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Mirrors > Home > ILE Home > Th. List > mulneg1 | Unicode version |
Description: Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
mulneg1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 7966 |
. . . 4
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2 | subdir 8360 |
. . . 4
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3 | 1, 2 | mp3an1 1334 |
. . 3
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4 | simpr 110 |
. . . . 5
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5 | 4 | mul02d 8366 |
. . . 4
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6 | 5 | oveq1d 5905 |
. . 3
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7 | 3, 6 | eqtrd 2221 |
. 2
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8 | df-neg 8148 |
. . 3
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9 | 8 | oveq1i 5900 |
. 2
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10 | df-neg 8148 |
. 2
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11 | 7, 9, 10 | 3eqtr4g 2246 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-pow 4188 ax-pr 4223 ax-setind 4550 ax-resscn 7920 ax-1cn 7921 ax-icn 7923 ax-addcl 7924 ax-addrcl 7925 ax-mulcl 7926 ax-addcom 7928 ax-mulcom 7929 ax-addass 7930 ax-distr 7932 ax-i2m1 7933 ax-0id 7936 ax-rnegex 7937 ax-cnre 7939 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-ral 2472 df-rex 2473 df-reu 2474 df-rab 2476 df-v 2753 df-sbc 2977 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-br 4018 df-opab 4079 df-id 4307 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-iota 5192 df-fun 5232 df-fv 5238 df-riota 5846 df-ov 5893 df-oprab 5894 df-mpo 5895 df-sub 8147 df-neg 8148 |
This theorem is referenced by: mulneg2 8370 mulneg12 8371 mulm1 8374 mulneg1i 8378 mulneg1d 8385 divnegap 8680 zmulcl 9323 cjreim 10929 tanval3ap 11739 dvdsnegb 11832 odd2np1 11895 modgcd 12009 pcexp 12326 cnfldmulg 13839 sinperlem 14612 |
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