![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > acsmred | Structured version Visualization version GIF version |
Description: An algebraic closure system is also a Moore system. Deduction form of acsmre 16915. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
acsmred.1 | ⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋)) |
Ref | Expression |
---|---|
acsmred | ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | acsmred.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋)) | |
2 | acsmre 16915 | . 2 ⊢ (𝐴 ∈ (ACS‘𝑋) → 𝐴 ∈ (Moore‘𝑋)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ‘cfv 6324 Moorecmre 16845 ACScacs 16848 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-acs 16852 |
This theorem is referenced by: mreacs 16921 acsficl2d 17778 acsfiindd 17779 acsmapd 17780 acsmap2d 17781 acsinfdimd 17784 acsexdimd 17785 mndind 17984 gsumwspan 18003 cycsubg2 18345 cycsubg2cl 18346 cntzspan 18957 dprdz 19145 pgpfac1lem2 19190 pgpfac1lem3a 19191 pgpfaclem1 19196 lidlincl 31015 lvecdimfi 31086 isnacs3 39651 |
Copyright terms: Public domain | W3C validator |