Theorem cntz2ss 19305

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 2737	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
2		cntzrec.z	. . . . . 6 ⊢ 𝑍 = (Cntz‘𝑀)
3	1, 2	cntzi 19299	. . . . 5 ⊢ ((𝑥 ∈ (𝑍‘𝑆) ∧ 𝑦 ∈ 𝑆) → (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))
4	3	ralrimiva 3130	. . . 4 ⊢ (𝑥 ∈ (𝑍‘𝑆) → ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))
5		ssralv 3991	. . . . 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 3129	. 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 19298	. . 3 ⊢ (𝑍‘𝑆) ⊆ 𝐵
11		sstr 3931	. . . 4 ⊢ ((𝑇 ⊆ 𝑆 ∧ 𝑆 ⊆ 𝐵) → 𝑇 ⊆ 𝐵)
12	11	ancoms 458	. . 3 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝐵)
13	9, 1, 2	sscntz 19296	. . 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 588	. 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 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3890  ‘cfv 6494  (class class class)co 7362  Basecbs 17174  +_gcplusg 17215  Cntzccntz 19285

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-cntz 19287

This theorem is referenced by:  cntzidss  19310  gsumzadd  19892  dprdfadd  19992  dprdss  20001  dprd2da  20014  dmdprdsplit2lem  20017  cntzsdrg  20774