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Mirrors > Home > ILE Home > Th. List > negsubd | Unicode version |
Description: Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
negidd.1 |
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pncand.2 |
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Ref | Expression |
---|---|
negsubd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidd.1 |
. 2
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2 | pncand.2 |
. 2
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3 | negsub 8034 |
. 2
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4 | 1, 2, 3 | syl2anc 409 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 ax-resscn 7736 ax-1cn 7737 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-sub 7959 df-neg 7960 |
This theorem is referenced by: mulsub 8187 apsub1 8428 subap0d 8430 divsubdirap 8492 divsubdivap 8512 div2subap 8620 zaddcllemneg 9117 icoshftf1o 9804 fzosubel 10002 ceiqm1l 10115 modqcyc2 10164 qnegmod 10173 modqsub12d 10185 modsumfzodifsn 10200 expaddzaplem 10367 binom2sub 10436 seq3shft 10642 cjreb 10670 recj 10671 remullem 10675 imcj 10679 resqrexlemover 10814 resqrexlemcalc1 10818 resqrexlemcalc3 10820 bdtri 11043 subcn2 11112 fsumshftm 11246 fsumsub 11253 geosergap 11307 efmival 11476 cosadd 11480 sinsub 11483 sincossq 11491 cos12dec 11510 moddvds 11538 dvdsadd2b 11576 dvmptsubcn 12893 cosq34lt1 12979 rpcxpsub 13037 rpabscxpbnd 13067 rprelogbdiv 13082 apdifflemr 13415 |
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