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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 17635. (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 17635 | . 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 2098 ‘cfv 6549 Moorecmre 17565 ACScacs 17568 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-acs 17572 |
This theorem is referenced by: mreacs 17641 acsficl2d 18547 acsfiindd 18548 acsmapd 18549 acsmap2d 18550 acsinfdimd 18553 acsexdimd 18554 mndind 18788 gsumwspan 18806 cycsubg2 19173 cycsubg2cl 19174 cntzspan 19811 dprdz 19999 pgpfac1lem2 20044 pgpfac1lem3a 20045 pgpfaclem1 20050 lidlincl 33242 lvecdimfi 33426 isnacs3 42272 |
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