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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrslt | Structured version Visualization version GIF version |
Description: The "strictly less than" relation for the extended real structure. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
Ref | Expression |
---|---|
xrslt | β’ < = (ltββ*π ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dflt2 13131 | . 2 β’ < = ( β€ β I ) | |
2 | xrsex 21160 | . . 3 β’ β*π β V | |
3 | xrsle 21165 | . . . 4 β’ β€ = (leββ*π ) | |
4 | eqid 2732 | . . . 4 β’ (ltββ*π ) = (ltββ*π ) | |
5 | 3, 4 | pltfval 18288 | . . 3 β’ (β*π β V β (ltββ*π ) = ( β€ β I )) |
6 | 2, 5 | ax-mp 5 | . 2 β’ (ltββ*π ) = ( β€ β I ) |
7 | 1, 6 | eqtr4i 2763 | 1 β’ < = (ltββ*π ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 β wcel 2106 Vcvv 3474 β cdif 3945 I cid 5573 βcfv 6543 < clt 11252 β€ cle 11253 β*π cxrs 17450 ltcplt 18265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-struct 17084 df-slot 17119 df-ndx 17131 df-base 17149 df-plusg 17214 df-mulr 17215 df-tset 17220 df-ple 17221 df-ds 17223 df-xrs 17452 df-plt 18287 |
This theorem is referenced by: xrsclat 32436 xrsp0 32437 xrsp1 32438 pnfinf 32587 |
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