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Mirrors > Home > MPE Home > Th. List > mulgnn0subcl | Structured version Visualization version Unicode version |
Description: Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.) |
Ref | Expression |
---|---|
mulgnnsubcl.b |
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mulgnnsubcl.t |
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mulgnnsubcl.p |
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mulgnnsubcl.g |
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mulgnnsubcl.s |
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mulgnnsubcl.c |
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mulgnn0subcl.z |
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mulgnn0subcl.c |
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Ref | Expression |
---|---|
mulgnn0subcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulgnnsubcl.b |
. . . . . 6
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2 | mulgnnsubcl.t |
. . . . . 6
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3 | mulgnnsubcl.p |
. . . . . 6
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4 | mulgnnsubcl.g |
. . . . . 6
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5 | mulgnnsubcl.s |
. . . . . 6
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6 | mulgnnsubcl.c |
. . . . . 6
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7 | 1, 2, 3, 4, 5, 6 | mulgnnsubcl 16782 |
. . . . 5
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8 | 7 | 3expa 1209 |
. . . 4
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9 | 8 | an32s 814 |
. . 3
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10 | 9 | 3adantl2 1166 |
. 2
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11 | oveq1 6302 |
. . . 4
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12 | 5 | 3ad2ant1 1030 |
. . . . . 6
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13 | simp3 1011 |
. . . . . 6
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14 | 12, 13 | sseldd 3435 |
. . . . 5
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15 | mulgnn0subcl.z |
. . . . . 6
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16 | 1, 15, 2 | mulg0 16775 |
. . . . 5
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17 | 14, 16 | syl 17 |
. . . 4
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18 | 11, 17 | sylan9eqr 2509 |
. . 3
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19 | mulgnn0subcl.c |
. . . . 5
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20 | 19 | 3ad2ant1 1030 |
. . . 4
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21 | 20 | adantr 467 |
. . 3
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22 | 18, 21 | eqeltrd 2531 |
. 2
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23 | simp2 1010 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | elnn0 10878 |
. . 3
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25 | 23, 24 | sylib 200 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 10, 22, 25 | mpjaodan 796 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-rep 4518 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 ax-inf2 8151 ax-cnex 9600 ax-resscn 9601 ax-1cn 9602 ax-icn 9603 ax-addcl 9604 ax-addrcl 9605 ax-mulcl 9606 ax-mulrcl 9607 ax-mulcom 9608 ax-addass 9609 ax-mulass 9610 ax-distr 9611 ax-i2m1 9612 ax-1ne0 9613 ax-1rid 9614 ax-rnegex 9615 ax-rrecex 9616 ax-cnre 9617 ax-pre-lttri 9618 ax-pre-lttrn 9619 ax-pre-ltadd 9620 ax-pre-mulgt0 9621 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 987 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-nel 2627 df-ral 2744 df-rex 2745 df-reu 2746 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-pss 3422 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-tp 3975 df-op 3977 df-uni 4202 df-iun 4283 df-br 4406 df-opab 4465 df-mpt 4466 df-tr 4501 df-eprel 4748 df-id 4752 df-po 4758 df-so 4759 df-fr 4796 df-we 4798 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-pred 5383 df-ord 5429 df-on 5430 df-lim 5431 df-suc 5432 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-riota 6257 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-om 6698 df-1st 6798 df-2nd 6799 df-wrecs 7033 df-recs 7095 df-rdg 7133 df-er 7368 df-en 7575 df-dom 7576 df-sdom 7577 df-pnf 9682 df-mnf 9683 df-xr 9684 df-ltxr 9685 df-le 9686 df-sub 9867 df-neg 9868 df-nn 10617 df-n0 10877 df-z 10945 df-uz 11167 df-fz 11792 df-seq 12221 df-mulg 16688 |
This theorem is referenced by: mulgsubcl 16784 mulgnn0cl 16786 submmulgcl 16804 mplbas2 18706 |
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