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Mirrors > Home > ILE Home > Th. List > resubcl | Unicode version |
Description: Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.) |
Ref | Expression |
---|---|
resubcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 7979 |
. . 3
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2 | recn 7979 |
. . 3
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3 | negsub 8240 |
. . 3
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4 | 1, 2, 3 | syl2an 289 |
. 2
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5 | renegcl 8253 |
. . 3
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6 | readdcl 7972 |
. . 3
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7 | 5, 6 | sylan2 286 |
. 2
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8 | 4, 7 | eqeltrrd 2267 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-setind 4557 ax-resscn 7938 ax-1cn 7939 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-addcom 7946 ax-addass 7948 ax-distr 7950 ax-i2m1 7951 ax-0id 7954 ax-rnegex 7955 ax-cnre 7957 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-br 4022 df-opab 4083 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-iota 5199 df-fun 5240 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-sub 8165 df-neg 8166 |
This theorem is referenced by: peano2rem 8259 resubcld 8373 posdif 8447 lt2sub 8452 le2sub 8453 cju 8953 elz2 9359 difrp 9728 iooshf 9988 iccshftl 10032 lincmb01cmp 10039 uzsubsubfz 10083 difelfzle 10170 fzonmapblen 10223 eluzgtdifelfzo 10233 subfzo0 10278 modfzo0difsn 10432 expubnd 10617 absdiflt 11142 absdifle 11143 elicc4abs 11144 abssubge0 11152 abs2difabs 11158 maxabsle 11254 resin4p 11767 recos4p 11768 cos01bnd 11807 cos01gt0 11811 pythagtriplem12 12318 pythagtriplem14 12320 pythagtriplem16 12322 fldivp1 12391 bl2ioo 14527 ioo2bl 14528 ioo2blex 14529 blssioo 14530 sincosq1sgn 14732 sincosq2sgn 14733 sincosq3sgn 14734 sincosq4sgn 14735 sinq12gt0 14736 cosq14gt0 14738 tangtx 14744 relogdiv 14776 logdivlti 14787 redc0 15293 reap0 15294 |
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