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Mirrors > Home > ILE Home > Th. List > xpncan | Unicode version |
Description: Extended real version of pncan 7885. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xpncan |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexneg 9500 |
. . . 4
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2 | 1 | adantl 273 |
. . 3
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3 | 2 | oveq2d 5742 |
. 2
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4 | renegcl 7940 |
. . . . . 6
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5 | 4 | ad2antlr 478 |
. . . . 5
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6 | rexr 7729 |
. . . . . 6
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7 | renepnf 7731 |
. . . . . 6
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8 | xaddmnf2 9519 |
. . . . . 6
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9 | 6, 7, 8 | syl2anc 406 |
. . . . 5
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10 | 5, 9 | syl 14 |
. . . 4
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11 | oveq1 5733 |
. . . . . 6
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12 | rexr 7729 |
. . . . . . . 8
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13 | renepnf 7731 |
. . . . . . . 8
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14 | xaddmnf2 9519 |
. . . . . . . 8
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15 | 12, 13, 14 | syl2anc 406 |
. . . . . . 7
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16 | 15 | adantl 273 |
. . . . . 6
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17 | 11, 16 | sylan9eqr 2167 |
. . . . 5
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18 | 17 | oveq1d 5741 |
. . . 4
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19 | simpr 109 |
. . . 4
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20 | 10, 18, 19 | 3eqtr4d 2155 |
. . 3
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21 | simpll 501 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | simpr 109 |
. . . . 5
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23 | 12 | ad2antlr 478 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | renemnf 7732 |
. . . . . 6
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25 | 24 | ad2antlr 478 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 4 | ad2antlr 478 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 26, 6 | syl 14 |
. . . . 5
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28 | renemnf 7732 |
. . . . . 6
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29 | 26, 28 | syl 14 |
. . . . 5
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30 | xaddass 9539 |
. . . . 5
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31 | 21, 22, 23, 25, 27, 29, 30 | syl222anc 1213 |
. . . 4
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32 | simplr 502 |
. . . . . . . 8
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33 | 32, 26 | rexaddd 9524 |
. . . . . . 7
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34 | 32 | recnd 7712 |
. . . . . . . 8
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35 | 34 | negidd 7980 |
. . . . . . 7
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36 | 33, 35 | eqtrd 2145 |
. . . . . 6
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37 | 36 | oveq2d 5742 |
. . . . 5
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38 | xaddid1 9532 |
. . . . . 6
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39 | 38 | ad2antrr 477 |
. . . . 5
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40 | 37, 39 | eqtrd 2145 |
. . . 4
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41 | 31, 40 | eqtrd 2145 |
. . 3
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42 | xrmnfdc 9513 |
. . . . . 6
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43 | exmiddc 804 |
. . . . . 6
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44 | 42, 43 | syl 14 |
. . . . 5
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45 | df-ne 2281 |
. . . . . 6
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46 | 45 | orbi2i 734 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
47 | 44, 46 | sylibr 133 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | 47 | adantr 272 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
49 | 20, 41, 48 | mpjaodan 770 |
. 2
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50 | 3, 49 | eqtrd 2145 |
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 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-cnex 7630 ax-resscn 7631 ax-1cn 7632 ax-1re 7633 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-addcom 7639 ax-addass 7641 ax-distr 7643 ax-i2m1 7644 ax-0id 7647 ax-rnegex 7648 ax-cnre 7650 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-if 3439 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5990 df-2nd 5991 df-pnf 7720 df-mnf 7721 df-xr 7722 df-sub 7852 df-neg 7853 df-xneg 9446 df-xadd 9447 |
This theorem is referenced by: xnpcan 9542 xleadd1 9545 xrmaxaddlem 10915 |
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