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Mirrors > Home > ILE Home > Th. List > ltadd1 | Unicode version |
Description: Addition to both sides of 'less than'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 12-Nov-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
ltadd1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltadd2 8174 | . 2 | |
2 | simp3 983 | . . . . 5 | |
3 | 2 | recnd 7787 | . . . 4 |
4 | simp1 981 | . . . . 5 | |
5 | 4 | recnd 7787 | . . . 4 |
6 | 3, 5 | addcomd 7906 | . . 3 |
7 | simp2 982 | . . . . 5 | |
8 | 7 | recnd 7787 | . . . 4 |
9 | 3, 8 | addcomd 7906 | . . 3 |
10 | 6, 9 | breq12d 3937 | . 2 |
11 | 1, 10 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 w3a 962 wcel 1480 class class class wbr 3924 (class class class)co 5767 cr 7612 caddc 7616 clt 7793 |
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-iota 5083 df-fv 5126 df-ov 5770 df-pnf 7795 df-mnf 7796 df-ltxr 7798 |
This theorem is referenced by: leadd1 8185 ltsubadd 8187 ltaddsub 8191 lt2add 8200 ltleadd 8201 ltadd1i 8257 ltadd1d 8293 reapadd1 8351 addltmul 8949 avglt2 8952 xltadd1 9652 icoshft 9766 fzonn0p1p1 9983 caucvgre 10746 |
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