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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltadd2 8100 |
. 2
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2 | simp3 966 |
. . . . 5
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3 | 2 | recnd 7718 |
. . . 4
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4 | simp1 964 |
. . . . 5
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5 | 4 | recnd 7718 |
. . . 4
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6 | 3, 5 | addcomd 7836 |
. . 3
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7 | simp2 965 |
. . . . 5
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8 | 7 | recnd 7718 |
. . . 4
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9 | 3, 8 | addcomd 7836 |
. . 3
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10 | 6, 9 | breq12d 3908 |
. 2
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11 | 1, 10 | bitrd 187 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-1cn 7638 ax-icn 7640 ax-addcl 7641 ax-addrcl 7642 ax-mulcl 7643 ax-addcom 7645 ax-addass 7647 ax-i2m1 7650 ax-0id 7653 ax-rnegex 7654 ax-pre-ltadd 7661 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-rab 2399 df-v 2659 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-xp 4505 df-iota 5046 df-fv 5089 df-ov 5731 df-pnf 7726 df-mnf 7727 df-ltxr 7729 |
This theorem is referenced by: leadd1 8111 ltsubadd 8113 ltaddsub 8117 lt2add 8126 ltleadd 8127 ltadd1i 8183 ltadd1d 8218 reapadd1 8276 addltmul 8860 avglt2 8863 xltadd1 9552 icoshft 9666 fzonn0p1p1 9883 caucvgre 10645 |
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