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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ablo4pnp | Structured version Visualization version Unicode version |
Description: A commutative/associative law for Abelian groups. (Contributed by Jeff Madsen, 11-Jun-2010.) |
Ref | Expression |
---|---|
abl4pnp.1 |
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abl4pnp.2 |
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Ref | Expression |
---|---|
ablo4pnp |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3an 986 |
. . . . 5
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2 | abl4pnp.1 |
. . . . . 6
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3 | abl4pnp.2 |
. . . . . 6
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4 | 2, 3 | ablomuldiv 26010 |
. . . . 5
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5 | 1, 4 | sylan2br 479 |
. . . 4
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6 | 5 | adantrrr 730 |
. . 3
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7 | 6 | oveq1d 6303 |
. 2
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8 | ablogrpo 26005 |
. . . . . . 7
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9 | 2 | grpocl 25921 |
. . . . . . . 8
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10 | 9 | 3expib 1210 |
. . . . . . 7
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11 | 8, 10 | syl 17 |
. . . . . 6
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12 | 11 | anim1d 567 |
. . . . 5
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13 | 3anass 988 |
. . . . 5
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14 | 12, 13 | syl6ibr 231 |
. . . 4
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15 | 14 | imp 431 |
. . 3
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16 | 2, 3 | ablodivdiv4 26012 |
. . 3
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17 | 15, 16 | syldan 473 |
. 2
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18 | 2, 3 | grpodivcl 25968 |
. . . . . . . 8
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19 | 18 | 3expib 1210 |
. . . . . . 7
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20 | 19 | anim1d 567 |
. . . . . 6
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21 | an4 832 |
. . . . . 6
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22 | 3anass 988 |
. . . . . 6
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23 | 20, 21, 22 | 3imtr4g 274 |
. . . . 5
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24 | 23 | imp 431 |
. . . 4
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25 | 2, 3 | grpomuldivass 25970 |
. . . 4
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26 | 24, 25 | syldan 473 |
. . 3
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27 | 8, 26 | sylan 474 |
. 2
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28 | 7, 17, 27 | 3eqtr3d 2492 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-rep 4514 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-ral 2741 df-rex 2742 df-reu 2743 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-op 3974 df-uni 4198 df-iun 4279 df-br 4402 df-opab 4461 df-mpt 4462 df-id 4748 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-fv 5589 df-riota 6250 df-ov 6291 df-oprab 6292 df-mpt2 6293 df-1st 6790 df-2nd 6791 df-grpo 25912 df-gid 25913 df-ginv 25914 df-gdiv 25915 df-ablo 26003 |
This theorem is referenced by: (None) |
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