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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 8235 |
. 2
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4 | 1, 2, 3 | syl2anc 411 |
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 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 ax-resscn 7933 ax-1cn 7934 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-addass 7943 ax-distr 7945 ax-i2m1 7946 ax-0id 7949 ax-rnegex 7950 ax-cnre 7952 |
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 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-sub 8160 df-neg 8161 |
This theorem is referenced by: mulsub 8388 apsub1 8629 divsubdirap 8695 divsubdivap 8715 div2subap 8824 zaddcllemneg 9322 icoshftf1o 10021 fzosubel 10224 ceiqm1l 10342 modqcyc2 10391 qnegmod 10400 modqsub12d 10412 modsumfzodifsn 10427 expaddzaplem 10594 binom2sub 10665 seq3shft 10879 cjreb 10907 recj 10908 remullem 10912 imcj 10916 resqrexlemover 11051 resqrexlemcalc1 11055 resqrexlemcalc3 11057 bdtri 11280 subcn2 11351 fsumshftm 11485 fsumsub 11492 geosergap 11546 efmival 11773 cosadd 11777 sinsub 11780 sincossq 11788 cos12dec 11807 moddvds 11838 dvdsadd2b 11879 pythagtriplem4 12300 mulgdirlem 13093 mulgmodid 13101 mulgsubdir 13102 dvmptsubcn 14645 cosq34lt1 14731 rpcxpsub 14789 rpabscxpbnd 14819 rprelogbdiv 14835 lgseisenlem1 14911 2sqlem4 14926 apdifflemr 15257 |
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