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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 4400 | . . 3 ⊢ ∅ ⊆ 𝐵 | |
| 2 | sseq1 4009 | . . 3 ⊢ ((𝑍‘𝑆) = ∅ → ((𝑍‘𝑆) ⊆ 𝐵 ↔ ∅ ⊆ 𝐵)) | |
| 3 | 1, 2 | mpbiri 258 | . 2 ⊢ ((𝑍‘𝑆) = ∅ → (𝑍‘𝑆) ⊆ 𝐵) |
| 4 | n0 4353 | . . 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 19345 | . . . . . 6 ⊢ (𝑥 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵)) |
| 8 | eqid 2737 | . . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
| 9 | 5, 8, 6 | cntzval 19339 | . . . . . 6 ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)}) |
| 10 | 7, 9 | simpl2im 503 | . . . . 5 ⊢ (𝑥 ∈ (𝑍‘𝑆) → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)}) |
| 11 | ssrab2 4080 | . . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)} ⊆ 𝐵 | |
| 12 | 10, 11 | eqsstrdi 4028 | . . . 4 ⊢ (𝑥 ∈ (𝑍‘𝑆) → (𝑍‘𝑆) ⊆ 𝐵) |
| 13 | 12 | exlimiv 1930 | . . 3 ⊢ (∃𝑥 𝑥 ∈ (𝑍‘𝑆) → (𝑍‘𝑆) ⊆ 𝐵) |
| 14 | 4, 13 | sylbi 217 | . 2 ⊢ ((𝑍‘𝑆) ≠ ∅ → (𝑍‘𝑆) ⊆ 𝐵) |
| 15 | 3, 14 | pm2.61ine 3025 | 1 ⊢ (𝑍‘𝑆) ⊆ 𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 {crab 3436 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Cntzccntz 19333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-cntz 19335 |
| This theorem is referenced by: cntrss 19349 cntzsgrpcl 19352 cntz2ss 19353 cntzsubm 19356 cntzsubg 19357 cntzidss 19358 cntzmhm 19359 cntzmhm2 19360 cntzcmn 19858 cntzspan 19862 cntzsubrng 20567 cntzsubr 20606 cntzsdrg 20803 |
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