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Mirrors > Home > MPE Home > Th. List > Mathboxes > ismndo1 | Structured version Visualization version Unicode version |
Description: The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ismndo1.1 |
Ref | Expression |
---|---|
ismndo1 | MndOp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismndo1.1 | . . 3 | |
2 | 1 | ismndo 33671 | . 2 MndOp |
3 | 1 | smgrpmgm 33663 | . . . . 5 |
4 | 3 | ad2antrl 764 | . . . 4 |
5 | 1 | smgrpassOLD 33664 | . . . . 5 |
6 | 5 | ad2antrl 764 | . . . 4 |
7 | simprr 796 | . . . 4 | |
8 | 4, 6, 7 | 3jca 1242 | . . 3 |
9 | 3simpa 1058 | . . . . . 6 | |
10 | 1 | issmgrpOLD 33662 | . . . . . 6 |
11 | 9, 10 | syl5ibr 236 | . . . . 5 |
12 | 11 | imp 445 | . . . 4 |
13 | simpr3 1069 | . . . 4 | |
14 | 12, 13 | jca 554 | . . 3 |
15 | 8, 14 | impbida 877 | . 2 |
16 | 2, 15 | bitrd 268 | 1 MndOp |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 wrex 2913 cxp 5112 cdm 5114 wf 5884 (class class class)co 6650 csem 33659 MndOpcmndo 33665 |
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-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-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-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-ass 33642 df-exid 33644 df-mgmOLD 33648 df-sgrOLD 33660 df-mndo 33666 |
This theorem is referenced by: ismndo2 33673 rngomndo 33734 |
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