Theorem sscntz 17956

Description: A centralizer expression for two sets elementwise commuting. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Hypotheses

Ref	Expression
cntzfval.b	⊢ 𝐵 = (Base‘𝑀)
cntzfval.p	⊢ + = (+g‘𝑀)
cntzfval.z	⊢ 𝑍 = (Cntz‘𝑀)

Assertion

Ref	Expression
sscntz	⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑆 ⊆ (𝑍‘𝑇) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))

Distinct variable groups:   𝑥,𝑦, +   𝑥,𝐵   𝑥,𝑀,𝑦   𝑥,𝑇,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝐵(𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem sscntz

Step	Hyp	Ref	Expression
1		cntzfval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑀)
2		cntzfval.p	. . . . 5 ⊢ + = (+g‘𝑀)
3		cntzfval.z	. . . . 5 ⊢ 𝑍 = (Cntz‘𝑀)
4	1, 2, 3	cntzval 17951	. . . 4 ⊢ (𝑇 ⊆ 𝐵 → (𝑍‘𝑇) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
5	4	sseq2d 3830	. . 3 ⊢ (𝑇 ⊆ 𝐵 → (𝑆 ⊆ (𝑍‘𝑇) ↔ 𝑆 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
6		ssrab 3877	. . 3 ⊢ (𝑆 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
7	5, 6	syl6bb 278	. 2 ⊢ (𝑇 ⊆ 𝐵 → (𝑆 ⊆ (𝑍‘𝑇) ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥))))
8		ibar 520	. . 3 ⊢ (𝑆 ⊆ 𝐵 → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥))))
9	8	bicomd 214	. 2 ⊢ (𝑆 ⊆ 𝐵 → ((𝑆 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
10	7, 9	sylan9bbr 502	1 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑆 ⊆ (𝑍‘𝑇) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   = wceq 1637  ∀wral 3096  {crab 3100   ⊆ wss 3769  ‘cfv 6097  (class class class)co 6870  Basecbs 16064  +_gcplusg 16149  Cntzccntz 17945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-ov 6873  df-cntz 17947

This theorem is referenced by:  cntz2ss  17962  cntzrec  17963  submcmn2  18441  mplcoe5lem  19672