![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > acsficl | Structured version Visualization version GIF version |
Description: A closure in an algebraic closure system is the union of the closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
Ref | Expression |
---|---|
acsdrscl.f | β’ πΉ = (mrClsβπΆ) |
Ref | Expression |
---|---|
acsficl | β’ ((πΆ β (ACSβπ) β§ π β π) β (πΉβπ) = βͺ (πΉ β (π« π β© Fin))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6884 | . . 3 β’ (π = π β (πΉβπ ) = (πΉβπ)) | |
2 | pweq 4611 | . . . . . 6 β’ (π = π β π« π = π« π) | |
3 | 2 | ineq1d 4206 | . . . . 5 β’ (π = π β (π« π β© Fin) = (π« π β© Fin)) |
4 | 3 | imaeq2d 6052 | . . . 4 β’ (π = π β (πΉ β (π« π β© Fin)) = (πΉ β (π« π β© Fin))) |
5 | 4 | unieqd 4915 | . . 3 β’ (π = π β βͺ (πΉ β (π« π β© Fin)) = βͺ (πΉ β (π« π β© Fin))) |
6 | 1, 5 | eqeq12d 2742 | . 2 β’ (π = π β ((πΉβπ ) = βͺ (πΉ β (π« π β© Fin)) β (πΉβπ) = βͺ (πΉ β (π« π β© Fin)))) |
7 | isacs3lem 18504 | . . . . 5 β’ (πΆ β (ACSβπ) β (πΆ β (Mooreβπ) β§ βπ β π« πΆ((toIncβπ ) β Dirset β βͺ π β πΆ))) | |
8 | acsdrscl.f | . . . . . 6 β’ πΉ = (mrClsβπΆ) | |
9 | 8 | isacs4lem 18506 | . . . . 5 β’ ((πΆ β (Mooreβπ) β§ βπ β π« πΆ((toIncβπ ) β Dirset β βͺ π β πΆ)) β (πΆ β (Mooreβπ) β§ βπ‘ β π« π« π((toIncβπ‘) β Dirset β (πΉββͺ π‘) = βͺ (πΉ β π‘)))) |
10 | 8 | isacs5lem 18507 | . . . . 5 β’ ((πΆ β (Mooreβπ) β§ βπ‘ β π« π« π((toIncβπ‘) β Dirset β (πΉββͺ π‘) = βͺ (πΉ β π‘))) β (πΆ β (Mooreβπ) β§ βπ β π« π(πΉβπ ) = βͺ (πΉ β (π« π β© Fin)))) |
11 | 7, 9, 10 | 3syl 18 | . . . 4 β’ (πΆ β (ACSβπ) β (πΆ β (Mooreβπ) β§ βπ β π« π(πΉβπ ) = βͺ (πΉ β (π« π β© Fin)))) |
12 | 11 | simprd 495 | . . 3 β’ (πΆ β (ACSβπ) β βπ β π« π(πΉβπ ) = βͺ (πΉ β (π« π β© Fin))) |
13 | 12 | adantr 480 | . 2 β’ ((πΆ β (ACSβπ) β§ π β π) β βπ β π« π(πΉβπ ) = βͺ (πΉ β (π« π β© Fin))) |
14 | elfvdm 6921 | . . . 4 β’ (πΆ β (ACSβπ) β π β dom ACS) | |
15 | elpw2g 5337 | . . . 4 β’ (π β dom ACS β (π β π« π β π β π)) | |
16 | 14, 15 | syl 17 | . . 3 β’ (πΆ β (ACSβπ) β (π β π« π β π β π)) |
17 | 16 | biimpar 477 | . 2 β’ ((πΆ β (ACSβπ) β§ π β π) β π β π« π) |
18 | 6, 13, 17 | rspcdva 3607 | 1 β’ ((πΆ β (ACSβπ) β§ π β π) β (πΉβπ) = βͺ (πΉ β (π« π β© Fin))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3055 β© cin 3942 β wss 3943 π« cpw 4597 βͺ cuni 4902 dom cdm 5669 β cima 5672 βcfv 6536 Fincfn 8938 Moorecmre 17532 mrClscmrc 17533 ACScacs 17535 Dirsetcdrs 18256 toInccipo 18489 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17151 df-tset 17222 df-ple 17223 df-ocomp 17224 df-mre 17536 df-mrc 17537 df-acs 17539 df-proset 18257 df-drs 18258 df-poset 18275 df-ipo 18490 |
This theorem is referenced by: acsficld 18513 |
Copyright terms: Public domain | W3C validator |