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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 8010 | . 2 | |
4 | 1, 2, 3 | syl2anc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 wcel 1480 (class class class)co 5774 cc 7618 caddc 7623 cmin 7933 cneg 7934 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-resscn 7712 ax-1cn 7713 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7935 df-neg 7936 |
This theorem is referenced by: mulsub 8163 apsub1 8404 subap0d 8406 divsubdirap 8468 divsubdivap 8488 div2subap 8596 zaddcllemneg 9093 icoshftf1o 9774 fzosubel 9971 ceiqm1l 10084 modqcyc2 10133 qnegmod 10142 modqsub12d 10154 modsumfzodifsn 10169 expaddzaplem 10336 binom2sub 10405 seq3shft 10610 cjreb 10638 recj 10639 remullem 10643 imcj 10647 resqrexlemover 10782 resqrexlemcalc1 10786 resqrexlemcalc3 10788 bdtri 11011 subcn2 11080 fsumshftm 11214 fsumsub 11221 geosergap 11275 efmival 11440 cosadd 11444 sinsub 11447 sincossq 11455 cos12dec 11474 moddvds 11502 dvdsadd2b 11540 dvmptsubcn 12854 cosq34lt1 12931 |
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