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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 3924 | . . 3 ⊢ ∅ ⊆ 𝐵 | |
2 | sseq1 3589 | . . 3 ⊢ ((𝑍‘𝑆) = ∅ → ((𝑍‘𝑆) ⊆ 𝐵 ↔ ∅ ⊆ 𝐵)) | |
3 | 1, 2 | mpbiri 247 | . 2 ⊢ ((𝑍‘𝑆) = ∅ → (𝑍‘𝑆) ⊆ 𝐵) |
4 | n0 3890 | . . 3 ⊢ ((𝑍‘𝑆) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑍‘𝑆)) | |
5 | cntzrcl.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑀) | |
6 | cntzrcl.z | . . . . . . . 8 ⊢ 𝑍 = (Cntz‘𝑀) | |
7 | 5, 6 | cntzrcl 17583 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵)) |
8 | 7 | simprd 478 | . . . . . 6 ⊢ (𝑥 ∈ (𝑍‘𝑆) → 𝑆 ⊆ 𝐵) |
9 | eqid 2610 | . . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
10 | 5, 9, 6 | cntzval 17577 | . . . . . 6 ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)}) |
11 | 8, 10 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ (𝑍‘𝑆) → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)}) |
12 | ssrab2 3650 | . . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)} ⊆ 𝐵 | |
13 | 11, 12 | syl6eqss 3618 | . . . 4 ⊢ (𝑥 ∈ (𝑍‘𝑆) → (𝑍‘𝑆) ⊆ 𝐵) |
14 | 13 | exlimiv 1845 | . . 3 ⊢ (∃𝑥 𝑥 ∈ (𝑍‘𝑆) → (𝑍‘𝑆) ⊆ 𝐵) |
15 | 4, 14 | sylbi 206 | . 2 ⊢ ((𝑍‘𝑆) ≠ ∅ → (𝑍‘𝑆) ⊆ 𝐵) |
16 | 3, 15 | pm2.61ine 2865 | 1 ⊢ (𝑍‘𝑆) ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 {crab 2900 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Cntzccntz 17571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-cntz 17573 |
This theorem is referenced by: cntz2ss 17588 cntzsubm 17591 cntzsubg 17592 cntzidss 17593 cntzmhm 17594 cntzmhm2 17595 cntzcmn 18068 cntzspan 18070 cntzsubr 18635 cntzsdrg 36791 |
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