![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cntzsnval | Structured version Visualization version GIF version |
Description: Special substitution for the centralizer of a singleton. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
cntzfval.b | โข ๐ต = (Baseโ๐) |
cntzfval.p | โข + = (+gโ๐) |
cntzfval.z | โข ๐ = (Cntzโ๐) |
Ref | Expression |
---|---|
cntzsnval | โข (๐ โ ๐ต โ (๐โ{๐}) = {๐ฅ โ ๐ต โฃ (๐ฅ + ๐) = (๐ + ๐ฅ)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 4773 | . . 3 โข (๐ โ ๐ต โ {๐} โ ๐ต) | |
2 | cntzfval.b | . . . 4 โข ๐ต = (Baseโ๐) | |
3 | cntzfval.p | . . . 4 โข + = (+gโ๐) | |
4 | cntzfval.z | . . . 4 โข ๐ = (Cntzโ๐) | |
5 | 2, 3, 4 | cntzval 19108 | . . 3 โข ({๐} โ ๐ต โ (๐โ{๐}) = {๐ฅ โ ๐ต โฃ โ๐ฆ โ {๐} (๐ฅ + ๐ฆ) = (๐ฆ + ๐ฅ)}) |
6 | 1, 5 | syl 17 | . 2 โข (๐ โ ๐ต โ (๐โ{๐}) = {๐ฅ โ ๐ต โฃ โ๐ฆ โ {๐} (๐ฅ + ๐ฆ) = (๐ฆ + ๐ฅ)}) |
7 | oveq2 7370 | . . . . 5 โข (๐ฆ = ๐ โ (๐ฅ + ๐ฆ) = (๐ฅ + ๐)) | |
8 | oveq1 7369 | . . . . 5 โข (๐ฆ = ๐ โ (๐ฆ + ๐ฅ) = (๐ + ๐ฅ)) | |
9 | 7, 8 | eqeq12d 2753 | . . . 4 โข (๐ฆ = ๐ โ ((๐ฅ + ๐ฆ) = (๐ฆ + ๐ฅ) โ (๐ฅ + ๐) = (๐ + ๐ฅ))) |
10 | 9 | ralsng 4639 | . . 3 โข (๐ โ ๐ต โ (โ๐ฆ โ {๐} (๐ฅ + ๐ฆ) = (๐ฆ + ๐ฅ) โ (๐ฅ + ๐) = (๐ + ๐ฅ))) |
11 | 10 | rabbidv 3418 | . 2 โข (๐ โ ๐ต โ {๐ฅ โ ๐ต โฃ โ๐ฆ โ {๐} (๐ฅ + ๐ฆ) = (๐ฆ + ๐ฅ)} = {๐ฅ โ ๐ต โฃ (๐ฅ + ๐) = (๐ + ๐ฅ)}) |
12 | 6, 11 | eqtrd 2777 | 1 โข (๐ โ ๐ต โ (๐โ{๐}) = {๐ฅ โ ๐ต โฃ (๐ฅ + ๐) = (๐ + ๐ฅ)}) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 โwral 3065 {crab 3410 โ wss 3915 {csn 4591 โcfv 6501 (class class class)co 7362 Basecbs 17090 +gcplusg 17140 Cntzccntz 19102 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-cntz 19104 |
This theorem is referenced by: elcntzsn 19112 cntziinsn 19122 |
Copyright terms: Public domain | W3C validator |