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Mirrors > Home > MPE Home > Th. List > xaddcld | Structured version Visualization version Unicode version |
Description: The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
xnegcld.1 | |
xaddcld.2 |
Ref | Expression |
---|---|
xaddcld |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xnegcld.1 | . 2 | |
2 | xaddcld.2 | . 2 | |
3 | xaddcl 12070 | . 2 | |
4 | 1, 2, 3 | syl2anc 693 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wcel 1990 (class class class)co 6650 cxr 10073 cxad 11944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-pnf 10076 df-mnf 10077 df-xr 10078 df-xadd 11947 |
This theorem is referenced by: xadd4d 12133 imasdsf1olem 22178 bldisj 22203 xblss2ps 22206 xblss2 22207 blcld 22310 comet 22318 stdbdxmet 22320 metdstri 22654 metdscnlem 22658 iscau3 23076 xlt2addrd 29523 xrge0addcld 29527 xrge0subcld 29528 xrofsup 29533 xrsmulgzz 29678 xrge0adddir 29692 xrge0adddi 29693 esumle 30120 esumlef 30124 omssubadd 30362 inelcarsg 30373 carsgclctunlem2 30381 carsgclctunlem3 30382 carsgclctun 30383 xle2addd 39552 infrpge 39567 xrlexaddrp 39568 infleinflem1 39586 infleinflem2 39587 limsupgtlem 40009 ismbl3 40203 ismbl4 40210 sge0prle 40618 sge0split 40626 sge0iunmptlemre 40632 sge0xaddlem1 40650 omeunle 40730 carageniuncl 40737 ovnsubaddlem1 40784 hspmbl 40843 ovolval5lem1 40866 |
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