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| Mirrors > Home > MPE Home > Th. List > mreexfidimd | Structured version Visualization version GIF version | ||
| Description: In a Moore system whose closure operator has the exchange property, if two independent sets have equal closure and one is finite, then they are equinumerous. Proven by using mreexdomd 17692 twice. This implies a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.) | 
| Ref | Expression | 
|---|---|
| mreexfidimd.1 | ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) | 
| mreexfidimd.2 | ⊢ 𝑁 = (mrCls‘𝐴) | 
| mreexfidimd.3 | ⊢ 𝐼 = (mrInd‘𝐴) | 
| mreexfidimd.4 | ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) | 
| mreexfidimd.5 | ⊢ (𝜑 → 𝑆 ∈ 𝐼) | 
| mreexfidimd.6 | ⊢ (𝜑 → 𝑇 ∈ 𝐼) | 
| mreexfidimd.7 | ⊢ (𝜑 → 𝑆 ∈ Fin) | 
| mreexfidimd.8 | ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) | 
| Ref | Expression | 
|---|---|
| mreexfidimd | ⊢ (𝜑 → 𝑆 ≈ 𝑇) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mreexfidimd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) | |
| 2 | mreexfidimd.2 | . . 3 ⊢ 𝑁 = (mrCls‘𝐴) | |
| 3 | mreexfidimd.3 | . . 3 ⊢ 𝐼 = (mrInd‘𝐴) | |
| 4 | mreexfidimd.4 | . . 3 ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) | |
| 5 | mreexfidimd.5 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝐼) | |
| 6 | 3, 1, 5 | mrissd 17679 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) | 
| 7 | 1, 2, 6 | mrcssidd 17668 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑆)) | 
| 8 | mreexfidimd.8 | . . . 4 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) | |
| 9 | 7, 8 | sseqtrd 4020 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) | 
| 10 | mreexfidimd.6 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝐼) | |
| 11 | 3, 1, 10 | mrissd 17679 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) | 
| 12 | mreexfidimd.7 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ Fin) | |
| 13 | 12 | orcd 874 | . . 3 ⊢ (𝜑 → (𝑆 ∈ Fin ∨ 𝑇 ∈ Fin)) | 
| 14 | 1, 2, 3, 4, 9, 11, 13, 5 | mreexdomd 17692 | . 2 ⊢ (𝜑 → 𝑆 ≼ 𝑇) | 
| 15 | 1, 2, 11 | mrcssidd 17668 | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ (𝑁‘𝑇)) | 
| 16 | 15, 8 | sseqtrrd 4021 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝑁‘𝑆)) | 
| 17 | 12 | olcd 875 | . . 3 ⊢ (𝜑 → (𝑇 ∈ Fin ∨ 𝑆 ∈ Fin)) | 
| 18 | 1, 2, 3, 4, 16, 6, 17, 10 | mreexdomd 17692 | . 2 ⊢ (𝜑 → 𝑇 ≼ 𝑆) | 
| 19 | sbth 9133 | . 2 ⊢ ((𝑆 ≼ 𝑇 ∧ 𝑇 ≼ 𝑆) → 𝑆 ≈ 𝑇) | |
| 20 | 14, 18, 19 | syl2anc 584 | 1 ⊢ (𝜑 → 𝑆 ≈ 𝑇) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∖ cdif 3948 ∪ cun 3949 𝒫 cpw 4600 {csn 4626 class class class wbr 5143 ‘cfv 6561 ≈ cen 8982 ≼ cdom 8983 Fincfn 8985 Moorecmre 17625 mrClscmrc 17626 mrIndcmri 17627 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-mre 17629 df-mrc 17630 df-mri 17631 | 
| This theorem is referenced by: acsexdimd 18604 lvecdimfi 33646 | 
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