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Mirrors > Home > MPE Home > Th. List > cntzssv | Structured version Visualization version Unicode version |
Description: The centralizer is unconditionally a subset. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Ref | Expression |
---|---|
cntzrcl.b | |
cntzrcl.z | Cntz |
Ref | Expression |
---|---|
cntzssv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 3972 | . . 3 | |
2 | sseq1 3626 | . . 3 | |
3 | 1, 2 | mpbiri 248 | . 2 |
4 | n0 3931 | . . 3 | |
5 | cntzrcl.b | . . . . . . . 8 | |
6 | cntzrcl.z | . . . . . . . 8 Cntz | |
7 | 5, 6 | cntzrcl 17760 | . . . . . . 7 |
8 | 7 | simprd 479 | . . . . . 6 |
9 | eqid 2622 | . . . . . . 7 | |
10 | 5, 9, 6 | cntzval 17754 | . . . . . 6 |
11 | 8, 10 | syl 17 | . . . . 5 |
12 | ssrab2 3687 | . . . . 5 | |
13 | 11, 12 | syl6eqss 3655 | . . . 4 |
14 | 13 | exlimiv 1858 | . . 3 |
15 | 4, 14 | sylbi 207 | . 2 |
16 | 3, 15 | pm2.61ine 2877 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 wex 1704 wcel 1990 wne 2794 wral 2912 crab 2916 cvv 3200 wss 3574 c0 3915 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 Cntzccntz 17748 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-cntz 17750 |
This theorem is referenced by: cntz2ss 17765 cntzsubm 17768 cntzsubg 17769 cntzidss 17770 cntzmhm 17771 cntzmhm2 17772 cntzcmn 18245 cntzspan 18247 cntzsubr 18812 cntzsdrg 37772 |
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