![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > elcntr | Structured version Visualization version GIF version |
Description: Elementhood in the center of a magma. (Contributed by SN, 21-Mar-2025.) |
Ref | Expression |
---|---|
elcntr.b | โข ๐ต = (Baseโ๐) |
elcntr.p | โข + = (+gโ๐) |
elcntr.z | โข ๐ = (Cntrโ๐) |
Ref | Expression |
---|---|
elcntr | โข (๐ด โ ๐ โ (๐ด โ ๐ต โง โ๐ฆ โ ๐ต (๐ด + ๐ฆ) = (๐ฆ + ๐ด))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elcntr.z | . . . 4 โข ๐ = (Cntrโ๐) | |
2 | elcntr.b | . . . . 5 โข ๐ต = (Baseโ๐) | |
3 | eqid 2728 | . . . . 5 โข (Cntzโ๐) = (Cntzโ๐) | |
4 | 2, 3 | cntrval 19270 | . . . 4 โข ((Cntzโ๐)โ๐ต) = (Cntrโ๐) |
5 | 1, 4 | eqtr4i 2759 | . . 3 โข ๐ = ((Cntzโ๐)โ๐ต) |
6 | 5 | eleq2i 2821 | . 2 โข (๐ด โ ๐ โ ๐ด โ ((Cntzโ๐)โ๐ต)) |
7 | ssid 4002 | . . 3 โข ๐ต โ ๐ต | |
8 | elcntr.p | . . . 4 โข + = (+gโ๐) | |
9 | 2, 8, 3 | elcntz 19273 | . . 3 โข (๐ต โ ๐ต โ (๐ด โ ((Cntzโ๐)โ๐ต) โ (๐ด โ ๐ต โง โ๐ฆ โ ๐ต (๐ด + ๐ฆ) = (๐ฆ + ๐ด)))) |
10 | 7, 9 | ax-mp 5 | . 2 โข (๐ด โ ((Cntzโ๐)โ๐ต) โ (๐ด โ ๐ต โง โ๐ฆ โ ๐ต (๐ด + ๐ฆ) = (๐ฆ + ๐ด))) |
11 | 6, 10 | bitri 275 | 1 โข (๐ด โ ๐ โ (๐ด โ ๐ต โง โ๐ฆ โ ๐ต (๐ด + ๐ฆ) = (๐ฆ + ๐ด))) |
Colors of variables: wff setvar class |
Syntax hints: โ wb 205 โง wa 395 = wceq 1534 โ wcel 2099 โwral 3058 โ wss 3947 โcfv 6548 (class class class)co 7420 Basecbs 17180 +gcplusg 17233 Cntzccntz 19266 Cntrccntr 19267 |
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 19268 df-cntr 19269 |
This theorem is referenced by: sraassab 21801 |
Copyright terms: Public domain | W3C validator |