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Mirrors > Home > ILE Home > Th. List > xaddf | Unicode version |
Description: The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
xaddf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 7812 | . . . . . . 7 | |
2 | 1 | a1i 9 | . . . . . 6 |
3 | pnfxr 7818 | . . . . . . 7 | |
4 | 3 | a1i 9 | . . . . . 6 |
5 | xrmnfdc 9626 | . . . . . . 7 DECID | |
6 | 5 | adantl 275 | . . . . . 6 DECID |
7 | 2, 4, 6 | ifcldcd 3507 | . . . . 5 |
8 | 7 | adantr 274 | . . . 4 |
9 | 1 | a1i 9 | . . . . . 6 |
10 | mnfxr 7822 | . . . . . . 7 | |
11 | 10 | a1i 9 | . . . . . 6 |
12 | xrpnfdc 9625 | . . . . . . 7 DECID | |
13 | 12 | ad3antlr 484 | . . . . . 6 DECID |
14 | 9, 11, 13 | ifcldcd 3507 | . . . . 5 |
15 | 3 | a1i 9 | . . . . . 6 |
16 | 10 | a1i 9 | . . . . . . 7 |
17 | simp-4r 531 | . . . . . . . . . 10 | |
18 | simp-5l 532 | . . . . . . . . . . 11 | |
19 | simpllr 523 | . . . . . . . . . . . 12 | |
20 | 19 | neqned 2315 | . . . . . . . . . . 11 |
21 | xrnemnf 9564 | . . . . . . . . . . . 12 | |
22 | 21 | biimpi 119 | . . . . . . . . . . 11 |
23 | 18, 20, 22 | syl2anc 408 | . . . . . . . . . 10 |
24 | 17, 23 | ecased 1327 | . . . . . . . . 9 |
25 | simplr 519 | . . . . . . . . . 10 | |
26 | simp-5r 533 | . . . . . . . . . . 11 | |
27 | neqne 2316 | . . . . . . . . . . . 12 | |
28 | 27 | adantl 275 | . . . . . . . . . . 11 |
29 | xrnemnf 9564 | . . . . . . . . . . . 12 | |
30 | 29 | biimpi 119 | . . . . . . . . . . 11 |
31 | 26, 28, 30 | syl2anc 408 | . . . . . . . . . 10 |
32 | 25, 31 | ecased 1327 | . . . . . . . . 9 |
33 | 24, 32 | readdcld 7795 | . . . . . . . 8 |
34 | 33 | rexrd 7815 | . . . . . . 7 |
35 | 6 | ad3antrrr 483 | . . . . . . 7 DECID |
36 | 16, 34, 35 | ifcldadc 3501 | . . . . . 6 |
37 | 12 | ad3antlr 484 | . . . . . 6 DECID |
38 | 15, 36, 37 | ifcldadc 3501 | . . . . 5 |
39 | xrmnfdc 9626 | . . . . . 6 DECID | |
40 | 39 | ad2antrr 479 | . . . . 5 DECID |
41 | 14, 38, 40 | ifcldadc 3501 | . . . 4 |
42 | xrpnfdc 9625 | . . . . 5 DECID | |
43 | 42 | adantr 274 | . . . 4 DECID |
44 | 8, 41, 43 | ifcldadc 3501 | . . 3 |
45 | 44 | rgen2a 2486 | . 2 |
46 | df-xadd 9560 | . . 3 | |
47 | 46 | fmpo 6099 | . 2 |
48 | 45, 47 | mpbi 144 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wo 697 DECID wdc 819 wceq 1331 wcel 1480 wne 2308 wral 2416 cif 3474 cxp 4537 wf 5119 (class class class)co 5774 cr 7619 cc0 7620 caddc 7623 cpnf 7797 cmnf 7798 cxr 7799 cxad 9557 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 ax-rnegex 7729 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-pnf 7802 df-mnf 7803 df-xr 7804 df-xadd 9560 |
This theorem is referenced by: xaddcl 9643 |
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