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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 16912 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 16899 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
7 | 1, 2, 6 | mrcssidd 16888 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑆)) |
8 | mreexfidimd.8 | . . . 4 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) | |
9 | 7, 8 | sseqtrd 4005 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) |
10 | mreexfidimd.6 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝐼) | |
11 | 3, 1, 10 | mrissd 16899 | . . 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 16912 | . 2 ⊢ (𝜑 → 𝑆 ≼ 𝑇) |
15 | 1, 2, 11 | mrcssidd 16888 | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ (𝑁‘𝑇)) |
16 | 15, 8 | sseqtrrd 4006 | . . 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 16912 | . 2 ⊢ (𝜑 → 𝑇 ≼ 𝑆) |
19 | sbth 8629 | . 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 1531 ∈ wcel 2108 ∀wral 3136 ∖ cdif 3931 ∪ cun 3932 𝒫 cpw 4537 {csn 4559 class class class wbr 5057 ‘cfv 6348 ≈ cen 8498 ≼ cdom 8499 Fincfn 8501 Moorecmre 16845 mrClscmrc 16846 mrIndcmri 16847 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7573 df-1o 8094 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-card 9360 df-mre 16849 df-mrc 16850 df-mri 16851 |
This theorem is referenced by: acsexdimd 17785 lvecdimfi 30991 |
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