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Mirrors > Home > MPE Home > Th. List > cntzi | Structured version Visualization version GIF version |
Description: Membership in a centralizer (inference). (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.) |
Ref | Expression |
---|---|
cntzi.p | โข + = (+gโ๐) |
cntzi.z | โข ๐ = (Cntzโ๐) |
Ref | Expression |
---|---|
cntzi | โข ((๐ โ (๐โ๐) โง ๐ โ ๐) โ (๐ + ๐) = (๐ + ๐)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2724 | . . . . . 6 โข (Baseโ๐) = (Baseโ๐) | |
2 | cntzi.z | . . . . . 6 โข ๐ = (Cntzโ๐) | |
3 | 1, 2 | cntzrcl 19239 | . . . . 5 โข (๐ โ (๐โ๐) โ (๐ โ V โง ๐ โ (Baseโ๐))) |
4 | cntzi.p | . . . . . 6 โข + = (+gโ๐) | |
5 | 1, 4, 2 | elcntz 19234 | . . . . 5 โข (๐ โ (Baseโ๐) โ (๐ โ (๐โ๐) โ (๐ โ (Baseโ๐) โง โ๐ฆ โ ๐ (๐ + ๐ฆ) = (๐ฆ + ๐)))) |
6 | 3, 5 | simpl2im 503 | . . . 4 โข (๐ โ (๐โ๐) โ (๐ โ (๐โ๐) โ (๐ โ (Baseโ๐) โง โ๐ฆ โ ๐ (๐ + ๐ฆ) = (๐ฆ + ๐)))) |
7 | 6 | simplbda 499 | . . 3 โข ((๐ โ (๐โ๐) โง ๐ โ (๐โ๐)) โ โ๐ฆ โ ๐ (๐ + ๐ฆ) = (๐ฆ + ๐)) |
8 | 7 | anidms 566 | . 2 โข (๐ โ (๐โ๐) โ โ๐ฆ โ ๐ (๐ + ๐ฆ) = (๐ฆ + ๐)) |
9 | oveq2 7410 | . . . 4 โข (๐ฆ = ๐ โ (๐ + ๐ฆ) = (๐ + ๐)) | |
10 | oveq1 7409 | . . . 4 โข (๐ฆ = ๐ โ (๐ฆ + ๐) = (๐ + ๐)) | |
11 | 9, 10 | eqeq12d 2740 | . . 3 โข (๐ฆ = ๐ โ ((๐ + ๐ฆ) = (๐ฆ + ๐) โ (๐ + ๐) = (๐ + ๐))) |
12 | 11 | rspccva 3603 | . 2 โข ((โ๐ฆ โ ๐ (๐ + ๐ฆ) = (๐ฆ + ๐) โง ๐ โ ๐) โ (๐ + ๐) = (๐ + ๐)) |
13 | 8, 12 | sylan 579 | 1 โข ((๐ โ (๐โ๐) โง ๐ โ ๐) โ (๐ + ๐) = (๐ + ๐)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wb 205 โง wa 395 = wceq 1533 โ wcel 2098 โwral 3053 Vcvv 3466 โ wss 3941 โcfv 6534 (class class class)co 7402 Basecbs 17149 +gcplusg 17202 Cntzccntz 19227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-cntz 19229 |
This theorem is referenced by: cntri 19244 cntzsgrpcl 19246 cntz2ss 19247 cntzsubm 19250 cntzsubg 19251 cntzmhm 19253 cntrsubgnsg 19255 lsmsubm 19569 lsmsubg 19570 lsmcom2 19571 subgdisj1 19607 subgdisj2 19608 pj1id 19615 pj1ghm 19619 gsumval3eu 19820 gsumval3 19823 gsumzaddlem 19837 gsumzoppg 19860 dprdfcntz 19933 cntzsubrng 20463 cntzsubr 20504 cntzsdrg 20649 |
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