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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 10667 |
. . 3
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2 | 1 | 3ad2ant1 1003 |
. 2
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3 | recl 10657 |
. . . . 5
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4 | 3 | recnd 7818 |
. . . 4
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5 | 4 | 3ad2ant1 1003 |
. . 3
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6 | simp1 982 |
. . 3
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7 | recn 7777 |
. . . . 5
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8 | 7 | anim1i 338 |
. . . 4
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9 | 8 | 3adant1 1000 |
. . 3
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10 | mulcanap 8450 |
. . 3
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11 | 5, 6, 9, 10 | syl3anc 1217 |
. 2
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12 | 7 | adantr 274 |
. . . . . . . . . 10
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13 | 4 | adantl 275 |
. . . . . . . . . 10
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14 | ax-icn 7739 |
. . . . . . . . . . . 12
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15 | imcl 10658 |
. . . . . . . . . . . . 13
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16 | 15 | recnd 7818 |
. . . . . . . . . . . 12
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17 | mulcl 7771 |
. . . . . . . . . . . 12
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18 | 14, 16, 17 | sylancr 411 |
. . . . . . . . . . 11
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19 | 18 | adantl 275 |
. . . . . . . . . 10
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20 | 12, 13, 19 | adddid 7814 |
. . . . . . . . 9
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21 | replim 10663 |
. . . . . . . . . . 11
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22 | 21 | adantl 275 |
. . . . . . . . . 10
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23 | 22 | oveq2d 5798 |
. . . . . . . . 9
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24 | mul12 7915 |
. . . . . . . . . . . 12
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25 | 14, 24 | mp3an1 1303 |
. . . . . . . . . . 11
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26 | 7, 16, 25 | syl2an 287 |
. . . . . . . . . 10
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27 | 26 | oveq2d 5798 |
. . . . . . . . 9
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28 | 20, 23, 27 | 3eqtr4d 2183 |
. . . . . . . 8
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29 | 28 | fveq2d 5433 |
. . . . . . 7
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30 | remulcl 7772 |
. . . . . . . . 9
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31 | 3, 30 | sylan2 284 |
. . . . . . . 8
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32 | remulcl 7772 |
. . . . . . . . 9
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33 | 15, 32 | sylan2 284 |
. . . . . . . 8
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34 | crre 10661 |
. . . . . . . 8
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35 | 31, 33, 34 | syl2anc 409 |
. . . . . . 7
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36 | 29, 35 | eqtr2d 2174 |
. . . . . 6
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37 | 36 | eqeq1d 2149 |
. . . . 5
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38 | mulcl 7771 |
. . . . . . 7
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39 | 7, 38 | sylan 281 |
. . . . . 6
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40 | rereb 10667 |
. . . . . 6
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41 | 39, 40 | syl 14 |
. . . . 5
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42 | 37, 41 | bitr4d 190 |
. . . 4
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43 | 42 | ancoms 266 |
. . 3
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44 | 43 | 3adant3 1002 |
. 2
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45 | 2, 11, 44 | 3bitr2d 215 |
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 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-po 4226 df-iso 4227 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-2 8803 df-cj 10646 df-re 10647 df-im 10648 |
This theorem is referenced by: (None) |
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