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Mirrors > Home > MPE Home > Th. List > cntzssv | Structured version Visualization version GIF version |
Description: The centralizer is unconditionally a subset. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Ref | Expression |
---|---|
cntzrcl.b | ⊢ 𝐵 = (Base‘𝑀) |
cntzrcl.z | ⊢ 𝑍 = (Cntz‘𝑀) |
Ref | Expression |
---|---|
cntzssv | ⊢ (𝑍‘𝑆) ⊆ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4355 | . . 3 ⊢ ∅ ⊆ 𝐵 | |
2 | sseq1 3968 | . . 3 ⊢ ((𝑍‘𝑆) = ∅ → ((𝑍‘𝑆) ⊆ 𝐵 ↔ ∅ ⊆ 𝐵)) | |
3 | 1, 2 | mpbiri 257 | . 2 ⊢ ((𝑍‘𝑆) = ∅ → (𝑍‘𝑆) ⊆ 𝐵) |
4 | n0 4305 | . . 3 ⊢ ((𝑍‘𝑆) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑍‘𝑆)) | |
5 | cntzrcl.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑀) | |
6 | cntzrcl.z | . . . . . . 7 ⊢ 𝑍 = (Cntz‘𝑀) | |
7 | 5, 6 | cntzrcl 19103 | . . . . . 6 ⊢ (𝑥 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵)) |
8 | eqid 2736 | . . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
9 | 5, 8, 6 | cntzval 19097 | . . . . . 6 ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)}) |
10 | 7, 9 | simpl2im 504 | . . . . 5 ⊢ (𝑥 ∈ (𝑍‘𝑆) → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)}) |
11 | ssrab2 4036 | . . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)} ⊆ 𝐵 | |
12 | 10, 11 | eqsstrdi 3997 | . . . 4 ⊢ (𝑥 ∈ (𝑍‘𝑆) → (𝑍‘𝑆) ⊆ 𝐵) |
13 | 12 | exlimiv 1933 | . . 3 ⊢ (∃𝑥 𝑥 ∈ (𝑍‘𝑆) → (𝑍‘𝑆) ⊆ 𝐵) |
14 | 4, 13 | sylbi 216 | . 2 ⊢ ((𝑍‘𝑆) ≠ ∅ → (𝑍‘𝑆) ⊆ 𝐵) |
15 | 3, 14 | pm2.61ine 3027 | 1 ⊢ (𝑍‘𝑆) ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2942 ∀wral 3063 {crab 3406 Vcvv 3444 ⊆ wss 3909 ∅c0 4281 ‘cfv 6494 (class class class)co 7354 Basecbs 17080 +gcplusg 17130 Cntzccntz 19091 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-ov 7357 df-cntz 19093 |
This theorem is referenced by: cntrss 19106 cntz2ss 19109 cntzsubm 19112 cntzsubg 19113 cntzidss 19114 cntzmhm 19115 cntzmhm2 19116 cntzcmn 19614 cntzspan 19618 cntzsubr 20251 cntzsdrg 20265 |
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