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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 16662 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 16649 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
7 | 1, 2, 6 | mrcssidd 16638 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑆)) |
8 | mreexfidimd.8 | . . . 4 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) | |
9 | 7, 8 | sseqtrd 3866 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) |
10 | mreexfidimd.6 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝐼) | |
11 | 3, 1, 10 | mrissd 16649 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) |
12 | mreexfidimd.7 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ Fin) | |
13 | 12 | orcd 906 | . . 3 ⊢ (𝜑 → (𝑆 ∈ Fin ∨ 𝑇 ∈ Fin)) |
14 | 1, 2, 3, 4, 9, 11, 13, 5 | mreexdomd 16662 | . 2 ⊢ (𝜑 → 𝑆 ≼ 𝑇) |
15 | 1, 2, 11 | mrcssidd 16638 | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ (𝑁‘𝑇)) |
16 | 15, 8 | sseqtr4d 3867 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝑁‘𝑆)) |
17 | 12 | olcd 907 | . . 3 ⊢ (𝜑 → (𝑇 ∈ Fin ∨ 𝑆 ∈ Fin)) |
18 | 1, 2, 3, 4, 16, 6, 17, 10 | mreexdomd 16662 | . 2 ⊢ (𝜑 → 𝑇 ≼ 𝑆) |
19 | sbth 8349 | . 2 ⊢ ((𝑆 ≼ 𝑇 ∧ 𝑇 ≼ 𝑆) → 𝑆 ≈ 𝑇) | |
20 | 14, 18, 19 | syl2anc 581 | 1 ⊢ (𝜑 → 𝑆 ≈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ∀wral 3117 ∖ cdif 3795 ∪ cun 3796 𝒫 cpw 4378 {csn 4397 class class class wbr 4873 ‘cfv 6123 ≈ cen 8219 ≼ cdom 8220 Fincfn 8222 Moorecmre 16595 mrClscmrc 16596 mrIndcmri 16597 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 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 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-om 7327 df-1o 7826 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-card 9078 df-mre 16599 df-mrc 16600 df-mri 16601 |
This theorem is referenced by: acsexdimd 17536 |
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