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Mirrors > Home > ILE Home > Th. List > submmulgcl | GIF version |
Description: Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 13-Jan-2015.) |
Ref | Expression |
---|---|
submmulgcl.t | β’ β = (.gβπΊ) |
Ref | Expression |
---|---|
submmulgcl | β’ ((π β (SubMndβπΊ) β§ π β β0 β§ π β π) β (π β π) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2187 | . 2 β’ (BaseβπΊ) = (BaseβπΊ) | |
2 | submmulgcl.t | . 2 β’ β = (.gβπΊ) | |
3 | eqid 2187 | . 2 β’ (+gβπΊ) = (+gβπΊ) | |
4 | submrcl 12883 | . 2 β’ (π β (SubMndβπΊ) β πΊ β Mnd) | |
5 | 1 | submss 12888 | . 2 β’ (π β (SubMndβπΊ) β π β (BaseβπΊ)) |
6 | 3 | submcl 12891 | . 2 β’ ((π β (SubMndβπΊ) β§ π₯ β π β§ π¦ β π) β (π₯(+gβπΊ)π¦) β π) |
7 | eqid 2187 | . 2 β’ (0gβπΊ) = (0gβπΊ) | |
8 | 7 | subm0cl 12890 | . 2 β’ (π β (SubMndβπΊ) β (0gβπΊ) β π) |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | mulgnn0subcl 13027 | 1 β’ ((π β (SubMndβπΊ) β§ π β β0 β§ π β π) β (π β π) β π) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ w3a 979 = wceq 1363 β wcel 2158 βcfv 5228 (class class class)co 5888 β0cn0 9189 Basecbs 12475 +gcplusg 12550 0gc0g 12722 Mndcmnd 12838 SubMndcsubmnd 12871 .gcmg 13013 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-0id 7932 ax-rnegex 7933 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-frec 6405 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-inn 8933 df-2 8991 df-n0 9190 df-z 9267 df-uz 9542 df-seqfrec 10459 df-ndx 12478 df-slot 12479 df-base 12481 df-plusg 12563 df-0g 12724 df-submnd 12873 df-minusg 12902 df-mulg 13014 |
This theorem is referenced by: (None) |
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