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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 16520. (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 16520 | . 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 2145 ‘cfv 6031 Moorecmre 16450 ACScacs 16453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-fv 6039 df-acs 16457 |
This theorem is referenced by: mreacs 16526 acsficl2d 17384 acsfiindd 17385 acsmapd 17386 acsmap2d 17387 acsinfdimd 17390 acsexdimd 17391 mrcmndind 17574 gsumwspan 17591 cycsubg2 17839 cycsubg2cl 17840 cntzspan 18454 dprdz 18637 pgpfac1lem2 18682 pgpfac1lem3a 18683 isnacs3 37799 |
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