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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 7710 |
. . . 4
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2 | 1 | oveq2i 5677 |
. . 3
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3 | 2 | a1i 9 |
. 2
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4 | 0cn 7534 |
. . 3
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5 | addsubass 7746 |
. . 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 7685 |
. . 3
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9 | 8 | oveq1d 5681 |
. 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 3963 ax-pow 4015 ax-pr 4045 ax-setind 4366 ax-resscn 7491 ax-1cn 7492 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-addcom 7499 ax-addass 7501 ax-distr 7503 ax-i2m1 7504 ax-0id 7507 ax-rnegex 7508 ax-cnre 7510 |
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 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-id 4129 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-sub 7709 df-neg 7710 |
This theorem is referenced by: negdi2 7794 negsubdi2 7795 resubcli 7799 resubcl 7800 negsubi 7814 negsubd 7853 submul2 7931 mulsub 7933 subap0 8172 divsubdirap 8229 zsubcl 8845 elz2 8872 qsubcl 9177 fzsubel 9528 expsubap 10057 binom2sub 10121 resub 10358 imsub 10366 cjsub 10380 cjreim 10391 absdiflt 10579 absdifle 10580 abs2dif2 10594 subcn2 10754 efsub 11025 efi4p 11062 sinsub 11085 cossub 11086 demoivreALT 11117 dvdssub 11173 modgcd 11314 |
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