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| Mirrors > Home > MPE Home > Th. List > resmndismnd | Structured version Visualization version GIF version | ||
| Description: If the base set of a monoid is contained in the base set of another monoid, and the group operation of the monoid is the restriction of the group operation of the other monoid to its base set, and the identity element of the other monoid is contained in the base set of the monoid, then the other monoid restricted to the base set of the monoid is a monoid. Analogous to resgrpisgrp 19074. (Contributed by AV, 17-Feb-2024.) |
| Ref | Expression |
|---|---|
| mndissubm.b | β’ π΅ = (BaseβπΊ) |
| mndissubm.s | β’ π = (Baseβπ») |
| mndissubm.z | β’ 0 = (0gβπΊ) |
| Ref | Expression |
|---|---|
| resmndismnd | β’ ((πΊ β Mnd β§ π» β Mnd) β ((π β π΅ β§ 0 β π β§ (+gβπ») = ((+gβπΊ) βΎ (π Γ π))) β (πΊ βΎs π) β Mnd)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mndissubm.b | . . . . 5 β’ π΅ = (BaseβπΊ) | |
| 2 | mndissubm.s | . . . . 5 β’ π = (Baseβπ») | |
| 3 | mndissubm.z | . . . . 5 β’ 0 = (0gβπΊ) | |
| 4 | 1, 2, 3 | mndissubm 18732 | . . . 4 β’ ((πΊ β Mnd β§ π» β Mnd) β ((π β π΅ β§ 0 β π β§ (+gβπ») = ((+gβπΊ) βΎ (π Γ π))) β π β (SubMndβπΊ))) |
| 5 | 4 | imp 406 | . . 3 β’ (((πΊ β Mnd β§ π» β Mnd) β§ (π β π΅ β§ 0 β π β§ (+gβπ») = ((+gβπΊ) βΎ (π Γ π)))) β π β (SubMndβπΊ)) |
| 6 | simpl 482 | . . . . . . 7 β’ ((πΊ β Mnd β§ π» β Mnd) β πΊ β Mnd) | |
| 7 | 3simpa 1145 | . . . . . . 7 β’ ((π β π΅ β§ 0 β π β§ (+gβπ») = ((+gβπΊ) βΎ (π Γ π))) β (π β π΅ β§ 0 β π)) | |
| 8 | 6, 7 | anim12i 612 | . . . . . 6 β’ (((πΊ β Mnd β§ π» β Mnd) β§ (π β π΅ β§ 0 β π β§ (+gβπ») = ((+gβπΊ) βΎ (π Γ π)))) β (πΊ β Mnd β§ (π β π΅ β§ 0 β π))) |
| 9 | 8 | biantrud 531 | . . . . 5 β’ (((πΊ β Mnd β§ π» β Mnd) β§ (π β π΅ β§ 0 β π β§ (+gβπ») = ((+gβπΊ) βΎ (π Γ π)))) β ((πΊ βΎs π) β Mnd β ((πΊ βΎs π) β Mnd β§ (πΊ β Mnd β§ (π β π΅ β§ 0 β π))))) |
| 10 | an21 641 | . . . . 5 β’ (((πΊ β Mnd β§ (πΊ βΎs π) β Mnd) β§ (π β π΅ β§ 0 β π)) β ((πΊ βΎs π) β Mnd β§ (πΊ β Mnd β§ (π β π΅ β§ 0 β π)))) | |
| 11 | 9, 10 | bitr4di 289 | . . . 4 β’ (((πΊ β Mnd β§ π» β Mnd) β§ (π β π΅ β§ 0 β π β§ (+gβπ») = ((+gβπΊ) βΎ (π Γ π)))) β ((πΊ βΎs π) β Mnd β ((πΊ β Mnd β§ (πΊ βΎs π) β Mnd) β§ (π β π΅ β§ 0 β π)))) |
| 12 | 1, 3 | issubmndb 18730 | . . . 4 β’ (π β (SubMndβπΊ) β ((πΊ β Mnd β§ (πΊ βΎs π) β Mnd) β§ (π β π΅ β§ 0 β π))) |
| 13 | 11, 12 | bitr4di 289 | . . 3 β’ (((πΊ β Mnd β§ π» β Mnd) β§ (π β π΅ β§ 0 β π β§ (+gβπ») = ((+gβπΊ) βΎ (π Γ π)))) β ((πΊ βΎs π) β Mnd β π β (SubMndβπΊ))) |
| 14 | 5, 13 | mpbird 257 | . 2 β’ (((πΊ β Mnd β§ π» β Mnd) β§ (π β π΅ β§ 0 β π β§ (+gβπ») = ((+gβπΊ) βΎ (π Γ π)))) β (πΊ βΎs π) β Mnd) |
| 15 | 14 | ex 412 | 1 β’ ((πΊ β Mnd β§ π» β Mnd) β ((π β π΅ β§ 0 β π β§ (+gβπ») = ((+gβπΊ) βΎ (π Γ π))) β (πΊ βΎs π) β Mnd)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wss 3943 Γ cxp 5667 βΎ cres 5671 βcfv 6537 (class class class)co 7405 Basecbs 17153 βΎs cress 17182 +gcplusg 17206 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: (None) |
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