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Theorem cntz2ss 18458
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 2801 . . . . . 6 (+g𝑀) = (+g𝑀)
2 cntzrec.z . . . . . 6 𝑍 = (Cntz‘𝑀)
31, 2cntzi 18454 . . . . 5 ((𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆) → (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
43ralrimiva 3152 . . . 4 (𝑥 ∈ (𝑍𝑆) → ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
5 ssralv 3984 . . . . 5 (𝑇𝑆 → (∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) → ∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
65adantl 485 . . . 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 3151 . 2 ((𝑆𝐵𝑇𝑆) → ∀𝑥 ∈ (𝑍𝑆)∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
9 cntzrec.b . . . 4 𝐵 = (Base‘𝑀)
109, 2cntzssv 18453 . . 3 (𝑍𝑆) ⊆ 𝐵
11 sstr 3926 . . . 4 ((𝑇𝑆𝑆𝐵) → 𝑇𝐵)
1211ancoms 462 . . 3 ((𝑆𝐵𝑇𝑆) → 𝑇𝐵)
139, 1, 2sscntz 18451 . . 3 (((𝑍𝑆) ⊆ 𝐵𝑇𝐵) → ((𝑍𝑆) ⊆ (𝑍𝑇) ↔ ∀𝑥 ∈ (𝑍𝑆)∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
1410, 12, 13sylancr 590 . 2 ((𝑆𝐵𝑇𝑆) → ((𝑍𝑆) ⊆ (𝑍𝑇) ↔ ∀𝑥 ∈ (𝑍𝑆)∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
158, 14mpbird 260 1 ((𝑆𝐵𝑇𝑆) → (𝑍𝑆) ⊆ (𝑍𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wral 3109  wss 3884  cfv 6328  (class class class)co 7139  Basecbs 16478  +gcplusg 16560  Cntzccntz 18440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-cntz 18442
This theorem is referenced by:  cntzidss  18463  gsumzadd  19038  dprdfadd  19138  dprdss  19147  dprd2da  19160  dmdprdsplit2lem  19163  cntzsdrg  19577
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