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Mirrors > Home > MPE Home > Th. List > mulgnn0z | Structured version Unicode version |
Description: A group multiple of the identity, for nonnegative multiple. (Contributed by Mario Carneiro, 13-Dec-2014.) |
Ref | Expression |
---|---|
mulgnn0z.b |
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mulgnn0z.t |
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mulgnn0z.o |
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Ref | Expression |
---|---|
mulgnn0z |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 10687 |
. 2
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2 | id 22 |
. . . . 5
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3 | mulgnn0z.b |
. . . . . 6
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4 | mulgnn0z.o |
. . . . . 6
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5 | 3, 4 | mndidcl 15553 |
. . . . 5
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6 | eqid 2452 |
. . . . . 6
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7 | mulgnn0z.t |
. . . . . 6
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8 | eqid 2452 |
. . . . . 6
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9 | 3, 6, 7, 8 | mulgnn 15747 |
. . . . 5
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10 | 2, 5, 9 | syl2anr 478 |
. . . 4
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11 | 3, 6, 4 | mndlid 15555 |
. . . . . . 7
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12 | 5, 11 | mpdan 668 |
. . . . . 6
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13 | 12 | adantr 465 |
. . . . 5
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14 | simpr 461 |
. . . . . 6
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15 | nnuz 11002 |
. . . . . 6
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16 | 14, 15 | syl6eleq 2550 |
. . . . 5
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17 | 5 | adantr 465 |
. . . . . 6
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18 | elfznn 11590 |
. . . . . 6
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19 | fvconst2g 6035 |
. . . . . 6
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20 | 17, 18, 19 | syl2an 477 |
. . . . 5
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21 | 13, 16, 20 | seqid3 11962 |
. . . 4
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22 | 10, 21 | eqtrd 2493 |
. . 3
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23 | oveq1 6202 |
. . . 4
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24 | 3, 4, 7 | mulg0 15746 |
. . . . 5
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25 | 5, 24 | syl 16 |
. . . 4
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26 | 23, 25 | sylan9eqr 2515 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 22, 26 | jaodan 783 |
. 2
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28 | 1, 27 | sylan2b 475 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1954 ax-ext 2431 ax-rep 4506 ax-sep 4516 ax-nul 4524 ax-pow 4573 ax-pr 4634 ax-un 6477 ax-inf2 7953 ax-cnex 9444 ax-resscn 9445 ax-1cn 9446 ax-icn 9447 ax-addcl 9448 ax-addrcl 9449 ax-mulcl 9450 ax-mulrcl 9451 ax-mulcom 9452 ax-addass 9453 ax-mulass 9454 ax-distr 9455 ax-i2m1 9456 ax-1ne0 9457 ax-1rid 9458 ax-rnegex 9459 ax-rrecex 9460 ax-cnre 9461 ax-pre-lttri 9462 ax-pre-lttrn 9463 ax-pre-ltadd 9464 ax-pre-mulgt0 9465 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2265 df-mo 2266 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2602 df-ne 2647 df-nel 2648 df-ral 2801 df-rex 2802 df-reu 2803 df-rmo 2804 df-rab 2805 df-v 3074 df-sbc 3289 df-csb 3391 df-dif 3434 df-un 3436 df-in 3438 df-ss 3445 df-pss 3447 df-nul 3741 df-if 3895 df-pw 3965 df-sn 3981 df-pr 3983 df-tp 3985 df-op 3987 df-uni 4195 df-iun 4276 df-br 4396 df-opab 4454 df-mpt 4455 df-tr 4489 df-eprel 4735 df-id 4739 df-po 4744 df-so 4745 df-fr 4782 df-we 4784 df-ord 4825 df-on 4826 df-lim 4827 df-suc 4828 df-xp 4949 df-rel 4950 df-cnv 4951 df-co 4952 df-dm 4953 df-rn 4954 df-res 4955 df-ima 4956 df-iota 5484 df-fun 5523 df-fn 5524 df-f 5525 df-f1 5526 df-fo 5527 df-f1o 5528 df-fv 5529 df-riota 6156 df-ov 6198 df-oprab 6199 df-mpt2 6200 df-om 6582 df-1st 6682 df-2nd 6683 df-recs 6937 df-rdg 6971 df-er 7206 df-en 7416 df-dom 7417 df-sdom 7418 df-pnf 9526 df-mnf 9527 df-xr 9528 df-ltxr 9529 df-le 9530 df-sub 9703 df-neg 9704 df-nn 10429 df-n0 10686 df-z 10753 df-uz 10968 df-fz 11550 df-seq 11919 df-0g 14494 df-mnd 15529 df-mulg 15662 |
This theorem is referenced by: mulgz 15762 mulgnn0ass 15770 odmodnn0 16159 mulgmhm 16431 srg1expzeq1 16755 tsmsxp 19856 lply1binomsc 31005 |
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