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Mirrors > Home > MPE Home > Th. List > mulgval | Structured version Visualization version GIF version |
Description: Value of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.) |
Ref | Expression |
---|---|
mulgval.b | โข ๐ต = (Baseโ๐บ) |
mulgval.p | โข + = (+gโ๐บ) |
mulgval.o | โข 0 = (0gโ๐บ) |
mulgval.i | โข ๐ผ = (invgโ๐บ) |
mulgval.t | โข ยท = (.gโ๐บ) |
mulgval.s | โข ๐ = seq1( + , (โ ร {๐})) |
Ref | Expression |
---|---|
mulgval | โข ((๐ โ โค โง ๐ โ ๐ต) โ (๐ ยท ๐) = if(๐ = 0, 0 , if(0 < ๐, (๐โ๐), (๐ผโ(๐โ-๐))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 484 | . . . 4 โข ((๐ = ๐ โง ๐ฅ = ๐) โ ๐ = ๐) | |
2 | 1 | eqeq1d 2735 | . . 3 โข ((๐ = ๐ โง ๐ฅ = ๐) โ (๐ = 0 โ ๐ = 0)) |
3 | 1 | breq2d 5121 | . . . 4 โข ((๐ = ๐ โง ๐ฅ = ๐) โ (0 < ๐ โ 0 < ๐)) |
4 | simpr 486 | . . . . . . . . 9 โข ((๐ = ๐ โง ๐ฅ = ๐) โ ๐ฅ = ๐) | |
5 | 4 | sneqd 4602 | . . . . . . . 8 โข ((๐ = ๐ โง ๐ฅ = ๐) โ {๐ฅ} = {๐}) |
6 | 5 | xpeq2d 5667 | . . . . . . 7 โข ((๐ = ๐ โง ๐ฅ = ๐) โ (โ ร {๐ฅ}) = (โ ร {๐})) |
7 | 6 | seqeq3d 13923 | . . . . . 6 โข ((๐ = ๐ โง ๐ฅ = ๐) โ seq1( + , (โ ร {๐ฅ})) = seq1( + , (โ ร {๐}))) |
8 | mulgval.s | . . . . . 6 โข ๐ = seq1( + , (โ ร {๐})) | |
9 | 7, 8 | eqtr4di 2791 | . . . . 5 โข ((๐ = ๐ โง ๐ฅ = ๐) โ seq1( + , (โ ร {๐ฅ})) = ๐) |
10 | 9, 1 | fveq12d 6853 | . . . 4 โข ((๐ = ๐ โง ๐ฅ = ๐) โ (seq1( + , (โ ร {๐ฅ}))โ๐) = (๐โ๐)) |
11 | 1 | negeqd 11403 | . . . . . 6 โข ((๐ = ๐ โง ๐ฅ = ๐) โ -๐ = -๐) |
12 | 9, 11 | fveq12d 6853 | . . . . 5 โข ((๐ = ๐ โง ๐ฅ = ๐) โ (seq1( + , (โ ร {๐ฅ}))โ-๐) = (๐โ-๐)) |
13 | 12 | fveq2d 6850 | . . . 4 โข ((๐ = ๐ โง ๐ฅ = ๐) โ (๐ผโ(seq1( + , (โ ร {๐ฅ}))โ-๐)) = (๐ผโ(๐โ-๐))) |
14 | 3, 10, 13 | ifbieq12d 4518 | . . 3 โข ((๐ = ๐ โง ๐ฅ = ๐) โ if(0 < ๐, (seq1( + , (โ ร {๐ฅ}))โ๐), (๐ผโ(seq1( + , (โ ร {๐ฅ}))โ-๐))) = if(0 < ๐, (๐โ๐), (๐ผโ(๐โ-๐)))) |
15 | 2, 14 | ifbieq2d 4516 | . 2 โข ((๐ = ๐ โง ๐ฅ = ๐) โ if(๐ = 0, 0 , if(0 < ๐, (seq1( + , (โ ร {๐ฅ}))โ๐), (๐ผโ(seq1( + , (โ ร {๐ฅ}))โ-๐)))) = if(๐ = 0, 0 , if(0 < ๐, (๐โ๐), (๐ผโ(๐โ-๐))))) |
16 | mulgval.b | . . 3 โข ๐ต = (Baseโ๐บ) | |
17 | mulgval.p | . . 3 โข + = (+gโ๐บ) | |
18 | mulgval.o | . . 3 โข 0 = (0gโ๐บ) | |
19 | mulgval.i | . . 3 โข ๐ผ = (invgโ๐บ) | |
20 | mulgval.t | . . 3 โข ยท = (.gโ๐บ) | |
21 | 16, 17, 18, 19, 20 | mulgfval 18882 | . 2 โข ยท = (๐ โ โค, ๐ฅ โ ๐ต โฆ if(๐ = 0, 0 , if(0 < ๐, (seq1( + , (โ ร {๐ฅ}))โ๐), (๐ผโ(seq1( + , (โ ร {๐ฅ}))โ-๐))))) |
22 | 18 | fvexi 6860 | . . 3 โข 0 โ V |
23 | fvex 6859 | . . . 4 โข (๐โ๐) โ V | |
24 | fvex 6859 | . . . 4 โข (๐ผโ(๐โ-๐)) โ V | |
25 | 23, 24 | ifex 4540 | . . 3 โข if(0 < ๐, (๐โ๐), (๐ผโ(๐โ-๐))) โ V |
26 | 22, 25 | ifex 4540 | . 2 โข if(๐ = 0, 0 , if(0 < ๐, (๐โ๐), (๐ผโ(๐โ-๐)))) โ V |
27 | 15, 21, 26 | ovmpoa 7514 | 1 โข ((๐ โ โค โง ๐ โ ๐ต) โ (๐ ยท ๐) = if(๐ = 0, 0 , if(0 < ๐, (๐โ๐), (๐ผโ(๐โ-๐))))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 397 = wceq 1542 โ wcel 2107 ifcif 4490 {csn 4590 class class class wbr 5109 ร cxp 5635 โcfv 6500 (class class class)co 7361 0cc0 11059 1c1 11060 < clt 11197 -cneg 11394 โcn 12161 โคcz 12507 seqcseq 13915 Basecbs 17091 +gcplusg 17141 0gc0g 17329 invgcminusg 18757 .gcmg 18880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-seq 13916 df-mulg 18881 |
This theorem is referenced by: mulg0 18887 mulgnn 18888 mulgnegnn 18894 subgmulg 18950 |
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