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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 16131. (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 16119 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
7 | dif0 3904 | . . . 4 ⊢ (𝑋 ∖ ∅) = 𝑋 | |
8 | 6, 7 | syl6sseqr 3615 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑋 ∖ ∅)) |
9 | mreexdomd.6 | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) | |
10 | 9, 7 | syl6sseqr 3615 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝑋 ∖ ∅)) |
11 | mreexdomd.5 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) | |
12 | un0 3919 | . . . . 5 ⊢ (𝑇 ∪ ∅) = 𝑇 | |
13 | 12 | fveq2i 6106 | . . . 4 ⊢ (𝑁‘(𝑇 ∪ ∅)) = (𝑁‘𝑇) |
14 | 11, 13 | syl6sseqr 3615 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘(𝑇 ∪ ∅))) |
15 | un0 3919 | . . . 4 ⊢ (𝑆 ∪ ∅) = 𝑆 | |
16 | 15, 5 | syl5eqel 2692 | . . 3 ⊢ (𝜑 → (𝑆 ∪ ∅) ∈ 𝐼) |
17 | mreexdomd.7 | . . 3 ⊢ (𝜑 → (𝑆 ∈ Fin ∨ 𝑇 ∈ Fin)) | |
18 | 1, 2, 3, 4, 8, 10, 14, 16, 17 | mreexexd 16131 | . 2 ⊢ (𝜑 → ∃𝑖 ∈ 𝒫 𝑇(𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼)) |
19 | simprrl 800 | . . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑆 ≈ 𝑖) | |
20 | simprl 790 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑖 ∈ 𝒫 𝑇) | |
21 | 20 | elpwid 4118 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑖 ⊆ 𝑇) |
22 | 1 | elfvexd 6132 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ V) |
23 | 22, 9 | ssexd 4733 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ V) |
24 | ssdomg 7887 | . . . . . 6 ⊢ (𝑇 ∈ V → (𝑖 ⊆ 𝑇 → 𝑖 ≼ 𝑇)) | |
25 | 23, 24 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑖 ⊆ 𝑇 → 𝑖 ≼ 𝑇)) |
26 | 25 | adantr 480 | . . . 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 7900 | . . 3 ⊢ ((𝑆 ≈ 𝑖 ∧ 𝑖 ≼ 𝑇) → 𝑆 ≼ 𝑇) | |
29 | 19, 27, 28 | syl2anc 691 | . 2 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑆 ≼ 𝑇) |
30 | 18, 29 | rexlimddv 3017 | 1 ⊢ (𝜑 → 𝑆 ≼ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 ‘cfv 5804 ≈ cen 7838 ≼ cdom 7839 Fincfn 7841 Moorecmre 16065 mrClscmrc 16066 mrIndcmri 16067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-mre 16069 df-mrc 16070 df-mri 16071 |
This theorem is referenced by: mreexfidimd 16134 |
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