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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 16919 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 16906 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
7 | 1, 2, 6 | mrcssidd 16895 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑆)) |
8 | mreexfidimd.8 | . . . 4 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) | |
9 | 7, 8 | sseqtrd 4006 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) |
10 | mreexfidimd.6 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝐼) | |
11 | 3, 1, 10 | mrissd 16906 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) |
12 | mreexfidimd.7 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ Fin) | |
13 | 12 | orcd 869 | . . 3 ⊢ (𝜑 → (𝑆 ∈ Fin ∨ 𝑇 ∈ Fin)) |
14 | 1, 2, 3, 4, 9, 11, 13, 5 | mreexdomd 16919 | . 2 ⊢ (𝜑 → 𝑆 ≼ 𝑇) |
15 | 1, 2, 11 | mrcssidd 16895 | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ (𝑁‘𝑇)) |
16 | 15, 8 | sseqtrrd 4007 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝑁‘𝑆)) |
17 | 12 | olcd 870 | . . 3 ⊢ (𝜑 → (𝑇 ∈ Fin ∨ 𝑆 ∈ Fin)) |
18 | 1, 2, 3, 4, 16, 6, 17, 10 | mreexdomd 16919 | . 2 ⊢ (𝜑 → 𝑇 ≼ 𝑆) |
19 | sbth 8636 | . 2 ⊢ ((𝑆 ≼ 𝑇 ∧ 𝑇 ≼ 𝑆) → 𝑆 ≈ 𝑇) | |
20 | 14, 18, 19 | syl2anc 586 | 1 ⊢ (𝜑 → 𝑆 ≈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∖ cdif 3932 ∪ cun 3933 𝒫 cpw 4538 {csn 4566 class class class wbr 5065 ‘cfv 6354 ≈ cen 8505 ≼ cdom 8506 Fincfn 8508 Moorecmre 16852 mrClscmrc 16853 mrIndcmri 16854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-om 7580 df-1o 8101 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 df-mre 16856 df-mrc 16857 df-mri 16858 |
This theorem is referenced by: acsexdimd 17792 lvecdimfi 30998 |
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