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Mirrors > Home > MPE Home > Th. List > acsdrscl | Structured version Visualization version GIF version |
Description: In an algebraic closure system, closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
Ref | Expression |
---|---|
acsdrscl.f | β’ πΉ = (mrClsβπΆ) |
Ref | Expression |
---|---|
acsdrscl | β’ ((πΆ β (ACSβπ) β§ π β π« π β§ (toIncβπ) β Dirset) β (πΉββͺ π) = βͺ (πΉ β π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6891 | . . . . 5 β’ (π‘ = π β (toIncβπ‘) = (toIncβπ)) | |
2 | 1 | eleq1d 2817 | . . . 4 β’ (π‘ = π β ((toIncβπ‘) β Dirset β (toIncβπ) β Dirset)) |
3 | unieq 4919 | . . . . . 6 β’ (π‘ = π β βͺ π‘ = βͺ π) | |
4 | 3 | fveq2d 6895 | . . . . 5 β’ (π‘ = π β (πΉββͺ π‘) = (πΉββͺ π)) |
5 | imaeq2 6055 | . . . . . 6 β’ (π‘ = π β (πΉ β π‘) = (πΉ β π)) | |
6 | 5 | unieqd 4922 | . . . . 5 β’ (π‘ = π β βͺ (πΉ β π‘) = βͺ (πΉ β π)) |
7 | 4, 6 | eqeq12d 2747 | . . . 4 β’ (π‘ = π β ((πΉββͺ π‘) = βͺ (πΉ β π‘) β (πΉββͺ π) = βͺ (πΉ β π))) |
8 | 2, 7 | imbi12d 344 | . . 3 β’ (π‘ = π β (((toIncβπ‘) β Dirset β (πΉββͺ π‘) = βͺ (πΉ β π‘)) β ((toIncβπ) β Dirset β (πΉββͺ π) = βͺ (πΉ β π)))) |
9 | isacs3lem 18500 | . . . . . 6 β’ (πΆ β (ACSβπ) β (πΆ β (Mooreβπ) β§ βπ β π« πΆ((toIncβπ ) β Dirset β βͺ π β πΆ))) | |
10 | acsdrscl.f | . . . . . . 7 β’ πΉ = (mrClsβπΆ) | |
11 | 10 | isacs4lem 18502 | . . . . . 6 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‘)))) |
12 | 9, 11 | syl 17 | . . . . 5 β’ (πΆ β (ACSβπ) β (πΆ β (Mooreβπ) β§ βπ‘ β π« π« π((toIncβπ‘) β Dirset β (πΉββͺ π‘) = βͺ (πΉ β π‘)))) |
13 | 12 | simprd 495 | . . . 4 β’ (πΆ β (ACSβπ) β βπ‘ β π« π« π((toIncβπ‘) β Dirset β (πΉββͺ π‘) = βͺ (πΉ β π‘))) |
14 | 13 | adantr 480 | . . 3 β’ ((πΆ β (ACSβπ) β§ π β π« π) β βπ‘ β π« π« π((toIncβπ‘) β Dirset β (πΉββͺ π‘) = βͺ (πΉ β π‘))) |
15 | elfvdm 6928 | . . . . 5 β’ (πΆ β (ACSβπ) β π β dom ACS) | |
16 | pwexg 5376 | . . . . 5 β’ (π β dom ACS β π« π β V) | |
17 | elpw2g 5344 | . . . . 5 β’ (π« π β V β (π β π« π« π β π β π« π)) | |
18 | 15, 16, 17 | 3syl 18 | . . . 4 β’ (πΆ β (ACSβπ) β (π β π« π« π β π β π« π)) |
19 | 18 | biimpar 477 | . . 3 β’ ((πΆ β (ACSβπ) β§ π β π« π) β π β π« π« π) |
20 | 8, 14, 19 | rspcdva 3613 | . 2 β’ ((πΆ β (ACSβπ) β§ π β π« π) β ((toIncβπ) β Dirset β (πΉββͺ π) = βͺ (πΉ β π))) |
21 | 20 | 3impia 1116 | 1 β’ ((πΆ β (ACSβπ) β§ π β π« π β§ (toIncβπ) β Dirset) β (πΉββͺ π) = βͺ (πΉ β π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 βwral 3060 Vcvv 3473 β wss 3948 π« cpw 4602 βͺ cuni 4908 dom cdm 5676 β cima 5679 βcfv 6543 Moorecmre 17531 mrClscmrc 17532 ACScacs 17534 Dirsetcdrs 18252 toInccipo 18485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-fz 13490 df-struct 17085 df-slot 17120 df-ndx 17132 df-base 17150 df-tset 17221 df-ple 17222 df-ocomp 17223 df-mre 17535 df-mrc 17536 df-acs 17538 df-proset 18253 df-drs 18254 df-poset 18271 df-ipo 18486 |
This theorem is referenced by: (None) |
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