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Mirrors > Home > MPE Home > Th. List > acsinfdimd | Structured version Visualization version GIF version |
Description: In an algebraic closure system, if two independent sets have equal closure and one is infinite, then they are equinumerous. This is proven by using acsdomd 17541 twice with acsinfd 17540. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
acsinfdimd.1 | ⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋)) |
acsinfdimd.2 | ⊢ 𝑁 = (mrCls‘𝐴) |
acsinfdimd.3 | ⊢ 𝐼 = (mrInd‘𝐴) |
acsinfdimd.4 | ⊢ (𝜑 → 𝑆 ∈ 𝐼) |
acsinfdimd.5 | ⊢ (𝜑 → 𝑇 ∈ 𝐼) |
acsinfdimd.6 | ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) |
acsinfdimd.7 | ⊢ (𝜑 → ¬ 𝑆 ∈ Fin) |
Ref | Expression |
---|---|
acsinfdimd | ⊢ (𝜑 → 𝑆 ≈ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | acsinfdimd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋)) | |
2 | acsinfdimd.2 | . . 3 ⊢ 𝑁 = (mrCls‘𝐴) | |
3 | acsinfdimd.3 | . . 3 ⊢ 𝐼 = (mrInd‘𝐴) | |
4 | acsinfdimd.4 | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝐼) | |
5 | 1 | acsmred 16676 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) |
6 | acsinfdimd.5 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝐼) | |
7 | 3, 5, 6 | mrissd 16656 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) |
8 | acsinfdimd.6 | . . 3 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) | |
9 | acsinfdimd.7 | . . 3 ⊢ (𝜑 → ¬ 𝑆 ∈ Fin) | |
10 | 1, 2, 3, 4, 7, 8, 9 | acsdomd 17541 | . 2 ⊢ (𝜑 → 𝑆 ≼ 𝑇) |
11 | 3, 5, 4 | mrissd 16656 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
12 | 8 | eqcomd 2831 | . . 3 ⊢ (𝜑 → (𝑁‘𝑇) = (𝑁‘𝑆)) |
13 | 1, 2, 3, 4, 7, 8, 9 | acsinfd 17540 | . . 3 ⊢ (𝜑 → ¬ 𝑇 ∈ Fin) |
14 | 1, 2, 3, 6, 11, 12, 13 | acsdomd 17541 | . 2 ⊢ (𝜑 → 𝑇 ≼ 𝑆) |
15 | sbth 8355 | . 2 ⊢ ((𝑆 ≼ 𝑇 ∧ 𝑇 ≼ 𝑆) → 𝑆 ≈ 𝑇) | |
16 | 10, 14, 15 | syl2anc 579 | 1 ⊢ (𝜑 → 𝑆 ≈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1656 ∈ wcel 2164 class class class wbr 4875 ‘cfv 6127 ≈ cen 8225 ≼ cdom 8226 Fincfn 8228 mrClscmrc 16603 mrIndcmri 16604 ACScacs 16605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-reg 8773 ax-inf2 8822 ax-ac2 9607 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-oi 8691 df-r1 8911 df-rank 8912 df-card 9085 df-acn 9088 df-ac 9259 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-tset 16331 df-ple 16332 df-ocomp 16333 df-mre 16606 df-mrc 16607 df-mri 16608 df-acs 16609 df-proset 17288 df-drs 17289 df-poset 17306 df-ipo 17512 |
This theorem is referenced by: acsexdimd 17543 |
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