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| Mirrors > Home > MPE Home > Th. List > acsdomd | Structured version Visualization version GIF version | ||
| Description: In an algebraic closure system, if π and π have the same closure and π is infinite independent, then π dominates π. This follows from applying acsinfd 18514 and then applying unirnfdomd 10565 to the map given in acsmap2d 18513. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.) |
| Ref | Expression |
|---|---|
| acsmap2d.1 | β’ (π β π΄ β (ACSβπ)) |
| acsmap2d.2 | β’ π = (mrClsβπ΄) |
| acsmap2d.3 | β’ πΌ = (mrIndβπ΄) |
| acsmap2d.4 | β’ (π β π β πΌ) |
| acsmap2d.5 | β’ (π β π β π) |
| acsmap2d.6 | β’ (π β (πβπ) = (πβπ)) |
| acsinfd.7 | β’ (π β Β¬ π β Fin) |
| Ref | Expression |
|---|---|
| acsdomd | β’ (π β π βΌ π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | acsmap2d.1 | . . 3 β’ (π β π΄ β (ACSβπ)) | |
| 2 | acsmap2d.2 | . . 3 β’ π = (mrClsβπ΄) | |
| 3 | acsmap2d.3 | . . 3 β’ πΌ = (mrIndβπ΄) | |
| 4 | acsmap2d.4 | . . 3 β’ (π β π β πΌ) | |
| 5 | acsmap2d.5 | . . 3 β’ (π β π β π) | |
| 6 | acsmap2d.6 | . . 3 β’ (π β (πβπ) = (πβπ)) | |
| 7 | 1, 2, 3, 4, 5, 6 | acsmap2d 18513 | . 2 β’ (π β βπ(π:πβΆ(π« π β© Fin) β§ π = βͺ ran π)) |
| 8 | simprr 770 | . . 3 β’ ((π β§ (π:πβΆ(π« π β© Fin) β§ π = βͺ ran π)) β π = βͺ ran π) | |
| 9 | simprl 768 | . . . . 5 β’ ((π β§ (π:πβΆ(π« π β© Fin) β§ π = βͺ ran π)) β π:πβΆ(π« π β© Fin)) | |
| 10 | inss2 4229 | . . . . 5 β’ (π« π β© Fin) β Fin | |
| 11 | fss 6734 | . . . . 5 β’ ((π:πβΆ(π« π β© Fin) β§ (π« π β© Fin) β Fin) β π:πβΆFin) | |
| 12 | 9, 10, 11 | sylancl 585 | . . . 4 β’ ((π β§ (π:πβΆ(π« π β© Fin) β§ π = βͺ ran π)) β π:πβΆFin) |
| 13 | acsinfd.7 | . . . . . 6 β’ (π β Β¬ π β Fin) | |
| 14 | 1, 2, 3, 4, 5, 6, 13 | acsinfd 18514 | . . . . 5 β’ (π β Β¬ π β Fin) |
| 15 | 14 | adantr 480 | . . . 4 β’ ((π β§ (π:πβΆ(π« π β© Fin) β§ π = βͺ ran π)) β Β¬ π β Fin) |
| 16 | 1 | adantr 480 | . . . . . 6 β’ ((π β§ (π:πβΆ(π« π β© Fin) β§ π = βͺ ran π)) β π΄ β (ACSβπ)) |
| 17 | 16 | elfvexd 6930 | . . . . 5 β’ ((π β§ (π:πβΆ(π« π β© Fin) β§ π = βͺ ran π)) β π β V) |
| 18 | 5 | adantr 480 | . . . . 5 β’ ((π β§ (π:πβΆ(π« π β© Fin) β§ π = βͺ ran π)) β π β π) |
| 19 | 17, 18 | ssexd 5324 | . . . 4 β’ ((π β§ (π:πβΆ(π« π β© Fin) β§ π = βͺ ran π)) β π β V) |
| 20 | 12, 15, 19 | unirnfdomd 10565 | . . 3 β’ ((π β§ (π:πβΆ(π« π β© Fin) β§ π = βͺ ran π)) β βͺ ran π βΌ π) |
| 21 | 8, 20 | eqbrtrd 5170 | . 2 β’ ((π β§ (π:πβΆ(π« π β© Fin) β§ π = βͺ ran π)) β π βΌ π) |
| 22 | 7, 21 | exlimddv 1937 | 1 β’ (π β π βΌ π) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 Vcvv 3473 β© cin 3947 β wss 3948 π« cpw 4602 βͺ cuni 4908 class class class wbr 5148 ran crn 5677 βΆwf 6539 βcfv 6543 βΌ cdom 8940 Fincfn 8942 mrClscmrc 17532 mrIndcmri 17533 ACScacs 17534 |
| 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-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 ax-reg 9590 ax-inf2 9639 ax-ac2 10461 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 |
| 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-rmo 3375 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-iin 5000 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-se 5632 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-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-oi 9508 df-r1 9762 df-rank 9763 df-card 9937 df-acn 9940 df-ac 10114 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-mri 17537 df-acs 17538 df-proset 18253 df-drs 18254 df-poset 18271 df-ipo 18486 |
| This theorem is referenced by: acsinfdimd 18516 |
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