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| Mirrors > Home > MPE Home > Th. List > issubmndb | Structured version Visualization version GIF version | ||
| Description: The submonoid predicate. Analogous to issubg 19053. (Contributed by AV, 1-Feb-2024.) |
| Ref | Expression |
|---|---|
| issubmndb.b | β’ π΅ = (BaseβπΊ) |
| issubmndb.z | β’ 0 = (0gβπΊ) |
| Ref | Expression |
|---|---|
| issubmndb | β’ (π β (SubMndβπΊ) β ((πΊ β Mnd β§ (πΊ βΎs π) β Mnd) β§ (π β π΅ β§ 0 β π))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | issubmndb.b | . . . . 5 β’ π΅ = (BaseβπΊ) | |
| 2 | issubmndb.z | . . . . 5 β’ 0 = (0gβπΊ) | |
| 3 | eqid 2726 | . . . . 5 β’ (πΊ βΎs π) = (πΊ βΎs π) | |
| 4 | 1, 2, 3 | issubm2 18729 | . . . 4 β’ (πΊ β Mnd β (π β (SubMndβπΊ) β (π β π΅ β§ 0 β π β§ (πΊ βΎs π) β Mnd))) |
| 5 | 3anrot 1097 | . . . . 5 β’ (((πΊ βΎs π) β Mnd β§ π β π΅ β§ 0 β π) β (π β π΅ β§ 0 β π β§ (πΊ βΎs π) β Mnd)) | |
| 6 | 3anass 1092 | . . . . 5 β’ (((πΊ βΎs π) β Mnd β§ π β π΅ β§ 0 β π) β ((πΊ βΎs π) β Mnd β§ (π β π΅ β§ 0 β π))) | |
| 7 | 5, 6 | bitr3i 277 | . . . 4 β’ ((π β π΅ β§ 0 β π β§ (πΊ βΎs π) β Mnd) β ((πΊ βΎs π) β Mnd β§ (π β π΅ β§ 0 β π))) |
| 8 | 4, 7 | bitrdi 287 | . . 3 β’ (πΊ β Mnd β (π β (SubMndβπΊ) β ((πΊ βΎs π) β Mnd β§ (π β π΅ β§ 0 β π)))) |
| 9 | 8 | pm5.32i 574 | . 2 β’ ((πΊ β Mnd β§ π β (SubMndβπΊ)) β (πΊ β Mnd β§ ((πΊ βΎs π) β Mnd β§ (π β π΅ β§ 0 β π)))) |
| 10 | submrcl 18727 | . . 3 β’ (π β (SubMndβπΊ) β πΊ β Mnd) | |
| 11 | 10 | pm4.71ri 560 | . 2 β’ (π β (SubMndβπΊ) β (πΊ β Mnd β§ π β (SubMndβπΊ))) |
| 12 | anass 468 | . 2 β’ (((πΊ β Mnd β§ (πΊ βΎs π) β Mnd) β§ (π β π΅ β§ 0 β π)) β (πΊ β Mnd β§ ((πΊ βΎs π) β Mnd β§ (π β π΅ β§ 0 β π)))) | |
| 13 | 9, 11, 12 | 3bitr4i 303 | 1 β’ (π β (SubMndβπΊ) β ((πΊ β Mnd β§ (πΊ βΎs π) β Mnd) β§ (π β π΅ β§ 0 β π))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wss 3943 βcfv 6537 (class class class)co 7405 Basecbs 17153 βΎs cress 17182 0gc0g 17394 Mndcmnd 18667 SubMndcsubmnd 18712 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 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 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 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-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 |
| This theorem is referenced by: resmndismnd 18733 nsmndex1 18838 symgsubmefmndALT 19323 primrootsunit1 41477 primrootscoprbij 41482 |
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