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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 17783 twice with acsinfd 17782. 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 16919 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) |
6 | acsinfdimd.5 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝐼) | |
7 | 3, 5, 6 | mrissd 16899 | . . 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 17783 | . 2 ⊢ (𝜑 → 𝑆 ≼ 𝑇) |
11 | 3, 5, 4 | mrissd 16899 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
12 | 8 | eqcomd 2804 | . . 3 ⊢ (𝜑 → (𝑁‘𝑇) = (𝑁‘𝑆)) |
13 | 1, 2, 3, 4, 7, 8, 9 | acsinfd 17782 | . . 3 ⊢ (𝜑 → ¬ 𝑇 ∈ Fin) |
14 | 1, 2, 3, 6, 11, 12, 13 | acsdomd 17783 | . 2 ⊢ (𝜑 → 𝑇 ≼ 𝑆) |
15 | sbth 8621 | . 2 ⊢ ((𝑆 ≼ 𝑇 ∧ 𝑇 ≼ 𝑆) → 𝑆 ≈ 𝑇) | |
16 | 10, 14, 15 | syl2anc 587 | 1 ⊢ (𝜑 → 𝑆 ≈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 ≈ cen 8489 ≼ cdom 8490 Fincfn 8492 mrClscmrc 16846 mrIndcmri 16847 ACScacs 16848 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-reg 9040 ax-inf2 9088 ax-ac2 9874 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-oi 8958 df-r1 9177 df-rank 9178 df-card 9352 df-acn 9355 df-ac 9527 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-tset 16576 df-ple 16577 df-ocomp 16578 df-mre 16849 df-mrc 16850 df-mri 16851 df-acs 16852 df-proset 17530 df-drs 17531 df-poset 17548 df-ipo 17754 |
This theorem is referenced by: acsexdimd 17785 |
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