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Mirrors > Home > MPE Home > Th. List > mulgnn | Structured version Visualization version GIF version |
Description: Group multiple (exponentiation) operation at a positive integer. (Contributed by Mario Carneiro, 11-Dec-2014.) |
Ref | Expression |
---|---|
mulgnn.b | โข ๐ต = (Baseโ๐บ) |
mulgnn.p | โข + = (+gโ๐บ) |
mulgnn.t | โข ยท = (.gโ๐บ) |
mulgnn.s | โข ๐ = seq1( + , (โ ร {๐})) |
Ref | Expression |
---|---|
mulgnn | โข ((๐ โ โ โง ๐ โ ๐ต) โ (๐ ยท ๐) = (๐โ๐)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnz 12610 | . . 3 โข (๐ โ โ โ ๐ โ โค) | |
2 | mulgnn.b | . . . 4 โข ๐ต = (Baseโ๐บ) | |
3 | mulgnn.p | . . . 4 โข + = (+gโ๐บ) | |
4 | eqid 2728 | . . . 4 โข (0gโ๐บ) = (0gโ๐บ) | |
5 | eqid 2728 | . . . 4 โข (invgโ๐บ) = (invgโ๐บ) | |
6 | mulgnn.t | . . . 4 โข ยท = (.gโ๐บ) | |
7 | mulgnn.s | . . . 4 โข ๐ = seq1( + , (โ ร {๐})) | |
8 | 2, 3, 4, 5, 6, 7 | mulgval 19027 | . . 3 โข ((๐ โ โค โง ๐ โ ๐ต) โ (๐ ยท ๐) = if(๐ = 0, (0gโ๐บ), if(0 < ๐, (๐โ๐), ((invgโ๐บ)โ(๐โ-๐))))) |
9 | 1, 8 | sylan 579 | . 2 โข ((๐ โ โ โง ๐ โ ๐ต) โ (๐ ยท ๐) = if(๐ = 0, (0gโ๐บ), if(0 < ๐, (๐โ๐), ((invgโ๐บ)โ(๐โ-๐))))) |
10 | nnne0 12277 | . . . . . 6 โข (๐ โ โ โ ๐ โ 0) | |
11 | 10 | neneqd 2942 | . . . . 5 โข (๐ โ โ โ ยฌ ๐ = 0) |
12 | 11 | iffalsed 4540 | . . . 4 โข (๐ โ โ โ if(๐ = 0, (0gโ๐บ), if(0 < ๐, (๐โ๐), ((invgโ๐บ)โ(๐โ-๐)))) = if(0 < ๐, (๐โ๐), ((invgโ๐บ)โ(๐โ-๐)))) |
13 | nngt0 12274 | . . . . 5 โข (๐ โ โ โ 0 < ๐) | |
14 | 13 | iftrued 4537 | . . . 4 โข (๐ โ โ โ if(0 < ๐, (๐โ๐), ((invgโ๐บ)โ(๐โ-๐))) = (๐โ๐)) |
15 | 12, 14 | eqtrd 2768 | . . 3 โข (๐ โ โ โ if(๐ = 0, (0gโ๐บ), if(0 < ๐, (๐โ๐), ((invgโ๐บ)โ(๐โ-๐)))) = (๐โ๐)) |
16 | 15 | adantr 480 | . 2 โข ((๐ โ โ โง ๐ โ ๐ต) โ if(๐ = 0, (0gโ๐บ), if(0 < ๐, (๐โ๐), ((invgโ๐บ)โ(๐โ-๐)))) = (๐โ๐)) |
17 | 9, 16 | eqtrd 2768 | 1 โข ((๐ โ โ โง ๐ โ ๐ต) โ (๐ ยท ๐) = (๐โ๐)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 395 = wceq 1534 โ wcel 2099 ifcif 4529 {csn 4629 class class class wbr 5148 ร cxp 5676 โcfv 6548 (class class class)co 7420 0cc0 11139 1c1 11140 < clt 11279 -cneg 11476 โcn 12243 โคcz 12589 seqcseq 13999 Basecbs 17180 +gcplusg 17233 0gc0g 17421 invgcminusg 18891 .gcmg 19023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-seq 14000 df-mulg 19024 |
This theorem is referenced by: ressmulgnn 19032 mulgnngsum 19034 mulg1 19036 mulgnnp1 19037 mulgnegnn 19039 mulgnnsubcl 19041 mulgnn0z 19056 mulgnndir 19058 submmulg 19073 subgmulg 19095 mulgnn0di 19780 gsumconst 19889 ressmulgnnd 41569 |
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