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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 19282 | . . 3 โข (๐ โ ๐ต โ (๐โ{๐}) = {๐ฅ โ ๐ต โฃ (๐ฅ + ๐) = (๐ + ๐ฅ)}) |
5 | 4 | eleq2d 2815 | . 2 โข (๐ โ ๐ต โ (๐ โ (๐โ{๐}) โ ๐ โ {๐ฅ โ ๐ต โฃ (๐ฅ + ๐) = (๐ + ๐ฅ)})) |
6 | oveq1 7433 | . . . 4 โข (๐ฅ = ๐ โ (๐ฅ + ๐) = (๐ + ๐)) | |
7 | oveq2 7434 | . . . 4 โข (๐ฅ = ๐ โ (๐ + ๐ฅ) = (๐ + ๐)) | |
8 | 6, 7 | eqeq12d 2744 | . . 3 โข (๐ฅ = ๐ โ ((๐ฅ + ๐) = (๐ + ๐ฅ) โ (๐ + ๐) = (๐ + ๐))) |
9 | 8 | elrab 3684 | . 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 286 | 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 394 = wceq 1533 โ wcel 2098 {crab 3430 {csn 4632 โcfv 6553 (class class class)co 7426 Basecbs 17187 +gcplusg 17240 Cntzccntz 19273 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-cntz 19275 |
This theorem is referenced by: gsumconst 19896 gsumpt 19924 cntzsnid 32796 |
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