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Theorem cntz2ss 18028
Description: Centralizers reverse the subset relation. (Contributed by Mario Carneiro, 3-Oct-2015.)
Hypotheses
Ref Expression
cntzrec.b 𝐵 = (Base‘𝑀)
cntzrec.z 𝑍 = (Cntz‘𝑀)
Assertion
Ref Expression
cntz2ss ((𝑆𝐵𝑇𝑆) → (𝑍𝑆) ⊆ (𝑍𝑇))

Proof of Theorem cntz2ss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2765 . . . . . 6 (+g𝑀) = (+g𝑀)
2 cntzrec.z . . . . . 6 𝑍 = (Cntz‘𝑀)
31, 2cntzi 18025 . . . . 5 ((𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆) → (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
43ralrimiva 3113 . . . 4 (𝑥 ∈ (𝑍𝑆) → ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
5 ssralv 3826 . . . . 5 (𝑇𝑆 → (∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) → ∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
65adantl 473 . . . 4 ((𝑆𝐵𝑇𝑆) → (∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) → ∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
74, 6syl5 34 . . 3 ((𝑆𝐵𝑇𝑆) → (𝑥 ∈ (𝑍𝑆) → ∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
87ralrimiv 3112 . 2 ((𝑆𝐵𝑇𝑆) → ∀𝑥 ∈ (𝑍𝑆)∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
9 cntzrec.b . . . 4 𝐵 = (Base‘𝑀)
109, 2cntzssv 18024 . . 3 (𝑍𝑆) ⊆ 𝐵
11 sstr 3769 . . . 4 ((𝑇𝑆𝑆𝐵) → 𝑇𝐵)
1211ancoms 450 . . 3 ((𝑆𝐵𝑇𝑆) → 𝑇𝐵)
139, 1, 2sscntz 18022 . . 3 (((𝑍𝑆) ⊆ 𝐵𝑇𝐵) → ((𝑍𝑆) ⊆ (𝑍𝑇) ↔ ∀𝑥 ∈ (𝑍𝑆)∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
1410, 12, 13sylancr 581 . 2 ((𝑆𝐵𝑇𝑆) → ((𝑍𝑆) ⊆ (𝑍𝑇) ↔ ∀𝑥 ∈ (𝑍𝑆)∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
158, 14mpbird 248 1 ((𝑆𝐵𝑇𝑆) → (𝑍𝑆) ⊆ (𝑍𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wss 3732  cfv 6068  (class class class)co 6842  Basecbs 16130  +gcplusg 16214  Cntzccntz 18011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-cntz 18013
This theorem is referenced by:  cntzidss  18033  gsumzadd  18588  dprdfadd  18686  dprdss  18695  dprd2da  18708  dmdprdsplit2lem  18711  cntzsdrg  38449
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