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Mirrors > Home > MPE Home > Th. List > submrcl | Structured version Visualization version GIF version |
Description: Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.) |
Ref | Expression |
---|---|
submrcl | ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-submnd 17949 | . 2 ⊢ SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0g‘𝑠) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡)}) | |
2 | 1 | mptrcl 6754 | 1 ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ∀wral 3106 {crab 3110 𝒫 cpw 4497 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 0gc0g 16705 Mndcmnd 17903 SubMndcsubmnd 17947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fv 6332 df-submnd 17949 |
This theorem is referenced by: issubmndb 17962 submss 17966 subm0cl 17968 submcl 17969 submmnd 17970 subm0 17972 subsubm 17973 insubm 17975 resmhm2 17978 gsumsubm 17991 gsumwsubmcl 17993 submmulgcl 18262 oppgsubm 18482 lsmub1x 18763 lsmub2x 18764 lsmsubm 18770 submarchi 30865 |
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