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Mirrors > Home > ILE Home > Th. List > subcl | Unicode version |
Description: Closure law for subtraction. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 21-Dec-2013.) |
Ref | Expression |
---|---|
subcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subval 8147 |
. 2
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2 | negeu 8146 |
. . . 4
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3 | 2 | ancoms 268 |
. . 3
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4 | riotacl 5844 |
. . 3
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5 | 3, 4 | syl 14 |
. 2
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6 | 1, 5 | eqeltrd 2254 |
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-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-setind 4536 ax-resscn 7902 ax-1cn 7903 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 |
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-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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-sub 8128 |
This theorem is referenced by: negcl 8155 subf 8157 pncan3 8163 npcan 8164 addsubass 8165 addsub 8166 addsub12 8168 addsubeq4 8170 npncan 8176 nppcan 8177 nnpcan 8178 nppcan3 8179 subcan2 8180 subsub2 8183 subsub4 8188 nnncan 8190 nnncan1 8191 nnncan2 8192 npncan3 8193 addsub4 8198 subadd4 8199 peano2cnm 8221 subcli 8231 subcld 8266 subeqrev 8331 subdi 8340 subdir 8341 mulsub2 8357 recextlem1 8606 recexap 8608 div2subap 8792 cju 8916 halfaddsubcl 9150 halfaddsub 9151 iccf1o 10002 ser3sub 10503 sqsubswap 10577 subsq 10623 subsq2 10624 bcn2 10739 shftval2 10830 2shfti 10835 sqabssub 11060 abssub 11105 abs3dif 11109 abs2dif 11110 abs2difabs 11112 climuni 11296 cjcn2 11319 recn2 11320 imcn2 11321 climsub 11331 fisum0diag2 11450 arisum2 11502 geosergap 11509 geolim 11514 geolim2 11515 georeclim 11516 geo2sum 11517 tanaddap 11742 addsin 11745 fzocongeq 11858 odd2np1 11872 phiprm 12217 pythagtriplem4 12262 pythagtriplem12 12269 pythagtriplem14 12271 fldivp1 12340 cnmet 13961 dveflem 14118 dvef 14119 efimpi 14171 ptolemy 14176 tangtx 14190 abssinper 14198 |
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