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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 16136. (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 16136 | . 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 1977 ‘cfv 5804 Moorecmre 16065 ACScacs 16068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-acs 16072 |
This theorem is referenced by: mreacs 16142 acsficl2d 16999 acsfiindd 17000 acsmapd 17001 acsmap2d 17002 acsinfdimd 17005 acsexdimd 17006 mrcmndind 17189 gsumwspan 17206 cycsubg2 17454 cycsubg2cl 17455 cntzspan 18070 dprdz 18252 pgpfac1lem2 18297 pgpfac1lem3a 18298 isnacs3 36291 |
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