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Mirrors > Home > MPE Home > Th. List > hashxrcl | Structured version Visualization version GIF version |
Description: Extended real closure of the āÆ function. (Contributed by Mario Carneiro, 22-Apr-2015.) |
Ref | Expression |
---|---|
hashxrcl | ā¢ (š“ ā š ā (āÆāš“) ā ā*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0ssre 12351 | . . . 4 ā¢ ā0 ā ā | |
2 | ressxr 11133 | . . . 4 ā¢ ā ā ā* | |
3 | 1, 2 | sstri 3952 | . . 3 ā¢ ā0 ā ā* |
4 | pnfxr 11143 | . . . 4 ā¢ +ā ā ā* | |
5 | snssi 4767 | . . . 4 ā¢ (+ā ā ā* ā {+ā} ā ā*) | |
6 | 4, 5 | ax-mp 5 | . . 3 ā¢ {+ā} ā ā* |
7 | 3, 6 | unssi 4144 | . 2 ā¢ (ā0 āŖ {+ā}) ā ā* |
8 | elex 3462 | . . 3 ā¢ (š“ ā š ā š“ ā V) | |
9 | hashf 14166 | . . . 4 ā¢ āÆ:Vā¶(ā0 āŖ {+ā}) | |
10 | 9 | ffvelcdmi 7029 | . . 3 ā¢ (š“ ā V ā (āÆāš“) ā (ā0 āŖ {+ā})) |
11 | 8, 10 | syl 17 | . 2 ā¢ (š“ ā š ā (āÆāš“) ā (ā0 āŖ {+ā})) |
12 | 7, 11 | sselid 3941 | 1 ā¢ (š“ ā š ā (āÆāš“) ā ā*) |
Colors of variables: wff setvar class |
Syntax hints: ā wi 4 ā wcel 2107 Vcvv 3444 āŖ cun 3907 ā wss 3909 {csn 4585 ācfv 6492 ācr 10984 +ācpnf 11120 ā*cxr 11122 ā0cn0 12347 āÆchash 14158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-card 9809 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-n0 12348 df-xnn0 12420 df-z 12434 df-uz 12697 df-hash 14159 |
This theorem is referenced by: nfile 14187 hashdom 14207 hashinfxadd 14213 hashunx 14214 hashgt0 14216 hashunsnggt 14222 hashle00 14228 hashgt0elex 14229 hashss 14237 hashgt12el 14250 hashgt12el2 14251 hashgt23el 14252 ramtlecl 16807 0ram 16827 isnzr2hash 20657 0ringnnzr 20662 ewlkle 28339 upgrewlkle2 28340 hashxpe 31491 esumcst 32423 esumpinfval 32433 lfuhgr2 33473 acycgr2v 33505 idomodle 41357 |
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