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Mirrors > Home > ILE Home > Th. List > subneg | Unicode version |
Description: Relationship between subtraction and negative. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
subneg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-neg 8166 |
. . . 4
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2 | 1 | oveq2i 5911 |
. . 3
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3 | 0cn 7984 |
. . . 4
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4 | subsub 8222 |
. . . 4
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5 | 3, 4 | mp3an2 1336 |
. . 3
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6 | 2, 5 | eqtrid 2234 |
. 2
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7 | subid1 8212 |
. . . 4
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8 | 7 | adantr 276 |
. . 3
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9 | 8 | oveq1d 5915 |
. 2
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10 | 6, 9 | eqtrd 2222 |
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 4139 ax-pow 4195 ax-pr 4230 ax-setind 4557 ax-resscn 7938 ax-1cn 7939 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-addcom 7946 ax-addass 7948 ax-distr 7950 ax-i2m1 7951 ax-0id 7954 ax-rnegex 7955 ax-cnre 7957 |
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 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-br 4022 df-opab 4083 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-iota 5199 df-fun 5240 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-sub 8165 df-neg 8166 |
This theorem is referenced by: negneg 8242 negdi 8249 neg2sub 8252 subnegi 8271 subnegd 8310 recextlem1 8643 fzshftral 10144 shftval4 10878 fsumshftm 11494 eftlub 11739 summodnegmod 11870 wilthlem1 14883 |
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