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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 17697. (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 17697 | . 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 2106 ‘cfv 6563 Moorecmre 17627 ACScacs 17630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-acs 17634 |
This theorem is referenced by: mreacs 17703 acsficl2d 18610 acsfiindd 18611 acsmapd 18612 acsmap2d 18613 acsinfdimd 18616 acsexdimd 18617 mndind 18854 gsumwspan 18872 cycsubg2 19241 cycsubg2cl 19242 cntzspan 19877 dprdz 20065 pgpfac1lem2 20110 pgpfac1lem3a 20111 pgpfaclem1 20116 lidlincl 33438 lvecdimfi 33625 isnacs3 42698 |
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