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Mirrors > Home > ILE Home > Th. List > xpncan | Unicode version |
Description: Extended real version of pncan 8095. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xpncan |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexneg 9757 | . . . 4 | |
2 | 1 | adantl 275 | . . 3 |
3 | 2 | oveq2d 5852 | . 2 |
4 | renegcl 8150 | . . . . . 6 | |
5 | 4 | ad2antlr 481 | . . . . 5 |
6 | rexr 7935 | . . . . . 6 | |
7 | renepnf 7937 | . . . . . 6 | |
8 | xaddmnf2 9776 | . . . . . 6 | |
9 | 6, 7, 8 | syl2anc 409 | . . . . 5 |
10 | 5, 9 | syl 14 | . . . 4 |
11 | oveq1 5843 | . . . . . 6 | |
12 | rexr 7935 | . . . . . . . 8 | |
13 | renepnf 7937 | . . . . . . . 8 | |
14 | xaddmnf2 9776 | . . . . . . . 8 | |
15 | 12, 13, 14 | syl2anc 409 | . . . . . . 7 |
16 | 15 | adantl 275 | . . . . . 6 |
17 | 11, 16 | sylan9eqr 2219 | . . . . 5 |
18 | 17 | oveq1d 5851 | . . . 4 |
19 | simpr 109 | . . . 4 | |
20 | 10, 18, 19 | 3eqtr4d 2207 | . . 3 |
21 | simpll 519 | . . . . 5 | |
22 | simpr 109 | . . . . 5 | |
23 | 12 | ad2antlr 481 | . . . . 5 |
24 | renemnf 7938 | . . . . . 6 | |
25 | 24 | ad2antlr 481 | . . . . 5 |
26 | 4 | ad2antlr 481 | . . . . . 6 |
27 | 26, 6 | syl 14 | . . . . 5 |
28 | renemnf 7938 | . . . . . 6 | |
29 | 26, 28 | syl 14 | . . . . 5 |
30 | xaddass 9796 | . . . . 5 | |
31 | 21, 22, 23, 25, 27, 29, 30 | syl222anc 1243 | . . . 4 |
32 | simplr 520 | . . . . . . . 8 | |
33 | 32, 26 | rexaddd 9781 | . . . . . . 7 |
34 | 32 | recnd 7918 | . . . . . . . 8 |
35 | 34 | negidd 8190 | . . . . . . 7 |
36 | 33, 35 | eqtrd 2197 | . . . . . 6 |
37 | 36 | oveq2d 5852 | . . . . 5 |
38 | xaddid1 9789 | . . . . . 6 | |
39 | 38 | ad2antrr 480 | . . . . 5 |
40 | 37, 39 | eqtrd 2197 | . . . 4 |
41 | 31, 40 | eqtrd 2197 | . . 3 |
42 | xrmnfdc 9770 | . . . . . 6 DECID | |
43 | exmiddc 826 | . . . . . 6 DECID | |
44 | 42, 43 | syl 14 | . . . . 5 |
45 | df-ne 2335 | . . . . . 6 | |
46 | 45 | orbi2i 752 | . . . . 5 |
47 | 44, 46 | sylibr 133 | . . . 4 |
48 | 47 | adantr 274 | . . 3 |
49 | 20, 41, 48 | mpjaodan 788 | . 2 |
50 | 3, 49 | eqtrd 2197 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 DECID wdc 824 wceq 1342 wcel 2135 wne 2334 (class class class)co 5836 cr 7743 cc0 7744 caddc 7747 cpnf 7921 cmnf 7922 cxr 7923 cneg 8061 cxne 9696 cxad 9697 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pnf 7926 df-mnf 7927 df-xr 7928 df-sub 8062 df-neg 8063 df-xneg 9699 df-xadd 9700 |
This theorem is referenced by: xnpcan 9799 xleadd1 9802 xrmaxaddlem 11187 |
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