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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 2728 | . . . . . 6 โข (Baseโ๐) = (Baseโ๐) | |
2 | cntzi.z | . . . . . 6 โข ๐ = (Cntzโ๐) | |
3 | 1, 2 | cntzrcl 19277 | . . . . 5 โข (๐ โ (๐โ๐) โ (๐ โ V โง ๐ โ (Baseโ๐))) |
4 | cntzi.p | . . . . . 6 โข + = (+gโ๐) | |
5 | 1, 4, 2 | elcntz 19272 | . . . . 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 7428 | . . . 4 โข (๐ฆ = ๐ โ (๐ + ๐ฆ) = (๐ + ๐)) | |
10 | oveq1 7427 | . . . 4 โข (๐ฆ = ๐ โ (๐ฆ + ๐) = (๐ + ๐)) | |
11 | 9, 10 | eqeq12d 2744 | . . 3 โข (๐ฆ = ๐ โ ((๐ + ๐ฆ) = (๐ฆ + ๐) โ (๐ + ๐) = (๐ + ๐))) |
12 | 11 | rspccva 3608 | . 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 1534 โ wcel 2099 โwral 3058 Vcvv 3471 โ wss 3947 โcfv 6548 (class class class)co 7420 Basecbs 17179 +gcplusg 17232 Cntzccntz 19265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-cntz 19267 |
This theorem is referenced by: cntri 19282 cntzsgrpcl 19284 cntz2ss 19285 cntzsubm 19288 cntzsubg 19289 cntzmhm 19291 cntrsubgnsg 19293 lsmsubm 19607 lsmsubg 19608 lsmcom2 19609 subgdisj1 19645 subgdisj2 19646 pj1id 19653 pj1ghm 19657 gsumval3eu 19858 gsumval3 19861 gsumzaddlem 19875 gsumzoppg 19898 dprdfcntz 19971 cntzsubrng 20503 cntzsubr 20544 cntzsdrg 20689 |
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