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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 16671 | . 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 493 | 1 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∃wex 1878 ∈ wcel 2164 ∀wral 3117 ∩ cin 3797 ⊆ wss 3798 𝒫 cpw 4380 ∪ cuni 4660 “ cima 5349 ⟶wf 6123 ‘cfv 6127 Fincfn 8228 Moorecmre 16602 ACScacs 16605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-fv 6135 df-acs 16609 |
This theorem is referenced by: acsfiel 16674 acsmred 16676 mreacs 16678 isacs3lem 17526 symggen 18247 odf1o1 18345 lsmmod 18446 gsumzsplit 18687 gsumzoppg 18704 gsumpt 18721 dmdprdd 18759 dprdfeq0 18782 dprdspan 18787 dprdres 18788 dprdss 18789 subgdmdprd 18794 subgdprd 18795 dprdsn 18796 dprd2dlem1 18801 dprd2da 18802 dmdprdsplit2lem 18805 ablfac1b 18830 pgpfac1lem1 18834 pgpfac1lem3 18837 pgpfac1lem4 18838 pgpfac1lem5 18839 pgpfaclem1 18841 pgpfaclem2 18842 isnacs2 38112 proot1mul 38619 proot1hash 38620 |
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