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Mirrors > Home > ILE Home > Th. List > mulclsr | Unicode version |
Description: Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) |
Ref | Expression |
---|---|
mulclsr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 7789 |
. . 3
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2 | oveq1 5926 |
. . . 4
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3 | 2 | eleq1d 2262 |
. . 3
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4 | oveq2 5927 |
. . . 4
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5 | 4 | eleq1d 2262 |
. . 3
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6 | mulsrpr 7808 |
. . . 4
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7 | mulclpr 7634 |
. . . . . . . 8
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8 | mulclpr 7634 |
. . . . . . . 8
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9 | addclpr 7599 |
. . . . . . . 8
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10 | 7, 8, 9 | syl2an 289 |
. . . . . . 7
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11 | 10 | an4s 588 |
. . . . . 6
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12 | mulclpr 7634 |
. . . . . . . 8
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13 | mulclpr 7634 |
. . . . . . . 8
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14 | addclpr 7599 |
. . . . . . . 8
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15 | 12, 13, 14 | syl2an 289 |
. . . . . . 7
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16 | 15 | an42s 589 |
. . . . . 6
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17 | 11, 16 | jca 306 |
. . . . 5
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18 | opelxpi 4692 |
. . . . 5
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19 | enrex 7799 |
. . . . . 6
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20 | 19 | ecelqsi 6645 |
. . . . 5
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21 | 17, 18, 20 | 3syl 17 |
. . . 4
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22 | 6, 21 | eqeltrd 2270 |
. . 3
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23 | 1, 3, 5, 22 | 2ecoptocl 6679 |
. 2
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24 | 23, 1 | eleqtrrdi 2287 |
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-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-eprel 4321 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-irdg 6425 df-1o 6471 df-2o 6472 df-oadd 6475 df-omul 6476 df-er 6589 df-ec 6591 df-qs 6595 df-ni 7366 df-pli 7367 df-mi 7368 df-lti 7369 df-plpq 7406 df-mpq 7407 df-enq 7409 df-nqqs 7410 df-plqqs 7411 df-mqqs 7412 df-1nqqs 7413 df-rq 7414 df-ltnqqs 7415 df-enq0 7486 df-nq0 7487 df-0nq0 7488 df-plq0 7489 df-mq0 7490 df-inp 7528 df-iplp 7530 df-imp 7531 df-enr 7788 df-nr 7789 df-mr 7791 |
This theorem is referenced by: pn0sr 7833 negexsr 7834 caucvgsrlemoffval 7858 caucvgsrlemofff 7859 map2psrprg 7867 mulcnsr 7897 mulresr 7900 mulcnsrec 7905 axmulcl 7928 axmulrcl 7929 axmulcom 7933 axmulass 7935 axdistr 7936 axrnegex 7941 |
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