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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 16921. (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 16909 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
7 | dif0 4334 | . . . 4 ⊢ (𝑋 ∖ ∅) = 𝑋 | |
8 | 6, 7 | sseqtrrdi 4020 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑋 ∖ ∅)) |
9 | mreexdomd.6 | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) | |
10 | 9, 7 | sseqtrrdi 4020 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝑋 ∖ ∅)) |
11 | mreexdomd.5 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) | |
12 | un0 4346 | . . . . 5 ⊢ (𝑇 ∪ ∅) = 𝑇 | |
13 | 12 | fveq2i 6675 | . . . 4 ⊢ (𝑁‘(𝑇 ∪ ∅)) = (𝑁‘𝑇) |
14 | 11, 13 | sseqtrrdi 4020 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘(𝑇 ∪ ∅))) |
15 | un0 4346 | . . . 4 ⊢ (𝑆 ∪ ∅) = 𝑆 | |
16 | 15, 5 | eqeltrid 2919 | . . 3 ⊢ (𝜑 → (𝑆 ∪ ∅) ∈ 𝐼) |
17 | mreexdomd.7 | . . 3 ⊢ (𝜑 → (𝑆 ∈ Fin ∨ 𝑇 ∈ Fin)) | |
18 | 1, 2, 3, 4, 8, 10, 14, 16, 17 | mreexexd 16921 | . 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 4552 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑖 ⊆ 𝑇) |
22 | 1 | elfvexd 6706 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ V) |
23 | 22, 9 | ssexd 5230 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ V) |
24 | ssdomg 8557 | . . . . . 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 8569 | . . 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 3293 | 1 ⊢ (𝜑 → 𝑆 ≼ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ∖ cdif 3935 ∪ cun 3936 ⊆ wss 3938 ∅c0 4293 𝒫 cpw 4541 {csn 4569 class class class wbr 5068 ‘cfv 6357 ≈ cen 8508 ≼ cdom 8509 Fincfn 8511 Moorecmre 16855 mrClscmrc 16856 mrIndcmri 16857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-1o 8104 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-mre 16859 df-mrc 16860 df-mri 16861 |
This theorem is referenced by: mreexfidimd 16923 |
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