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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 | |
pncand.2 |
Ref | Expression |
---|---|
negsubd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidd.1 | . 2 | |
2 | pncand.2 | . 2 | |
3 | negsub 8154 | . 2 | |
4 | 1, 2, 3 | syl2anc 409 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1348 wcel 2141 (class class class)co 5850 cc 7759 caddc 7764 cmin 8077 cneg 8078 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-setind 4519 ax-resscn 7853 ax-1cn 7854 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-addass 7863 ax-distr 7865 ax-i2m1 7866 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-sub 8079 df-neg 8080 |
This theorem is referenced by: mulsub 8307 apsub1 8548 divsubdirap 8612 divsubdivap 8632 div2subap 8741 zaddcllemneg 9238 icoshftf1o 9935 fzosubel 10137 ceiqm1l 10254 modqcyc2 10303 qnegmod 10312 modqsub12d 10324 modsumfzodifsn 10339 expaddzaplem 10506 binom2sub 10576 seq3shft 10789 cjreb 10817 recj 10818 remullem 10822 imcj 10826 resqrexlemover 10961 resqrexlemcalc1 10965 resqrexlemcalc3 10967 bdtri 11190 subcn2 11261 fsumshftm 11395 fsumsub 11402 geosergap 11456 efmival 11683 cosadd 11687 sinsub 11690 sincossq 11698 cos12dec 11717 moddvds 11748 dvdsadd2b 11789 pythagtriplem4 12209 dvmptsubcn 13438 cosq34lt1 13524 rpcxpsub 13582 rpabscxpbnd 13612 rprelogbdiv 13628 2sqlem4 13707 apdifflemr 14039 |
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