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Theorem cntz2ss 17811
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 2651 . . . . . 6 (+g𝑀) = (+g𝑀)
2 cntzrec.z . . . . . 6 𝑍 = (Cntz‘𝑀)
31, 2cntzi 17808 . . . . 5 ((𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆) → (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
43ralrimiva 2995 . . . 4 (𝑥 ∈ (𝑍𝑆) → ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
5 ssralv 3699 . . . . 5 (𝑇𝑆 → (∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) → ∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
65adantl 481 . . . 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 2994 . 2 ((𝑆𝐵𝑇𝑆) → ∀𝑥 ∈ (𝑍𝑆)∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
9 cntzrec.b . . . 4 𝐵 = (Base‘𝑀)
109, 2cntzssv 17807 . . 3 (𝑍𝑆) ⊆ 𝐵
11 sstr 3644 . . . 4 ((𝑇𝑆𝑆𝐵) → 𝑇𝐵)
1211ancoms 468 . . 3 ((𝑆𝐵𝑇𝑆) → 𝑇𝐵)
139, 1, 2sscntz 17805 . . 3 (((𝑍𝑆) ⊆ 𝐵𝑇𝐵) → ((𝑍𝑆) ⊆ (𝑍𝑇) ↔ ∀𝑥 ∈ (𝑍𝑆)∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
1410, 12, 13sylancr 696 . 2 ((𝑆𝐵𝑇𝑆) → ((𝑍𝑆) ⊆ (𝑍𝑇) ↔ ∀𝑥 ∈ (𝑍𝑆)∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
158, 14mpbird 247 1 ((𝑆𝐵𝑇𝑆) → (𝑍𝑆) ⊆ (𝑍𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wss 3607  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  Cntzccntz 17794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-cntz 17796
This theorem is referenced by:  cntzidss  17816  gsumzadd  18368  dprdfadd  18465  dprdss  18474  dprd2da  18487  dmdprdsplit2lem  18490  cntzsdrg  38089
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