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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 17159 | . . 3 ⊢ SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0g‘𝑠) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡)}) | |
2 | 1 | dmmptss 5548 | . 2 ⊢ dom SubMnd ⊆ Mnd |
3 | elfvdm 6130 | . 2 ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ dom SubMnd) | |
4 | 2, 3 | sseldi 3566 | 1 ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 {crab 2900 𝒫 cpw 4108 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Mndcmnd 17117 SubMndcsubmnd 17157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fv 5812 df-submnd 17159 |
This theorem is referenced by: submss 17173 subm0cl 17175 submcl 17176 submmnd 17177 subm0 17179 subsubm 17180 resmhm2 17183 gsumsubm 17196 gsumwsubmcl 17198 submmulgcl 17408 oppgsubm 17615 lsmub1x 17884 lsmub2x 17885 lsmsubm 17891 submarchi 29071 |
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