![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > acsmre | Structured version Visualization version GIF version |
Description: Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
Ref | Expression |
---|---|
acsmre | ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isacs 17630 | . 2 ⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)))) | |
2 | 1 | simplbi 497 | 1 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∃wex 1774 ∈ wcel 2099 ∀wral 3058 ∩ cin 3946 ⊆ wss 3947 𝒫 cpw 4603 ∪ cuni 4908 “ cima 5681 ⟶wf 6544 ‘cfv 6548 Fincfn 8963 Moorecmre 17561 ACScacs 17564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-acs 17568 |
This theorem is referenced by: acsfiel 17633 acsmred 17635 mreacs 17637 isacs3lem 18533 symggen 19424 odf1o1 19526 lsmmod 19629 gsumzsplit 19881 gsumzoppg 19898 gsumpt 19916 dmdprdd 19955 dprdfeq0 19978 dprdspan 19983 dprdres 19984 dprdss 19985 subgdmdprd 19990 subgdprd 19991 dprdsn 19992 dprd2dlem1 19997 dprd2da 19998 dmdprdsplit2lem 20001 ablfac1b 20026 pgpfac1lem1 20030 pgpfac1lem3 20033 pgpfac1lem4 20034 pgpfac1lem5 20035 pgpfaclem2 20038 isnacs2 42126 proot1mul 42622 proot1hash 42623 |
Copyright terms: Public domain | W3C validator |