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Mirrors > Home > ILE Home > Th. List > negsubi | Unicode version |
Description: Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
negidi.1 | |
pncan3i.2 |
Ref | Expression |
---|---|
negsubi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidi.1 | . 2 | |
2 | pncan3i.2 | . 2 | |
3 | negsub 8179 | . 2 | |
4 | 1, 2, 3 | mp2an 426 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1353 wcel 2146 (class class class)co 5865 cc 7784 caddc 7789 cmin 8102 cneg 8103 |
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 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-setind 4530 ax-resscn 7878 ax-1cn 7879 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-addcom 7886 ax-addass 7888 ax-distr 7890 ax-i2m1 7891 ax-0id 7894 ax-rnegex 7895 ax-cnre 7897 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-iota 5170 df-fun 5210 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-sub 8104 df-neg 8105 |
This theorem is referenced by: negsubdii 8216 negsubdi2i 8217 cosq14gt0 13746 lgsdir2lem1 13922 |
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