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Mirrors > Home > MPE Home > Th. List > acsmap2d | Structured version Visualization version GIF version |
Description: In an algebraic closure system, if 𝑆 and 𝑇 have the same closure and 𝑆 is independent, then there is a map 𝑓 from 𝑇 into the set of finite subsets of 𝑆 such that 𝑆 equals the union of ran 𝑓. This is proven by taking the map 𝑓 from acsmapd 17783 and observing that, since 𝑆 and 𝑇 have the same closure, the closure of ∪ ran 𝑓 must contain 𝑆. Since 𝑆 is independent, by mrissmrcd 16906, ∪ ran 𝑓 must equal 𝑆. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
acsmap2d.1 | ⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋)) |
acsmap2d.2 | ⊢ 𝑁 = (mrCls‘𝐴) |
acsmap2d.3 | ⊢ 𝐼 = (mrInd‘𝐴) |
acsmap2d.4 | ⊢ (𝜑 → 𝑆 ∈ 𝐼) |
acsmap2d.5 | ⊢ (𝜑 → 𝑇 ⊆ 𝑋) |
acsmap2d.6 | ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) |
Ref | Expression |
---|---|
acsmap2d | ⊢ (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | acsmap2d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋)) | |
2 | acsmap2d.2 | . . 3 ⊢ 𝑁 = (mrCls‘𝐴) | |
3 | acsmap2d.3 | . . . 4 ⊢ 𝐼 = (mrInd‘𝐴) | |
4 | 1 | acsmred 16922 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) |
5 | acsmap2d.4 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐼) | |
6 | 3, 4, 5 | mrissd 16902 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
7 | acsmap2d.5 | . . . . 5 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) | |
8 | 4, 2, 7 | mrcssidd 16891 | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ (𝑁‘𝑇)) |
9 | acsmap2d.6 | . . . 4 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) | |
10 | 8, 9 | sseqtrrd 4001 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝑁‘𝑆)) |
11 | 1, 2, 6, 10 | acsmapd 17783 | . 2 ⊢ (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) |
12 | simprl 769 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin)) | |
13 | 4 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝐴 ∈ (Moore‘𝑋)) |
14 | 5 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑆 ∈ 𝐼) |
15 | 3, 13, 14 | mrissd 16902 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑆 ⊆ 𝑋) |
16 | 13, 2, 15 | mrcssidd 16891 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑆 ⊆ (𝑁‘𝑆)) |
17 | 9 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘𝑆) = (𝑁‘𝑇)) |
18 | simprr 771 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑇 ⊆ (𝑁‘∪ ran 𝑓)) | |
19 | 13, 2 | mrcssvd 16889 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘∪ ran 𝑓) ⊆ 𝑋) |
20 | 13, 2, 18, 19 | mrcssd 16890 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘𝑇) ⊆ (𝑁‘(𝑁‘∪ ran 𝑓))) |
21 | frn 6513 | . . . . . . . . . . . . . 14 ⊢ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ran 𝑓 ⊆ (𝒫 𝑆 ∩ Fin)) | |
22 | 21 | unissd 4841 | . . . . . . . . . . . . 13 ⊢ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ∪ ran 𝑓 ⊆ ∪ (𝒫 𝑆 ∩ Fin)) |
23 | unifpw 8820 | . . . . . . . . . . . . 13 ⊢ ∪ (𝒫 𝑆 ∩ Fin) = 𝑆 | |
24 | 22, 23 | sseqtrdi 4010 | . . . . . . . . . . . 12 ⊢ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ∪ ran 𝑓 ⊆ 𝑆) |
25 | 24 | ad2antrl 726 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → ∪ ran 𝑓 ⊆ 𝑆) |
26 | 25, 15 | sstrd 3970 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → ∪ ran 𝑓 ⊆ 𝑋) |
27 | 13, 2, 26 | mrcidmd 16892 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘(𝑁‘∪ ran 𝑓)) = (𝑁‘∪ ran 𝑓)) |
28 | 20, 27 | sseqtrd 4000 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘𝑇) ⊆ (𝑁‘∪ ran 𝑓)) |
29 | 17, 28 | eqsstrd 3998 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘𝑆) ⊆ (𝑁‘∪ ran 𝑓)) |
30 | 16, 29 | sstrd 3970 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑆 ⊆ (𝑁‘∪ ran 𝑓)) |
31 | 13, 2, 3, 30, 25, 14 | mrissmrcd 16906 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑆 = ∪ ran 𝑓) |
32 | 12, 31 | jca 514 | . . . 4 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓)) |
33 | 32 | ex 415 | . . 3 ⊢ (𝜑 → ((𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓)) → (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓))) |
34 | 33 | eximdv 1917 | . 2 ⊢ (𝜑 → (∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓)) → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓))) |
35 | 11, 34 | mpd 15 | 1 ⊢ (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∃wex 1779 ∈ wcel 2113 ∩ cin 3928 ⊆ wss 3929 𝒫 cpw 4532 ∪ cuni 4831 ran crn 5549 ⟶wf 6344 ‘cfv 6348 Fincfn 8502 Moorecmre 16848 mrClscmrc 16849 mrIndcmri 16850 ACScacs 16851 |
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-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-reg 9049 ax-inf2 9097 ax-ac2 9878 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
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-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 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-iun 4914 df-iin 4915 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-se 5508 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-pred 6141 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-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-r1 9186 df-rank 9187 df-card 9361 df-ac 9535 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12890 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-tset 16579 df-ple 16580 df-ocomp 16581 df-mre 16852 df-mrc 16853 df-mri 16854 df-acs 16855 df-proset 17533 df-drs 17534 df-poset 17551 df-ipo 17757 |
This theorem is referenced by: acsinfd 17785 acsdomd 17786 |
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