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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 | ⊢ 𝐵 = (Base‘𝐺) |
mulgval.p | ⊢ + = (+g‘𝐺) |
mulgval.o | ⊢ 0 = (0g‘𝐺) |
mulgval.i | ⊢ 𝐼 = (invg‘𝐺) |
mulgval.t | ⊢ · = (.g‘𝐺) |
mulgval.s | ⊢ 𝑆 = seq1( + , (ℕ × {𝑋})) |
Ref | Expression |
---|---|
mulgval | ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 481 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → 𝑛 = 𝑁) | |
2 | 1 | eqeq1d 2728 | . . 3 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑛 = 0 ↔ 𝑁 = 0)) |
3 | 1 | breq2d 5165 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (0 < 𝑛 ↔ 0 < 𝑁)) |
4 | simpr 483 | . . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
5 | 4 | sneqd 4645 | . . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → {𝑥} = {𝑋}) |
6 | 5 | xpeq2d 5712 | . . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (ℕ × {𝑥}) = (ℕ × {𝑋})) |
7 | 6 | seqeq3d 14029 | . . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑋}))) |
8 | mulgval.s | . . . . . 6 ⊢ 𝑆 = seq1( + , (ℕ × {𝑋})) | |
9 | 7, 8 | eqtr4di 2784 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = 𝑆) |
10 | 9, 1 | fveq12d 6908 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (𝑆‘𝑁)) |
11 | 1 | negeqd 11504 | . . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → -𝑛 = -𝑁) |
12 | 9, 11 | fveq12d 6908 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘-𝑛) = (𝑆‘-𝑁)) |
13 | 12 | fveq2d 6905 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) = (𝐼‘(𝑆‘-𝑁))) |
14 | 3, 10, 13 | ifbieq12d 4561 | . . 3 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))) = if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁)))) |
15 | 2, 14 | ifbieq2d 4559 | . 2 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁))))) |
16 | mulgval.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
17 | mulgval.p | . . 3 ⊢ + = (+g‘𝐺) | |
18 | mulgval.o | . . 3 ⊢ 0 = (0g‘𝐺) | |
19 | mulgval.i | . . 3 ⊢ 𝐼 = (invg‘𝐺) | |
20 | mulgval.t | . . 3 ⊢ · = (.g‘𝐺) | |
21 | 16, 17, 18, 19, 20 | mulgfval 19063 | . 2 ⊢ · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) |
22 | 18 | fvexi 6915 | . . 3 ⊢ 0 ∈ V |
23 | fvex 6914 | . . . 4 ⊢ (𝑆‘𝑁) ∈ V | |
24 | fvex 6914 | . . . 4 ⊢ (𝐼‘(𝑆‘-𝑁)) ∈ V | |
25 | 23, 24 | ifex 4583 | . . 3 ⊢ if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁))) ∈ V |
26 | 22, 25 | ifex 4583 | . 2 ⊢ if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁)))) ∈ V |
27 | 15, 21, 26 | ovmpoa 7581 | 1 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ifcif 4533 {csn 4633 class class class wbr 5153 × cxp 5680 ‘cfv 6554 (class class class)co 7424 0cc0 11158 1c1 11159 < clt 11298 -cneg 11495 ℕcn 12264 ℤcz 12610 seqcseq 14021 Basecbs 17213 +gcplusg 17266 0gc0g 17454 invgcminusg 18929 .gcmg 19061 |
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 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-n0 12525 df-z 12611 df-uz 12875 df-seq 14022 df-mulg 19062 |
This theorem is referenced by: mulg0 19068 mulgnn 19069 mulgnegnn 19078 subgmulg 19134 |
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