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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 7719 |
. . . 4
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2 | 1 | oveq2i 5679 |
. . 3
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3 | 2 | a1i 9 |
. 2
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4 | 0cn 7543 |
. . 3
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5 | addsubass 7755 |
. . 3
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6 | 4, 5 | mp3an2 1262 |
. 2
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7 | simpl 108 |
. . . 4
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8 | 7 | addid1d 7694 |
. . 3
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9 | 8 | oveq1d 5683 |
. 2
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10 | 3, 6, 9 | 3eqtr2d 2127 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-pow 4017 ax-pr 4047 ax-setind 4368 ax-resscn 7500 ax-1cn 7501 ax-icn 7503 ax-addcl 7504 ax-addrcl 7505 ax-mulcl 7506 ax-addcom 7508 ax-addass 7510 ax-distr 7512 ax-i2m1 7513 ax-0id 7516 ax-rnegex 7517 ax-cnre 7519 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2624 df-sbc 2844 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-br 3854 df-opab 3908 df-id 4131 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-iota 4995 df-fun 5032 df-fv 5038 df-riota 5624 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-sub 7718 df-neg 7719 |
This theorem is referenced by: negdi2 7803 negsubdi2 7804 resubcli 7808 resubcl 7809 negsubi 7823 negsubd 7862 submul2 7940 mulsub 7942 subap0 8181 divsubdirap 8238 zsubcl 8854 elz2 8881 qsubcl 9186 fzsubel 9537 expsubap 10066 binom2sub 10130 resub 10367 imsub 10375 cjsub 10389 cjreim 10400 absdiflt 10588 absdifle 10589 abs2dif2 10603 subcn2 10763 efsub 11034 efi4p 11071 sinsub 11094 cossub 11095 demoivreALT 11126 dvdssub 11182 modgcd 11323 |
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