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Mirrors > Home > ILE Home > Th. List > mulreap | Unicode version |
Description: A product with a real multiplier apart from zero is real iff the multiplicand is real. (Contributed by Jim Kingdon, 14-Jun-2020.) |
Ref | Expression |
---|---|
mulreap |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rereb 10293 |
. . 3
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2 | 1 | 3ad2ant1 964 |
. 2
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3 | recl 10283 |
. . . . 5
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4 | 3 | recnd 7514 |
. . . 4
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5 | 4 | 3ad2ant1 964 |
. . 3
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6 | simp1 943 |
. . 3
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7 | recn 7473 |
. . . . 5
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8 | 7 | anim1i 333 |
. . . 4
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9 | 8 | 3adant1 961 |
. . 3
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10 | mulcanap 8132 |
. . 3
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11 | 5, 6, 9, 10 | syl3anc 1174 |
. 2
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12 | 7 | adantr 270 |
. . . . . . . . . 10
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13 | 4 | adantl 271 |
. . . . . . . . . 10
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14 | ax-icn 7438 |
. . . . . . . . . . . 12
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15 | imcl 10284 |
. . . . . . . . . . . . 13
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16 | 15 | recnd 7514 |
. . . . . . . . . . . 12
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17 | mulcl 7467 |
. . . . . . . . . . . 12
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18 | 14, 16, 17 | sylancr 405 |
. . . . . . . . . . 11
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19 | 18 | adantl 271 |
. . . . . . . . . 10
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20 | 12, 13, 19 | adddid 7510 |
. . . . . . . . 9
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21 | replim 10289 |
. . . . . . . . . . 11
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22 | 21 | adantl 271 |
. . . . . . . . . 10
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23 | 22 | oveq2d 5668 |
. . . . . . . . 9
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24 | mul12 7609 |
. . . . . . . . . . . 12
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25 | 14, 24 | mp3an1 1260 |
. . . . . . . . . . 11
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26 | 7, 16, 25 | syl2an 283 |
. . . . . . . . . 10
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27 | 26 | oveq2d 5668 |
. . . . . . . . 9
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28 | 20, 23, 27 | 3eqtr4d 2130 |
. . . . . . . 8
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29 | 28 | fveq2d 5309 |
. . . . . . 7
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30 | remulcl 7468 |
. . . . . . . . 9
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31 | 3, 30 | sylan2 280 |
. . . . . . . 8
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32 | remulcl 7468 |
. . . . . . . . 9
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33 | 15, 32 | sylan2 280 |
. . . . . . . 8
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34 | crre 10287 |
. . . . . . . 8
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35 | 31, 33, 34 | syl2anc 403 |
. . . . . . 7
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36 | 29, 35 | eqtr2d 2121 |
. . . . . 6
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37 | 36 | eqeq1d 2096 |
. . . . 5
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38 | mulcl 7467 |
. . . . . . 7
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39 | 7, 38 | sylan 277 |
. . . . . 6
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40 | rereb 10293 |
. . . . . 6
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41 | 39, 40 | syl 14 |
. . . . 5
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42 | 37, 41 | bitr4d 189 |
. . . 4
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43 | 42 | ancoms 264 |
. . 3
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44 | 43 | 3adant3 963 |
. 2
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45 | 2, 11, 44 | 3bitr2d 214 |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-cnex 7434 ax-resscn 7435 ax-1cn 7436 ax-1re 7437 ax-icn 7438 ax-addcl 7439 ax-addrcl 7440 ax-mulcl 7441 ax-mulrcl 7442 ax-addcom 7443 ax-mulcom 7444 ax-addass 7445 ax-mulass 7446 ax-distr 7447 ax-i2m1 7448 ax-0lt1 7449 ax-1rid 7450 ax-0id 7451 ax-rnegex 7452 ax-precex 7453 ax-cnre 7454 ax-pre-ltirr 7455 ax-pre-ltwlin 7456 ax-pre-lttrn 7457 ax-pre-apti 7458 ax-pre-ltadd 7459 ax-pre-mulgt0 7460 ax-pre-mulext 7461 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rmo 2367 df-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-mpt 3901 df-id 4120 df-po 4123 df-iso 4124 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-fv 5023 df-riota 5608 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-pnf 7522 df-mnf 7523 df-xr 7524 df-ltxr 7525 df-le 7526 df-sub 7653 df-neg 7654 df-reap 8050 df-ap 8057 df-div 8138 df-2 8479 df-cj 10272 df-re 10273 df-im 10274 |
This theorem is referenced by: (None) |
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