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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 4806 | . . 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 19237 | . . 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 7413 | . . . . 5 โข (๐ฆ = ๐ โ (๐ฅ + ๐ฆ) = (๐ฅ + ๐)) | |
8 | oveq1 7412 | . . . . 5 โข (๐ฆ = ๐ โ (๐ฆ + ๐ฅ) = (๐ + ๐ฅ)) | |
9 | 7, 8 | eqeq12d 2742 | . . . 4 โข (๐ฆ = ๐ โ ((๐ฅ + ๐ฆ) = (๐ฆ + ๐ฅ) โ (๐ฅ + ๐) = (๐ + ๐ฅ))) |
10 | 9 | ralsng 4672 | . . 3 โข (๐ โ ๐ต โ (โ๐ฆ โ {๐} (๐ฅ + ๐ฆ) = (๐ฆ + ๐ฅ) โ (๐ฅ + ๐) = (๐ + ๐ฅ))) |
11 | 10 | rabbidv 3434 | . 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 2766 | 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 1533 โ wcel 2098 โwral 3055 {crab 3426 โ wss 3943 {csn 4623 โcfv 6537 (class class class)co 7405 Basecbs 17153 +gcplusg 17206 Cntzccntz 19231 |
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 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-cntz 19233 |
This theorem is referenced by: elcntzsn 19241 cntziinsn 19253 |
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