Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ismndd | Structured version Visualization version Unicode version |
Description: Deduce a monoid from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.) |
Ref | Expression |
---|---|
ismndd.b | |
ismndd.p | |
ismndd.c | |
ismndd.a | |
ismndd.z | |
ismndd.i | |
ismndd.j |
Ref | Expression |
---|---|
ismndd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismndd.c | . . . . . 6 | |
2 | 1 | 3expb 1266 | . . . . 5 |
3 | simpll 790 | . . . . . . 7 | |
4 | simplrl 800 | . . . . . . 7 | |
5 | simplrr 801 | . . . . . . 7 | |
6 | simpr 477 | . . . . . . 7 | |
7 | ismndd.a | . . . . . . 7 | |
8 | 3, 4, 5, 6, 7 | syl13anc 1328 | . . . . . 6 |
9 | 8 | ralrimiva 2966 | . . . . 5 |
10 | 2, 9 | jca 554 | . . . 4 |
11 | 10 | ralrimivva 2971 | . . 3 |
12 | ismndd.b | . . . 4 | |
13 | ismndd.p | . . . . . . . 8 | |
14 | 13 | oveqd 6667 | . . . . . . 7 |
15 | 14, 12 | eleq12d 2695 | . . . . . 6 |
16 | eqidd 2623 | . . . . . . . . 9 | |
17 | 13, 14, 16 | oveq123d 6671 | . . . . . . . 8 |
18 | eqidd 2623 | . . . . . . . . 9 | |
19 | 13 | oveqd 6667 | . . . . . . . . 9 |
20 | 13, 18, 19 | oveq123d 6671 | . . . . . . . 8 |
21 | 17, 20 | eqeq12d 2637 | . . . . . . 7 |
22 | 12, 21 | raleqbidv 3152 | . . . . . 6 |
23 | 15, 22 | anbi12d 747 | . . . . 5 |
24 | 12, 23 | raleqbidv 3152 | . . . 4 |
25 | 12, 24 | raleqbidv 3152 | . . 3 |
26 | 11, 25 | mpbid 222 | . 2 |
27 | ismndd.z | . . . 4 | |
28 | 27, 12 | eleqtrd 2703 | . . 3 |
29 | 12 | eleq2d 2687 | . . . . . 6 |
30 | 29 | biimpar 502 | . . . . 5 |
31 | 13 | adantr 481 | . . . . . . . 8 |
32 | 31 | oveqd 6667 | . . . . . . 7 |
33 | ismndd.i | . . . . . . 7 | |
34 | 32, 33 | eqtr3d 2658 | . . . . . 6 |
35 | 31 | oveqd 6667 | . . . . . . 7 |
36 | ismndd.j | . . . . . . 7 | |
37 | 35, 36 | eqtr3d 2658 | . . . . . 6 |
38 | 34, 37 | jca 554 | . . . . 5 |
39 | 30, 38 | syldan 487 | . . . 4 |
40 | 39 | ralrimiva 2966 | . . 3 |
41 | oveq1 6657 | . . . . . . 7 | |
42 | 41 | eqeq1d 2624 | . . . . . 6 |
43 | oveq2 6658 | . . . . . . 7 | |
44 | 43 | eqeq1d 2624 | . . . . . 6 |
45 | 42, 44 | anbi12d 747 | . . . . 5 |
46 | 45 | ralbidv 2986 | . . . 4 |
47 | 46 | rspcev 3309 | . . 3 |
48 | 28, 40, 47 | syl2anc 693 | . 2 |
49 | eqid 2622 | . . 3 | |
50 | eqid 2622 | . . 3 | |
51 | 49, 50 | ismnd 17297 | . 2 |
52 | 26, 48, 51 | sylanbrc 698 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 wrex 2913 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 cmnd 17294 |
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-nul 4789 ax-pow 4843 |
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-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-iota 5851 df-fv 5896 df-ov 6653 df-mgm 17242 df-sgrp 17284 df-mnd 17295 |
This theorem is referenced by: issubmnd 17318 prdsmndd 17323 imasmnd2 17327 frmdmnd 17396 isgrpde 17443 oppgmnd 17784 isringd 18585 iscrngd 18586 xrsmcmn 19769 xrs1mnd 19784 |
Copyright terms: Public domain | W3C validator |