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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 17123. (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 17111 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
7 | dif0 4277 | . . . 4 ⊢ (𝑋 ∖ ∅) = 𝑋 | |
8 | 6, 7 | sseqtrrdi 3942 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑋 ∖ ∅)) |
9 | mreexdomd.6 | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) | |
10 | 9, 7 | sseqtrrdi 3942 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝑋 ∖ ∅)) |
11 | mreexdomd.5 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) | |
12 | un0 4295 | . . . . 5 ⊢ (𝑇 ∪ ∅) = 𝑇 | |
13 | 12 | fveq2i 6709 | . . . 4 ⊢ (𝑁‘(𝑇 ∪ ∅)) = (𝑁‘𝑇) |
14 | 11, 13 | sseqtrrdi 3942 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘(𝑇 ∪ ∅))) |
15 | un0 4295 | . . . 4 ⊢ (𝑆 ∪ ∅) = 𝑆 | |
16 | 15, 5 | eqeltrid 2838 | . . 3 ⊢ (𝜑 → (𝑆 ∪ ∅) ∈ 𝐼) |
17 | mreexdomd.7 | . . 3 ⊢ (𝜑 → (𝑆 ∈ Fin ∨ 𝑇 ∈ Fin)) | |
18 | 1, 2, 3, 4, 8, 10, 14, 16, 17 | mreexexd 17123 | . 2 ⊢ (𝜑 → ∃𝑖 ∈ 𝒫 𝑇(𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼)) |
19 | simprrl 781 | . . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑆 ≈ 𝑖) | |
20 | simprl 771 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑖 ∈ 𝒫 𝑇) | |
21 | 20 | elpwid 4514 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑖 ⊆ 𝑇) |
22 | 1 | elfvexd 6740 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ V) |
23 | 22, 9 | ssexd 5206 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ V) |
24 | ssdomg 8663 | . . . . . 6 ⊢ (𝑇 ∈ V → (𝑖 ⊆ 𝑇 → 𝑖 ≼ 𝑇)) | |
25 | 23, 24 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑖 ⊆ 𝑇 → 𝑖 ≼ 𝑇)) |
26 | 25 | adantr 484 | . . . 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 8675 | . . 3 ⊢ ((𝑆 ≈ 𝑖 ∧ 𝑖 ≼ 𝑇) → 𝑆 ≼ 𝑇) | |
29 | 19, 27, 28 | syl2anc 587 | . 2 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑆 ≼ 𝑇) |
30 | 18, 29 | rexlimddv 3203 | 1 ⊢ (𝜑 → 𝑆 ≼ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2110 ∀wral 3054 Vcvv 3401 ∖ cdif 3854 ∪ cun 3855 ⊆ wss 3857 ∅c0 4227 𝒫 cpw 4503 {csn 4531 class class class wbr 5043 ‘cfv 6369 ≈ cen 8612 ≼ cdom 8613 Fincfn 8615 Moorecmre 17057 mrClscmrc 17058 mrIndcmri 17059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-om 7634 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-card 9538 df-mre 17061 df-mrc 17062 df-mri 17063 |
This theorem is referenced by: mreexfidimd 17125 |
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