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Mirrors > Home > ILE Home > Th. List > lesubadd | Unicode version |
Description: 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
lesubadd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 992 | . . . 4 | |
2 | simp2 993 | . . . 4 | |
3 | 1, 2 | resubcld 8289 | . . 3 |
4 | simp3 994 | . . 3 | |
5 | leadd1 8338 | . . 3 | |
6 | 3, 4, 2, 5 | syl3anc 1233 | . 2 |
7 | 1 | recnd 7937 | . . . 4 |
8 | 2 | recnd 7937 | . . . 4 |
9 | 7, 8 | npcand 8223 | . . 3 |
10 | 9 | breq1d 3997 | . 2 |
11 | 6, 10 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 w3a 973 wcel 2141 class class class wbr 3987 (class class class)co 5851 cr 7762 caddc 7766 cle 7944 cmin 8079 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7854 ax-resscn 7855 ax-1cn 7856 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-addcom 7863 ax-addass 7865 ax-distr 7867 ax-i2m1 7868 ax-0id 7871 ax-rnegex 7872 ax-cnre 7874 ax-pre-ltadd 7879 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7945 df-mnf 7946 df-xr 7947 df-ltxr 7948 df-le 7949 df-sub 8081 df-neg 8082 |
This theorem is referenced by: lesubadd2 8343 suble 8348 lesub1 8364 lesub2 8365 subge02 8386 lesubaddi 8414 lesubaddd 8450 fzen 9988 abs2dif 11059 |
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