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Mirrors > Home > MPE Home > Th. List > mreexdomd | Structured version Visualization version GIF version |
Description: In a Moore system whose closure operator has the exchange property, if 𝑆 is independent and contained in the closure of 𝑇, and either 𝑆 or 𝑇 is finite, then 𝑇 dominates 𝑆. This is an immediate consequence of mreexexd 16914. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
mreexdomd.1 | ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) |
mreexdomd.2 | ⊢ 𝑁 = (mrCls‘𝐴) |
mreexdomd.3 | ⊢ 𝐼 = (mrInd‘𝐴) |
mreexdomd.4 | ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) |
mreexdomd.5 | ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) |
mreexdomd.6 | ⊢ (𝜑 → 𝑇 ⊆ 𝑋) |
mreexdomd.7 | ⊢ (𝜑 → (𝑆 ∈ Fin ∨ 𝑇 ∈ Fin)) |
mreexdomd.8 | ⊢ (𝜑 → 𝑆 ∈ 𝐼) |
Ref | Expression |
---|---|
mreexdomd | ⊢ (𝜑 → 𝑆 ≼ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mreexdomd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) | |
2 | mreexdomd.2 | . . 3 ⊢ 𝑁 = (mrCls‘𝐴) | |
3 | mreexdomd.3 | . . 3 ⊢ 𝐼 = (mrInd‘𝐴) | |
4 | mreexdomd.4 | . . 3 ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) | |
5 | mreexdomd.8 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝐼) | |
6 | 3, 1, 5 | mrissd 16902 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
7 | dif0 4325 | . . . 4 ⊢ (𝑋 ∖ ∅) = 𝑋 | |
8 | 6, 7 | sseqtrrdi 4011 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑋 ∖ ∅)) |
9 | mreexdomd.6 | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) | |
10 | 9, 7 | sseqtrrdi 4011 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝑋 ∖ ∅)) |
11 | mreexdomd.5 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) | |
12 | un0 4337 | . . . . 5 ⊢ (𝑇 ∪ ∅) = 𝑇 | |
13 | 12 | fveq2i 6666 | . . . 4 ⊢ (𝑁‘(𝑇 ∪ ∅)) = (𝑁‘𝑇) |
14 | 11, 13 | sseqtrrdi 4011 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘(𝑇 ∪ ∅))) |
15 | un0 4337 | . . . 4 ⊢ (𝑆 ∪ ∅) = 𝑆 | |
16 | 15, 5 | eqeltrid 2916 | . . 3 ⊢ (𝜑 → (𝑆 ∪ ∅) ∈ 𝐼) |
17 | mreexdomd.7 | . . 3 ⊢ (𝜑 → (𝑆 ∈ Fin ∨ 𝑇 ∈ Fin)) | |
18 | 1, 2, 3, 4, 8, 10, 14, 16, 17 | mreexexd 16914 | . 2 ⊢ (𝜑 → ∃𝑖 ∈ 𝒫 𝑇(𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼)) |
19 | simprrl 779 | . . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑆 ≈ 𝑖) | |
20 | simprl 769 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑖 ∈ 𝒫 𝑇) | |
21 | 20 | elpwid 4543 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑖 ⊆ 𝑇) |
22 | 1 | elfvexd 6697 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ V) |
23 | 22, 9 | ssexd 5221 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ V) |
24 | ssdomg 8548 | . . . . . 6 ⊢ (𝑇 ∈ V → (𝑖 ⊆ 𝑇 → 𝑖 ≼ 𝑇)) | |
25 | 23, 24 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑖 ⊆ 𝑇 → 𝑖 ≼ 𝑇)) |
26 | 25 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → (𝑖 ⊆ 𝑇 → 𝑖 ≼ 𝑇)) |
27 | 21, 26 | mpd 15 | . . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑖 ≼ 𝑇) |
28 | endomtr 8560 | . . 3 ⊢ ((𝑆 ≈ 𝑖 ∧ 𝑖 ≼ 𝑇) → 𝑆 ≼ 𝑇) | |
29 | 19, 27, 28 | syl2anc 586 | . 2 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑆 ≼ 𝑇) |
30 | 18, 29 | rexlimddv 3290 | 1 ⊢ (𝜑 → 𝑆 ≼ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 ∀wral 3137 Vcvv 3491 ∖ cdif 3926 ∪ cun 3927 ⊆ wss 3929 ∅c0 4284 𝒫 cpw 4532 {csn 4560 class class class wbr 5059 ‘cfv 6348 ≈ cen 8499 ≼ cdom 8500 Fincfn 8502 Moorecmre 16848 mrClscmrc 16849 mrIndcmri 16850 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 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 7574 df-1o 8095 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-card 9361 df-mre 16852 df-mrc 16853 df-mri 16854 |
This theorem is referenced by: mreexfidimd 16916 |
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