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Mirrors > Home > ILE Home > Th. List > divcanap1d | Unicode version |
Description: A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.) |
Ref | Expression |
---|---|
divcld.1 |
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divcld.2 |
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divclapd.3 |
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Ref | Expression |
---|---|
divcanap1d |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divcld.1 |
. 2
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2 | divcld.2 |
. 2
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3 | divclapd.3 |
. 2
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4 | divcanap1 8202 |
. 2
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5 | 1, 2, 3, 4 | syl3anc 1175 |
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-13 1450 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-un 4269 ax-setind 4366 ax-cnex 7490 ax-resscn 7491 ax-1cn 7492 ax-1re 7493 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-mulrcl 7498 ax-addcom 7499 ax-mulcom 7500 ax-addass 7501 ax-mulass 7502 ax-distr 7503 ax-i2m1 7504 ax-0lt1 7505 ax-1rid 7506 ax-0id 7507 ax-rnegex 7508 ax-precex 7509 ax-cnre 7510 ax-pre-ltirr 7511 ax-pre-ltwlin 7512 ax-pre-lttrn 7513 ax-pre-apti 7514 ax-pre-ltadd 7515 ax-pre-mulgt0 7516 ax-pre-mulext 7517 |
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-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 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-po 4132 df-iso 4133 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-pnf 7578 df-mnf 7579 df-xr 7580 df-ltxr 7581 df-le 7582 df-sub 7709 df-neg 7710 df-reap 8106 df-ap 8113 df-div 8194 |
This theorem is referenced by: apdivmuld 8334 ltdiv23 8407 lediv23 8408 recp1lt1 8414 ledivp1 8418 xp1d2m1eqxm1d2 8722 div4p1lem1div2 8723 qmulz 9162 iccf1o 9475 bcpasc 10228 resqrexlemcalc1 10501 sqrtdiv 10529 geo2sum 10962 dvdsval2 11131 flodddiv4t2lthalf 11269 mulgcddvds 11408 qredeq 11410 isprm6 11458 sqrt2irrlem 11472 qmuldeneqnum 11505 hashgcdlem 11535 |
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