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Mirrors > Home > ILE Home > Th. List > negsub | Unicode version |
Description: Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
negsub |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-neg 8162 |
. . . 4
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2 | 1 | oveq2i 5908 |
. . 3
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3 | 2 | a1i 9 |
. 2
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4 | 0cn 7980 |
. . 3
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5 | addsubass 8198 |
. . 3
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6 | 4, 5 | mp3an2 1336 |
. 2
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7 | simpl 109 |
. . . 4
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8 | 7 | addridd 8137 |
. . 3
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9 | 8 | oveq1d 5912 |
. 2
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10 | 3, 6, 9 | 3eqtr2d 2228 |
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 7934 ax-1cn 7935 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-addass 7944 ax-distr 7946 ax-i2m1 7947 ax-0id 7950 ax-rnegex 7951 ax-cnre 7953 |
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 5900 df-oprab 5901 df-mpo 5902 df-sub 8161 df-neg 8162 |
This theorem is referenced by: negdi2 8246 negsubdi2 8247 resubcli 8251 resubcl 8252 negsubi 8266 negsubd 8305 submul2 8387 mulsub 8389 subap0 8631 divsubdirap 8696 zsubcl 9325 difgtsumgt 9353 elz2 9355 qsubcl 9670 rexsub 9885 fzsubel 10092 expsubap 10602 binom2sub 10668 resub 10914 imsub 10922 cjsub 10936 cjreim 10947 absdiflt 11136 absdifle 11137 abs2dif2 11151 subcn2 11354 efsub 11724 efi4p 11760 sinsub 11783 cossub 11784 demoivreALT 11816 dvdssub 11880 modgcd 12027 gzsubcl 12415 cnfldsub 13895 wilthlem1 14875 lgsvalmod 14898 |
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