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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 4349 | . . 3 ⊢ ∅ ⊆ 𝐵 | |
2 | sseq1 3991 | . . 3 ⊢ ((𝑍‘𝑆) = ∅ → ((𝑍‘𝑆) ⊆ 𝐵 ↔ ∅ ⊆ 𝐵)) | |
3 | 1, 2 | mpbiri 260 | . 2 ⊢ ((𝑍‘𝑆) = ∅ → (𝑍‘𝑆) ⊆ 𝐵) |
4 | n0 4309 | . . 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 18451 | . . . . . 6 ⊢ (𝑥 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵)) |
8 | eqid 2821 | . . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
9 | 5, 8, 6 | cntzval 18445 | . . . . . 6 ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)}) |
10 | 7, 9 | simpl2im 506 | . . . . 5 ⊢ (𝑥 ∈ (𝑍‘𝑆) → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)}) |
11 | ssrab2 4055 | . . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)} ⊆ 𝐵 | |
12 | 10, 11 | eqsstrdi 4020 | . . . 4 ⊢ (𝑥 ∈ (𝑍‘𝑆) → (𝑍‘𝑆) ⊆ 𝐵) |
13 | 12 | exlimiv 1927 | . . 3 ⊢ (∃𝑥 𝑥 ∈ (𝑍‘𝑆) → (𝑍‘𝑆) ⊆ 𝐵) |
14 | 4, 13 | sylbi 219 | . 2 ⊢ ((𝑍‘𝑆) ≠ ∅ → (𝑍‘𝑆) ⊆ 𝐵) |
15 | 3, 14 | pm2.61ine 3100 | 1 ⊢ (𝑍‘𝑆) ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∃wex 1776 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 {crab 3142 Vcvv 3494 ⊆ wss 3935 ∅c0 4290 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 Cntzccntz 18439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-cntz 18441 |
This theorem is referenced by: cntrss 18454 cntz2ss 18457 cntzsubm 18460 cntzsubg 18461 cntzidss 18462 cntzmhm 18463 cntzmhm2 18464 cntzcmn 18954 cntzspan 18958 cntzsubr 19562 cntzsdrg 19575 |
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