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Mirrors > Home > ILE Home > Th. List > mulgnn | Unicode version |
Description: Group multiple (exponentiation) operation at a positive integer. (Contributed by Mario Carneiro, 11-Dec-2014.) |
Ref | Expression |
---|---|
mulgnn.b |
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mulgnn.p |
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mulgnn.t |
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mulgnn.s |
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Ref | Expression |
---|---|
mulgnn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnz 9248 |
. . 3
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2 | mulgnn.b |
. . . 4
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3 | mulgnn.p |
. . . 4
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4 | eqid 2177 |
. . . 4
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5 | eqid 2177 |
. . . 4
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6 | mulgnn.t |
. . . 4
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7 | mulgnn.s |
. . . 4
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8 | 2, 3, 4, 5, 6, 7 | mulgval 12862 |
. . 3
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9 | 1, 8 | sylan 283 |
. 2
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10 | nnne0 8923 |
. . . . . 6
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11 | 10 | neneqd 2368 |
. . . . 5
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12 | 11 | iffalsed 3544 |
. . . 4
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13 | nngt0 8920 |
. . . . 5
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14 | 13 | iftrued 3541 |
. . . 4
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15 | 12, 14 | eqtrd 2210 |
. . 3
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16 | 15 | adantr 276 |
. 2
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17 | 9, 16 | eqtrd 2210 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-ltadd 7905 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4289 df-iord 4362 df-on 4364 df-ilim 4365 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-recs 6299 df-frec 6385 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-inn 8896 df-2 8954 df-n0 9153 df-z 9230 df-uz 9505 df-seqfrec 10419 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-0g 12642 df-minusg 12758 df-mulg 12860 |
This theorem is referenced by: mulg1 12866 mulgnnp1 12867 mulgnegnn 12869 mulgnnsubcl 12871 mulgnn0z 12885 mulgnndir 12887 |
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