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Mirrors > Home > MPE Home > Th. List > sgrp2nmndlem1 | Structured version Visualization version Unicode version |
Description: Lemma 1 for sgrp2nmnd 17417: is a magma, even if ( is the trivial magma in this case, see mgmb1mgm1 17254). (Contributed by AV, 29-Jan-2020.) |
Ref | Expression |
---|---|
mgm2nsgrp.s | |
mgm2nsgrp.b | |
sgrp2nmnd.o |
Ref | Expression |
---|---|
sgrp2nmndlem1 | Mgm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prid1g 4295 | . . 3 | |
2 | mgm2nsgrp.s | . . 3 | |
3 | 1, 2 | syl6eleqr 2712 | . 2 |
4 | prid2g 4296 | . . 3 | |
5 | 4, 2 | syl6eleqr 2712 | . 2 |
6 | mgm2nsgrp.b | . . . 4 | |
7 | 6 | eqcomi 2631 | . . 3 |
8 | sgrp2nmnd.o | . . 3 | |
9 | ne0i 3921 | . . . 4 | |
10 | 9 | adantr 481 | . . 3 |
11 | simpll 790 | . . 3 | |
12 | simplr 792 | . . 3 | |
13 | 7, 8, 10, 11, 12 | opifismgm 17258 | . 2 Mgm |
14 | 3, 5, 13 | syl2an 494 | 1 Mgm |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wne 2794 c0 3915 cif 4086 cpr 4179 cfv 5888 cmpt2 6652 cbs 15857 cplusg 15941 Mgmcmgm 17240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-mgm 17242 |
This theorem is referenced by: sgrp2nmndlem4 17415 |
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