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Mirrors > Home > MPE Home > Th. List > mndprop | Structured version Visualization version Unicode version |
Description: If two structures have the same group components (properties), one is a monoid iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) |
Ref | Expression |
---|---|
mndprop.b | |
mndprop.p |
Ref | Expression |
---|---|
mndprop |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2623 | . . 3 | |
2 | mndprop.b | . . . 4 | |
3 | 2 | a1i 11 | . . 3 |
4 | mndprop.p | . . . . 5 | |
5 | 4 | oveqi 6663 | . . . 4 |
6 | 5 | a1i 11 | . . 3 |
7 | 1, 3, 6 | mndpropd 17316 | . 2 |
8 | 7 | trud 1493 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 wceq 1483 wtru 1484 wcel 1990 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: ring1 18602 |
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