Theorem cntz2ss 18854

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	⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝑆) → (𝑍‘𝑆) ⊆ (𝑍‘𝑇))

Proof of Theorem cntz2ss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
2		cntzrec.z	. . . . . 6 ⊢ 𝑍 = (Cntz‘𝑀)
3	1, 2	cntzi 18850	. . . . 5 ⊢ ((𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆) → (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))
4	3	ralrimiva 3107	. . . 4 ⊢ (𝑥 ∈ (𝑍‘𝑆) → ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))
5		ssralv 3983	. . . . 5 ⊢ (𝑇 ⊆ 𝑆 → (∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥) → ∀𝑦 ∈ 𝑇 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)))
6	5	adantl 481	. . . 4 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝑆) → (∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥) → ∀𝑦 ∈ 𝑇 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)))
7	4, 6	syl5 34	. . 3 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝑆) → (𝑥 ∈ (𝑍‘𝑆) → ∀𝑦 ∈ 𝑇 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)))
8	7	ralrimiv 3106	. 2 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝑆) → ∀𝑥 ∈ (𝑍‘𝑆)∀𝑦 ∈ 𝑇 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))
9		cntzrec.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
10	9, 2	cntzssv 18849	. . 3 ⊢ (𝑍‘𝑆) ⊆ 𝐵
11		sstr 3925	. . . 4 ⊢ ((𝑇 ⊆ 𝑆 ∧ 𝑆 ⊆ 𝐵) → 𝑇 ⊆ 𝐵)
12	11	ancoms 458	. . 3 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝐵)
13	9, 1, 2	sscntz 18847	. . 3 ⊢ (((𝑍‘𝑆) ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → ((𝑍‘𝑆) ⊆ (𝑍‘𝑇) ↔ ∀𝑥 ∈ (𝑍‘𝑆)∀𝑦 ∈ 𝑇 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)))
14	10, 12, 13	sylancr 586	. 2 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝑆) → ((𝑍‘𝑆) ⊆ (𝑍‘𝑇) ↔ ∀𝑥 ∈ (𝑍‘𝑆)∀𝑦 ∈ 𝑇 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)))
15	8, 14	mpbird 256	1 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝑆) → (𝑍‘𝑆) ⊆ (𝑍‘𝑇))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ⊆ wss 3883  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  Cntzccntz 18836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-cntz 18838

This theorem is referenced by:  cntzidss  18859  gsumzadd  19438  dprdfadd  19538  dprdss  19547  dprd2da  19560  dmdprdsplit2lem  19563  cntzsdrg  19985