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Mirrors > Home > ILE Home > Th. List > xpncan | Unicode version |
Description: Extended real version of pncan 8118. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xpncan |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexneg 9780 | . . . 4 | |
2 | 1 | adantl 275 | . . 3 |
3 | 2 | oveq2d 5867 | . 2 |
4 | renegcl 8173 | . . . . . 6 | |
5 | 4 | ad2antlr 486 | . . . . 5 |
6 | rexr 7958 | . . . . . 6 | |
7 | renepnf 7960 | . . . . . 6 | |
8 | xaddmnf2 9799 | . . . . . 6 | |
9 | 6, 7, 8 | syl2anc 409 | . . . . 5 |
10 | 5, 9 | syl 14 | . . . 4 |
11 | oveq1 5858 | . . . . . 6 | |
12 | rexr 7958 | . . . . . . . 8 | |
13 | renepnf 7960 | . . . . . . . 8 | |
14 | xaddmnf2 9799 | . . . . . . . 8 | |
15 | 12, 13, 14 | syl2anc 409 | . . . . . . 7 |
16 | 15 | adantl 275 | . . . . . 6 |
17 | 11, 16 | sylan9eqr 2225 | . . . . 5 |
18 | 17 | oveq1d 5866 | . . . 4 |
19 | simpr 109 | . . . 4 | |
20 | 10, 18, 19 | 3eqtr4d 2213 | . . 3 |
21 | simpll 524 | . . . . 5 | |
22 | simpr 109 | . . . . 5 | |
23 | 12 | ad2antlr 486 | . . . . 5 |
24 | renemnf 7961 | . . . . . 6 | |
25 | 24 | ad2antlr 486 | . . . . 5 |
26 | 4 | ad2antlr 486 | . . . . . 6 |
27 | 26, 6 | syl 14 | . . . . 5 |
28 | renemnf 7961 | . . . . . 6 | |
29 | 26, 28 | syl 14 | . . . . 5 |
30 | xaddass 9819 | . . . . 5 | |
31 | 21, 22, 23, 25, 27, 29, 30 | syl222anc 1249 | . . . 4 |
32 | simplr 525 | . . . . . . . 8 | |
33 | 32, 26 | rexaddd 9804 | . . . . . . 7 |
34 | 32 | recnd 7941 | . . . . . . . 8 |
35 | 34 | negidd 8213 | . . . . . . 7 |
36 | 33, 35 | eqtrd 2203 | . . . . . 6 |
37 | 36 | oveq2d 5867 | . . . . 5 |
38 | xaddid1 9812 | . . . . . 6 | |
39 | 38 | ad2antrr 485 | . . . . 5 |
40 | 37, 39 | eqtrd 2203 | . . . 4 |
41 | 31, 40 | eqtrd 2203 | . . 3 |
42 | xrmnfdc 9793 | . . . . . 6 DECID | |
43 | exmiddc 831 | . . . . . 6 DECID | |
44 | 42, 43 | syl 14 | . . . . 5 |
45 | df-ne 2341 | . . . . . 6 | |
46 | 45 | orbi2i 757 | . . . . 5 |
47 | 44, 46 | sylibr 133 | . . . 4 |
48 | 47 | adantr 274 | . . 3 |
49 | 20, 41, 48 | mpjaodan 793 | . 2 |
50 | 3, 49 | eqtrd 2203 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 703 DECID wdc 829 wceq 1348 wcel 2141 wne 2340 (class class class)co 5851 cr 7766 cc0 7767 caddc 7770 cpnf 7944 cmnf 7945 cxr 7946 cneg 8084 cxne 9719 cxad 9720 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-addcom 7867 ax-addass 7869 ax-distr 7871 ax-i2m1 7872 ax-0id 7875 ax-rnegex 7876 ax-cnre 7878 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-pnf 7949 df-mnf 7950 df-xr 7951 df-sub 8085 df-neg 8086 df-xneg 9722 df-xadd 9723 |
This theorem is referenced by: xnpcan 9822 xleadd1 9825 xrmaxaddlem 11216 |
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