Theorem sscntz 19338

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 19333	. . . 4 ⊢ (𝑇 ⊆ 𝐵 → (𝑍‘𝑇) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
5	4	sseq2d 3959	. . 3 ⊢ (𝑇 ⊆ 𝐵 → (𝑆 ⊆ (𝑍‘𝑇) ↔ 𝑆 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
6		ssrab 4015	. . 3 ⊢ (𝑆 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
7	5, 6	bitrdi 289	. 2 ⊢ (𝑇 ⊆ 𝐵 → (𝑆 ⊆ (𝑍‘𝑇) ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥))))
8		ibar 535	. . 3 ⊢ (𝑆 ⊆ 𝐵 → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥))))
9	8	bicomd 225	. 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 517	1 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑆 ⊆ (𝑍‘𝑇) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1550  ∀wral 3066  {crab 3404   ⊆ wss 3895  ‘cfv 6506  (class class class)co 7381  Basecbs 17217  +_gcplusg 17258  Cntzccntz 19327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-ov 7384  df-cntz 19329

This theorem is referenced by:  cntz2ss  19347  cntzrec  19348  submcmn2  19851  mplcoe5lem  22061  symgcntz  33215