is referenced by: leadd2 8186 lesubadd 8189 leaddsub 8193 le2add 8199 leadd1i 8258 leadd1d 8294 zleltp1 9102 eluzp1p1 9344 eluzaddi 9345 icoshft 9766 iccshftr 9770 fzen 9816 fzaddel 9832 fznatpl1 9849 fldiv4p1lem1div2 10071 faclbnd6 10483 |
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