Theorem cntz2ss 19318

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 2735	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
2		cntzrec.z	. . . . . 6 ⊢ 𝑍 = (Cntz‘𝑀)
3	1, 2	cntzi 19312	. . . . 5 ⊢ ((𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆) → (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))
4	3	ralrimiva 3132	. . . 4 ⊢ (𝑥 ∈ (𝑍‘𝑆) → ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))
5		ssralv 4027	. . . . 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 3131	. 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 19311	. . 3 ⊢ (𝑍‘𝑆) ⊆ 𝐵
11		sstr 3967	. . . 4 ⊢ ((𝑇 ⊆ 𝑆 ∧ 𝑆 ⊆ 𝐵) → 𝑇 ⊆ 𝐵)
12	11	ancoms 458	. . 3 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝐵)
13	9, 1, 2	sscntz 19309	. . 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 587	. 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 257	1 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝑆) → (𝑍‘𝑆) ⊆ (𝑍‘𝑇))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051   ⊆ wss 3926  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  +_gcplusg 17271  Cntzccntz 19298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-cntz 19300

This theorem is referenced by:  cntzidss  19323  gsumzadd  19903  dprdfadd  20003  dprdss  20012  dprd2da  20025  dmdprdsplit2lem  20028  cntzsdrg  20762