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Mirrors > Home > ILE Home > Th. List > ltsubadd2 | Unicode version |
Description: 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) |
Ref | Expression |
---|---|
ltsubadd2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltsubadd 8302 | . 2 | |
2 | simp2 983 | . . . . 5 | |
3 | 2 | recnd 7901 | . . . 4 |
4 | simp3 984 | . . . . 5 | |
5 | 4 | recnd 7901 | . . . 4 |
6 | 3, 5 | addcomd 8021 | . . 3 |
7 | 6 | breq2d 3977 | . 2 |
8 | 1, 7 | bitr4d 190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 w3a 963 wcel 2128 class class class wbr 3965 (class class class)co 5821 cr 7726 caddc 7730 clt 7907 cmin 8041 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-cnex 7818 ax-resscn 7819 ax-1cn 7820 ax-icn 7822 ax-addcl 7823 ax-addrcl 7824 ax-mulcl 7825 ax-addcom 7827 ax-addass 7829 ax-distr 7831 ax-i2m1 7832 ax-0id 7835 ax-rnegex 7836 ax-cnre 7838 ax-pre-ltadd 7843 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-iota 5134 df-fun 5171 df-fv 5177 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-pnf 7909 df-mnf 7910 df-ltxr 7912 df-sub 8043 df-neg 8044 |
This theorem is referenced by: ltsub23 8312 ltsubadd2i 8377 ltsubadd2d 8413 absdiflt 10987 caubnd2 11012 |
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