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Mirrors > Home > ILE Home > Th. List > absmul | Unicode version |
Description: Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
absmul |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cjmul 10762 |
. . . . . 6
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2 | 1 | oveq2d 5830 |
. . . . 5
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3 | simpl 108 |
. . . . . 6
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4 | simpr 109 |
. . . . . 6
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5 | 3 | cjcld 10817 |
. . . . . 6
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6 | 4 | cjcld 10817 |
. . . . . 6
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7 | 3, 4, 5, 6 | mul4d 8009 |
. . . . 5
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8 | 2, 7 | eqtrd 2187 |
. . . 4
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9 | 8 | fveq2d 5465 |
. . 3
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10 | cjmulrcl 10764 |
. . . . 5
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11 | cjmulge0 10766 |
. . . . 5
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12 | 10, 11 | jca 304 |
. . . 4
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13 | cjmulrcl 10764 |
. . . . 5
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14 | cjmulge0 10766 |
. . . . 5
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15 | 13, 14 | jca 304 |
. . . 4
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16 | sqrtmul 10912 |
. . . 4
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17 | 12, 15, 16 | syl2an 287 |
. . 3
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18 | 9, 17 | eqtrd 2187 |
. 2
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19 | mulcl 7838 |
. . 3
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20 | absval 10878 |
. . 3
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21 | 19, 20 | syl 14 |
. 2
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22 | absval 10878 |
. . 3
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23 | absval 10878 |
. . 3
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24 | 22, 23 | oveqan12d 5833 |
. 2
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25 | 18, 21, 24 | 3eqtr4d 2197 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-13 2127 ax-14 2128 ax-ext 2136 ax-coll 4075 ax-sep 4078 ax-nul 4086 ax-pow 4130 ax-pr 4164 ax-un 4388 ax-setind 4490 ax-iinf 4541 ax-cnex 7802 ax-resscn 7803 ax-1cn 7804 ax-1re 7805 ax-icn 7806 ax-addcl 7807 ax-addrcl 7808 ax-mulcl 7809 ax-mulrcl 7810 ax-addcom 7811 ax-mulcom 7812 ax-addass 7813 ax-mulass 7814 ax-distr 7815 ax-i2m1 7816 ax-0lt1 7817 ax-1rid 7818 ax-0id 7819 ax-rnegex 7820 ax-precex 7821 ax-cnre 7822 ax-pre-ltirr 7823 ax-pre-ltwlin 7824 ax-pre-lttrn 7825 ax-pre-apti 7826 ax-pre-ltadd 7827 ax-pre-mulgt0 7828 ax-pre-mulext 7829 ax-arch 7830 ax-caucvg 7831 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-nel 2420 df-ral 2437 df-rex 2438 df-reu 2439 df-rmo 2440 df-rab 2441 df-v 2711 df-sbc 2934 df-csb 3028 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-nul 3391 df-if 3502 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-int 3804 df-iun 3847 df-br 3962 df-opab 4022 df-mpt 4023 df-tr 4059 df-id 4248 df-po 4251 df-iso 4252 df-iord 4321 df-on 4323 df-ilim 4324 df-suc 4326 df-iom 4544 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-rn 4590 df-res 4591 df-ima 4592 df-iota 5128 df-fun 5165 df-fn 5166 df-f 5167 df-f1 5168 df-fo 5169 df-f1o 5170 df-fv 5171 df-riota 5770 df-ov 5817 df-oprab 5818 df-mpo 5819 df-1st 6078 df-2nd 6079 df-recs 6242 df-frec 6328 df-pnf 7893 df-mnf 7894 df-xr 7895 df-ltxr 7896 df-le 7897 df-sub 8027 df-neg 8028 df-reap 8429 df-ap 8436 df-div 8525 df-inn 8813 df-2 8871 df-3 8872 df-4 8873 df-n0 9070 df-z 9147 df-uz 9419 df-rp 9539 df-seqfrec 10323 df-exp 10397 df-cj 10719 df-re 10720 df-im 10721 df-rsqrt 10875 df-abs 10876 |
This theorem is referenced by: absdivap 10947 absexp 10956 absimle 10961 abstri 10981 absmuli 11028 absmuld 11071 ef01bndlem 11630 absmulgcd 11872 gcdmultiplez 11876 |
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