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Mirrors > Home > MPE Home > Th. List > submgmacs | Structured version Visualization version GIF version |
Description: Submagmas are an algebraic closure system. (Contributed by AV, 27-Feb-2020.) |
Ref | Expression |
---|---|
submgmacs.b | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
submgmacs | ⊢ (𝐺 ∈ Mgm → (SubMgm‘𝐺) ∈ (ACS‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | submgmacs.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2728 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | 1, 2 | issubmgm 18671 | . . . . 5 ⊢ (𝐺 ∈ Mgm → (𝑠 ∈ (SubMgm‘𝐺) ↔ (𝑠 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝐺)𝑦) ∈ 𝑠))) |
4 | velpw 4611 | . . . . . 6 ⊢ (𝑠 ∈ 𝒫 𝐵 ↔ 𝑠 ⊆ 𝐵) | |
5 | 4 | anbi1i 622 | . . . . 5 ⊢ ((𝑠 ∈ 𝒫 𝐵 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝐺)𝑦) ∈ 𝑠) ↔ (𝑠 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝐺)𝑦) ∈ 𝑠)) |
6 | 3, 5 | bitr4di 288 | . . . 4 ⊢ (𝐺 ∈ Mgm → (𝑠 ∈ (SubMgm‘𝐺) ↔ (𝑠 ∈ 𝒫 𝐵 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝐺)𝑦) ∈ 𝑠))) |
7 | 6 | eqabdv 2863 | . . 3 ⊢ (𝐺 ∈ Mgm → (SubMgm‘𝐺) = {𝑠 ∣ (𝑠 ∈ 𝒫 𝐵 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝐺)𝑦) ∈ 𝑠)}) |
8 | df-rab 3431 | . . 3 ⊢ {𝑠 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝐺)𝑦) ∈ 𝑠} = {𝑠 ∣ (𝑠 ∈ 𝒫 𝐵 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝐺)𝑦) ∈ 𝑠)} | |
9 | 7, 8 | eqtr4di 2786 | . 2 ⊢ (𝐺 ∈ Mgm → (SubMgm‘𝐺) = {𝑠 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝐺)𝑦) ∈ 𝑠}) |
10 | 1 | fvexi 6916 | . . 3 ⊢ 𝐵 ∈ V |
11 | 1, 2 | mgmcl 18612 | . . . . 5 ⊢ ((𝐺 ∈ Mgm ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
12 | 11 | 3expb 1117 | . . . 4 ⊢ ((𝐺 ∈ Mgm ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
13 | 12 | ralrimivva 3198 | . . 3 ⊢ (𝐺 ∈ Mgm → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
14 | acsfn2 17652 | . . 3 ⊢ ((𝐵 ∈ V ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) → {𝑠 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝐺)𝑦) ∈ 𝑠} ∈ (ACS‘𝐵)) | |
15 | 10, 13, 14 | sylancr 585 | . 2 ⊢ (𝐺 ∈ Mgm → {𝑠 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝐺)𝑦) ∈ 𝑠} ∈ (ACS‘𝐵)) |
16 | 9, 15 | eqeltrd 2829 | 1 ⊢ (𝐺 ∈ Mgm → (SubMgm‘𝐺) ∈ (ACS‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {cab 2705 ∀wral 3058 {crab 3430 Vcvv 3473 ⊆ wss 3949 𝒫 cpw 4606 ‘cfv 6553 (class class class)co 7426 Basecbs 17189 +gcplusg 17242 ACScacs 17574 Mgmcmgm 18607 SubMgmcsubmgm 18660 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-om 7879 df-1o 8495 df-en 8973 df-fin 8976 df-mre 17575 df-mrc 17576 df-acs 17578 df-mgm 18609 df-submgm 18662 |
This theorem is referenced by: (None) |
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