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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 8149 | . 2 | |
2 | simp3 968 | . . . . 5 | |
3 | 2 | recnd 7762 | . . . 4 |
4 | simp1 966 | . . . . 5 | |
5 | 4 | recnd 7762 | . . . 4 |
6 | 3, 5 | addcomd 7881 | . . 3 |
7 | simp2 967 | . . . . 5 | |
8 | 7 | recnd 7762 | . . . 4 |
9 | 3, 8 | addcomd 7881 | . . 3 |
10 | 6, 9 | breq12d 3912 | . 2 |
11 | 1, 10 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 w3a 947 wcel 1465 class class class wbr 3899 (class class class)co 5742 cr 7587 caddc 7591 clt 7768 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-iota 5058 df-fv 5101 df-ov 5745 df-pnf 7770 df-mnf 7771 df-ltxr 7773 |
This theorem is referenced by: leadd1 8160 ltsubadd 8162 ltaddsub 8166 lt2add 8175 ltleadd 8176 ltadd1i 8232 ltadd1d 8268 reapadd1 8326 addltmul 8924 avglt2 8927 xltadd1 9627 icoshft 9741 fzonn0p1p1 9958 caucvgre 10721 |
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