![]() |
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 17650 | . . 3 ⊢ SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0g‘𝑠) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡)}) | |
2 | 1 | dmmptss 5851 | . 2 ⊢ dom SubMnd ⊆ Mnd |
3 | elfvdm 6444 | . 2 ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ dom SubMnd) | |
4 | 2, 3 | sseldi 3797 | 1 ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 ∀wral 3090 {crab 3094 𝒫 cpw 4350 dom cdm 5313 ‘cfv 6102 (class class class)co 6879 Basecbs 16183 +gcplusg 16266 0gc0g 16414 Mndcmnd 17608 SubMndcsubmnd 17648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3388 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-br 4845 df-opab 4907 df-mpt 4924 df-xp 5319 df-rel 5320 df-cnv 5321 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-iota 6065 df-fv 6110 df-submnd 17650 |
This theorem is referenced by: submss 17664 subm0cl 17666 submcl 17667 submmnd 17668 subm0 17670 subsubm 17671 resmhm2 17674 gsumsubm 17687 gsumwsubmcl 17689 submmulgcl 17897 oppgsubm 18103 lsmub1x 18373 lsmub2x 18374 lsmsubm 18380 submarchi 30255 |
Copyright terms: Public domain | W3C validator |