![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > elcntzsn | Structured version Visualization version GIF version |
Description: Value of the centralizer of a singleton. (Contributed by Mario Carneiro, 25-Apr-2016.) |
Ref | Expression |
---|---|
cntzfval.b | โข ๐ต = (Baseโ๐) |
cntzfval.p | โข + = (+gโ๐) |
cntzfval.z | โข ๐ = (Cntzโ๐) |
Ref | Expression |
---|---|
elcntzsn | โข (๐ โ ๐ต โ (๐ โ (๐โ{๐}) โ (๐ โ ๐ต โง (๐ + ๐) = (๐ + ๐)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntzfval.b | . . . 4 โข ๐ต = (Baseโ๐) | |
2 | cntzfval.p | . . . 4 โข + = (+gโ๐) | |
3 | cntzfval.z | . . . 4 โข ๐ = (Cntzโ๐) | |
4 | 1, 2, 3 | cntzsnval 19237 | . . 3 โข (๐ โ ๐ต โ (๐โ{๐}) = {๐ฅ โ ๐ต โฃ (๐ฅ + ๐) = (๐ + ๐ฅ)}) |
5 | 4 | eleq2d 2813 | . 2 โข (๐ โ ๐ต โ (๐ โ (๐โ{๐}) โ ๐ โ {๐ฅ โ ๐ต โฃ (๐ฅ + ๐) = (๐ + ๐ฅ)})) |
6 | oveq1 7411 | . . . 4 โข (๐ฅ = ๐ โ (๐ฅ + ๐) = (๐ + ๐)) | |
7 | oveq2 7412 | . . . 4 โข (๐ฅ = ๐ โ (๐ + ๐ฅ) = (๐ + ๐)) | |
8 | 6, 7 | eqeq12d 2742 | . . 3 โข (๐ฅ = ๐ โ ((๐ฅ + ๐) = (๐ + ๐ฅ) โ (๐ + ๐) = (๐ + ๐))) |
9 | 8 | elrab 3678 | . 2 โข (๐ โ {๐ฅ โ ๐ต โฃ (๐ฅ + ๐) = (๐ + ๐ฅ)} โ (๐ โ ๐ต โง (๐ + ๐) = (๐ + ๐))) |
10 | 5, 9 | bitrdi 287 | 1 โข (๐ โ ๐ต โ (๐ โ (๐โ{๐}) โ (๐ โ ๐ต โง (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 {crab 3426 {csn 4623 โcfv 6536 (class class class)co 7404 Basecbs 17150 +gcplusg 17203 Cntzccntz 19228 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-cntz 19230 |
This theorem is referenced by: gsumconst 19851 gsumpt 19879 cntzsnid 32716 |
Copyright terms: Public domain | W3C validator |