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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 2189 | . 2 β’ (BaseβπΊ) = (BaseβπΊ) | |
2 | submmulgcl.t | . 2 β’ β = (.gβπΊ) | |
3 | eqid 2189 | . 2 β’ (+gβπΊ) = (+gβπΊ) | |
4 | submrcl 12895 | . 2 β’ (π β (SubMndβπΊ) β πΊ β Mnd) | |
5 | 1 | submss 12900 | . 2 β’ (π β (SubMndβπΊ) β π β (BaseβπΊ)) |
6 | 3 | submcl 12903 | . 2 β’ ((π β (SubMndβπΊ) β§ π₯ β π β§ π¦ β π) β (π₯(+gβπΊ)π¦) β π) |
7 | eqid 2189 | . 2 β’ (0gβπΊ) = (0gβπΊ) | |
8 | 7 | subm0cl 12902 | . 2 β’ (π β (SubMndβπΊ) β (0gβπΊ) β π) |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | mulgnn0subcl 13047 | 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 980 = wceq 1364 β wcel 2160 βcfv 5231 (class class class)co 5891 β0cn0 9195 Basecbs 12486 +gcplusg 12561 0gc0g 12733 Mndcmnd 12849 SubMndcsubmnd 12882 .gcmg 13033 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-addcom 7930 ax-addass 7932 ax-distr 7934 ax-i2m1 7935 ax-0lt1 7936 ax-0id 7938 ax-rnegex 7939 ax-cnre 7941 ax-pre-ltirr 7942 ax-pre-ltwlin 7943 ax-pre-lttrn 7944 ax-pre-ltadd 7946 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-frec 6410 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-le 8017 df-sub 8149 df-neg 8150 df-inn 8939 df-2 8997 df-n0 9196 df-z 9273 df-uz 9548 df-seqfrec 10465 df-ndx 12489 df-slot 12490 df-base 12492 df-plusg 12574 df-0g 12735 df-submnd 12884 df-minusg 12921 df-mulg 13034 |
This theorem is referenced by: (None) |
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