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Mirrors > Home > MPE Home > Th. List > xrsadd | Structured version Visualization version GIF version |
Description: The addition operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
xrsadd | β’ +π = (+gββ*π ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xaddf 13236 | . . 3 β’ +π :(β* Γ β*)βΆβ* | |
2 | xrex 13002 | . . . 4 β’ β* β V | |
3 | 2, 2 | xpex 7755 | . . 3 β’ (β* Γ β*) β V |
4 | fex2 7941 | . . 3 β’ (( +π :(β* Γ β*)βΆβ* β§ (β* Γ β*) β V β§ β* β V) β +π β V) | |
5 | 1, 3, 2, 4 | mp3an 1458 | . 2 β’ +π β V |
6 | df-xrs 17484 | . . 3 β’ β*π = ({β¨(Baseβndx), β*β©, β¨(+gβndx), +π β©, β¨(.rβndx), Β·e β©} βͺ {β¨(TopSetβndx), (ordTopβ β€ )β©, β¨(leβndx), β€ β©, β¨(distβndx), (π₯ β β*, π¦ β β* β¦ if(π₯ β€ π¦, (π¦ +π -ππ₯), (π₯ +π -ππ¦)))β©}) | |
7 | 6 | odrngplusg 17386 | . 2 β’ ( +π β V β +π = (+gββ*π )) |
8 | 5, 7 | ax-mp 5 | 1 β’ +π = (+gββ*π ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 β wcel 2099 Vcvv 3471 ifcif 4529 class class class wbr 5148 Γ cxp 5676 βΆwf 6544 βcfv 6548 (class class class)co 7420 β cmpo 7422 β*cxr 11278 β€ cle 11280 -πcxne 13122 +π cxad 13123 Β·e cxmu 13124 +gcplusg 17233 ordTopcordt 17481 β*π cxrs 17482 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-xadd 13126 df-fz 13518 df-struct 17116 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-mulr 17247 df-tset 17252 df-ple 17253 df-ds 17255 df-xrs 17484 |
This theorem is referenced by: xrsmgm 21334 xrsnsgrp 21335 xrs1mnd 21337 xrs10 21338 xrs1cmn 21339 xrge0subm 21340 imasdsf1olem 24292 xrge0gsumle 24762 xrs0 32746 xrsinvgval 32748 xrsmulgzz 32749 xrge0plusg 32756 xrge0tmdALT 33547 esumpfinvallem 33693 sge0tsms 45768 |
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