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Mirrors > Home > ILE Home > Th. List > divnegap | Unicode version |
Description: Move negative sign inside of a division. (Contributed by Jim Kingdon, 25-Feb-2020.) |
Ref | Expression |
---|---|
divnegap |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recclap 7870 |
. . . 4
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2 | mulneg1 7602 |
. . . 4
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3 | 1, 2 | sylan2 280 |
. . 3
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4 | 3 | 3impb 1135 |
. 2
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5 | negcl 7411 |
. . 3
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6 | divrecap 7879 |
. . 3
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7 | 5, 6 | syl3an1 1203 |
. 2
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8 | divrecap 7879 |
. . 3
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9 | 8 | negeqd 7406 |
. 2
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10 | 4, 7, 9 | 3eqtr4rd 2126 |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3917 ax-pow 3969 ax-pr 3993 ax-un 4217 ax-setind 4309 ax-cnex 7165 ax-resscn 7166 ax-1cn 7167 ax-1re 7168 ax-icn 7169 ax-addcl 7170 ax-addrcl 7171 ax-mulcl 7172 ax-mulrcl 7173 ax-addcom 7174 ax-mulcom 7175 ax-addass 7176 ax-mulass 7177 ax-distr 7178 ax-i2m1 7179 ax-0lt1 7180 ax-1rid 7181 ax-0id 7182 ax-rnegex 7183 ax-precex 7184 ax-cnre 7185 ax-pre-ltirr 7186 ax-pre-ltwlin 7187 ax-pre-lttrn 7188 ax-pre-apti 7189 ax-pre-ltadd 7190 ax-pre-mulgt0 7191 ax-pre-mulext 7192 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2612 df-sbc 2826 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-br 3807 df-opab 3861 df-id 4077 df-po 4080 df-iso 4081 df-xp 4398 df-rel 4399 df-cnv 4400 df-co 4401 df-dm 4402 df-iota 4918 df-fun 4955 df-fv 4961 df-riota 5520 df-ov 5567 df-oprab 5568 df-mpt2 5569 df-pnf 7253 df-mnf 7254 df-xr 7255 df-ltxr 7256 df-le 7257 df-sub 7384 df-neg 7385 df-reap 7778 df-ap 7785 df-div 7864 |
This theorem is referenced by: divsubdirap 7899 divsubdivap 7919 div2negap 7926 divneg2ap 7927 divnegapd 7993 zeo 8569 |
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