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Mirrors > Home > MPE Home > Th. List > acsdomd | Structured version Visualization version GIF version |
Description: In an algebraic closure system, if 𝑆 and 𝑇 have the same closure and 𝑆 is infinite independent, then 𝑇 dominates 𝑆. This follows from applying acsinfd 17784 and then applying unirnfdomd 9983 to the map given in acsmap2d 17783. 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)) |
acsinfd.7 | ⊢ (𝜑 → ¬ 𝑆 ∈ Fin) |
Ref | Expression |
---|---|
acsdomd | ⊢ (𝜑 → 𝑆 ≼ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | acsmap2d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋)) | |
2 | acsmap2d.2 | . . 3 ⊢ 𝑁 = (mrCls‘𝐴) | |
3 | acsmap2d.3 | . . 3 ⊢ 𝐼 = (mrInd‘𝐴) | |
4 | acsmap2d.4 | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝐼) | |
5 | acsmap2d.5 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) | |
6 | acsmap2d.6 | . . 3 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) | |
7 | 1, 2, 3, 4, 5, 6 | acsmap2d 17783 | . 2 ⊢ (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓)) |
8 | simprr 771 | . . 3 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓)) → 𝑆 = ∪ ran 𝑓) | |
9 | simprl 769 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓)) → 𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin)) | |
10 | inss2 4206 | . . . . 5 ⊢ (𝒫 𝑆 ∩ Fin) ⊆ Fin | |
11 | fss 6522 | . . . . 5 ⊢ ((𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ (𝒫 𝑆 ∩ Fin) ⊆ Fin) → 𝑓:𝑇⟶Fin) | |
12 | 9, 10, 11 | sylancl 588 | . . . 4 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓)) → 𝑓:𝑇⟶Fin) |
13 | acsinfd.7 | . . . . . 6 ⊢ (𝜑 → ¬ 𝑆 ∈ Fin) | |
14 | 1, 2, 3, 4, 5, 6, 13 | acsinfd 17784 | . . . . 5 ⊢ (𝜑 → ¬ 𝑇 ∈ Fin) |
15 | 14 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓)) → ¬ 𝑇 ∈ Fin) |
16 | 1 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓)) → 𝐴 ∈ (ACS‘𝑋)) |
17 | 16 | elfvexd 6699 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓)) → 𝑋 ∈ V) |
18 | 5 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓)) → 𝑇 ⊆ 𝑋) |
19 | 17, 18 | ssexd 5221 | . . . 4 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓)) → 𝑇 ∈ V) |
20 | 12, 15, 19 | unirnfdomd 9983 | . . 3 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓)) → ∪ ran 𝑓 ≼ 𝑇) |
21 | 8, 20 | eqbrtrd 5081 | . 2 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓)) → 𝑆 ≼ 𝑇) |
22 | 7, 21 | exlimddv 1932 | 1 ⊢ (𝜑 → 𝑆 ≼ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ∩ cin 3935 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4832 class class class wbr 5059 ran crn 5551 ⟶wf 6346 ‘cfv 6350 ≼ cdom 8501 Fincfn 8503 mrClscmrc 16848 mrIndcmri 16849 ACScacs 16850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-reg 9050 ax-inf2 9098 ax-ac2 9879 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-oi 8968 df-r1 9187 df-rank 9188 df-card 9362 df-acn 9365 df-ac 9536 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 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 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-tset 16578 df-ple 16579 df-ocomp 16580 df-mre 16851 df-mrc 16852 df-mri 16853 df-acs 16854 df-proset 17532 df-drs 17533 df-poset 17550 df-ipo 17756 |
This theorem is referenced by: acsinfdimd 17786 |
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