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Mirrors > Home > ILE Home > Th. List > leaddsub | Unicode version |
Description: 'Less than or equal to' relationship between addition and subtraction. (Contributed by NM, 6-Apr-2005.) |
Ref | Expression |
---|---|
leaddsub |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 997 |
. . 3
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2 | simp3 999 |
. . . 4
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3 | simp2 998 |
. . . 4
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4 | 2, 3 | resubcld 8315 |
. . 3
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5 | leadd1 8364 |
. . 3
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6 | 1, 4, 3, 5 | syl3anc 1238 |
. 2
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7 | 2 | recnd 7963 |
. . . 4
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8 | 3 | recnd 7963 |
. . . 4
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9 | 7, 8 | npcand 8249 |
. . 3
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10 | 9 | breq2d 4012 |
. 2
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11 | 6, 10 | bitr2d 189 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-distr 7893 ax-i2m1 7894 ax-0id 7897 ax-rnegex 7898 ax-cnre 7900 ax-pre-ltadd 7905 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-iota 5173 df-fun 5213 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 |
This theorem is referenced by: leaddsub2 8373 lesub 8375 lesub2 8391 subge0 8409 div4p1lem1div2 9148 eluzp1m1 9527 eluzsubi 9531 eluzsub 9533 fzen 10016 fznatpl1 10049 seq3f1olemqsumkj 10471 bcval5 10714 uzwodc 12008 hashdvds 12191 |
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