| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 18797 | . 2 ⊢ SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0g‘𝑠) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡)}) | |
| 2 | 1 | mptrcl 7025 | 1 ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 {crab 3436 𝒫 cpw 4600 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Mndcmnd 18747 SubMndcsubmnd 18795 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fv 6569 df-submnd 18797 |
| This theorem is referenced by: issubmndb 18818 submss 18822 subm0cl 18824 submcl 18825 submmnd 18826 subm0 18828 subsubm 18829 insubm 18831 resmhm2 18834 gsumsubm 18848 gsumwsubmcl 18850 submmulgcl 19135 oppgsubm 19381 lsmub1x 19664 lsmub2x 19665 lsmsubm 19671 submarchi 33193 |
| Copyright terms: Public domain | W3C validator |