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Mirrors > Home > ILE Home > Th. List > xpncan | Unicode version |
Description: Extended real version of pncan 8188. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xpncan |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexneg 9855 |
. . . 4
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2 | 1 | adantl 277 |
. . 3
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3 | 2 | oveq2d 5908 |
. 2
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4 | renegcl 8243 |
. . . . . 6
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5 | 4 | ad2antlr 489 |
. . . . 5
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6 | rexr 8028 |
. . . . . 6
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7 | renepnf 8030 |
. . . . . 6
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8 | xaddmnf2 9874 |
. . . . . 6
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9 | 6, 7, 8 | syl2anc 411 |
. . . . 5
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10 | 5, 9 | syl 14 |
. . . 4
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11 | oveq1 5899 |
. . . . . 6
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12 | rexr 8028 |
. . . . . . . 8
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13 | renepnf 8030 |
. . . . . . . 8
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14 | xaddmnf2 9874 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 12, 13, 14 | syl2anc 411 |
. . . . . . 7
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16 | 15 | adantl 277 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 11, 16 | sylan9eqr 2244 |
. . . . 5
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18 | 17 | oveq1d 5907 |
. . . 4
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19 | simpr 110 |
. . . 4
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20 | 10, 18, 19 | 3eqtr4d 2232 |
. . 3
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21 | simpll 527 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | simpr 110 |
. . . . 5
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23 | 12 | ad2antlr 489 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | renemnf 8031 |
. . . . . 6
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25 | 24 | ad2antlr 489 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 4 | ad2antlr 489 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 26, 6 | syl 14 |
. . . . 5
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28 | renemnf 8031 |
. . . . . 6
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29 | 26, 28 | syl 14 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | xaddass 9894 |
. . . . 5
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31 | 21, 22, 23, 25, 27, 29, 30 | syl222anc 1265 |
. . . 4
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32 | simplr 528 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
33 | 32, 26 | rexaddd 9879 |
. . . . . . 7
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34 | 32 | recnd 8011 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | 34 | negidd 8283 |
. . . . . . 7
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36 | 33, 35 | eqtrd 2222 |
. . . . . 6
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37 | 36 | oveq2d 5908 |
. . . . 5
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38 | xaddid1 9887 |
. . . . . 6
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39 | 38 | ad2antrr 488 |
. . . . 5
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40 | 37, 39 | eqtrd 2222 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | 31, 40 | eqtrd 2222 |
. . 3
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42 | xrmnfdc 9868 |
. . . . . 6
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43 | exmiddc 837 |
. . . . . 6
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44 | 42, 43 | syl 14 |
. . . . 5
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45 | df-ne 2361 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
46 | 45 | orbi2i 763 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
47 | 44, 46 | sylibr 134 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | 47 | adantr 276 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
49 | 20, 41, 48 | mpjaodan 799 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
50 | 3, 49 | eqtrd 2222 |
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 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7927 ax-resscn 7928 ax-1cn 7929 ax-1re 7930 ax-icn 7931 ax-addcl 7932 ax-addrcl 7933 ax-mulcl 7934 ax-addcom 7936 ax-addass 7938 ax-distr 7940 ax-i2m1 7941 ax-0id 7944 ax-rnegex 7945 ax-cnre 7947 |
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 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-fv 5240 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-1st 6160 df-2nd 6161 df-pnf 8019 df-mnf 8020 df-xr 8021 df-sub 8155 df-neg 8156 df-xneg 9797 df-xadd 9798 |
This theorem is referenced by: xnpcan 9897 xleadd1 9900 xrmaxaddlem 11295 |
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