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Theorem cntz2ss 19192
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 2733 . . . . . 6 (+g𝑀) = (+g𝑀)
2 cntzrec.z . . . . . 6 𝑍 = (Cntz‘𝑀)
31, 2cntzi 19187 . . . . 5 ((𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆) → (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
43ralrimiva 3147 . . . 4 (𝑥 ∈ (𝑍𝑆) → ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
5 ssralv 4049 . . . . 5 (𝑇𝑆 → (∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) → ∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
65adantl 483 . . . 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 3146 . 2 ((𝑆𝐵𝑇𝑆) → ∀𝑥 ∈ (𝑍𝑆)∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
9 cntzrec.b . . . 4 𝐵 = (Base‘𝑀)
109, 2cntzssv 19186 . . 3 (𝑍𝑆) ⊆ 𝐵
11 sstr 3989 . . . 4 ((𝑇𝑆𝑆𝐵) → 𝑇𝐵)
1211ancoms 460 . . 3 ((𝑆𝐵𝑇𝑆) → 𝑇𝐵)
139, 1, 2sscntz 19184 . . 3 (((𝑍𝑆) ⊆ 𝐵𝑇𝐵) → ((𝑍𝑆) ⊆ (𝑍𝑇) ↔ ∀𝑥 ∈ (𝑍𝑆)∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
1410, 12, 13sylancr 588 . 2 ((𝑆𝐵𝑇𝑆) → ((𝑍𝑆) ⊆ (𝑍𝑇) ↔ ∀𝑥 ∈ (𝑍𝑆)∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
158, 14mpbird 257 1 ((𝑆𝐵𝑇𝑆) → (𝑍𝑆) ⊆ (𝑍𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  wss 3947  cfv 6540  (class class class)co 7404  Basecbs 17140  +gcplusg 17193  Cntzccntz 19173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-cntz 19175
This theorem is referenced by:  cntzidss  19197  gsumzadd  19782  dprdfadd  19882  dprdss  19891  dprd2da  19904  dmdprdsplit2lem  19907  cntzsdrg  20406
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