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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 17945 | . 2 ⊢ SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0g‘𝑠) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡)}) | |
2 | 1 | mptrcl 6769 | 1 ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ∀wral 3135 {crab 3139 𝒫 cpw 4535 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 0gc0g 16701 Mndcmnd 17899 SubMndcsubmnd 17943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fv 6356 df-submnd 17945 |
This theorem is referenced by: issubmndb 17958 submss 17962 subm0cl 17964 submcl 17965 submmnd 17966 subm0 17968 subsubm 17969 insubm 17971 resmhm2 17974 gsumsubm 17987 gsumwsubmcl 17989 submmulgcl 18208 oppgsubm 18428 lsmub1x 18700 lsmub2x 18701 lsmsubm 18707 submarchi 30742 |
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