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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 486 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → 𝑛 = 𝑁) | |
2 | 1 | eqeq1d 2800 | . . 3 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑛 = 0 ↔ 𝑁 = 0)) |
3 | 1 | breq2d 5042 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (0 < 𝑛 ↔ 0 < 𝑁)) |
4 | simpr 488 | . . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
5 | 4 | sneqd 4537 | . . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → {𝑥} = {𝑋}) |
6 | 5 | xpeq2d 5549 | . . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (ℕ × {𝑥}) = (ℕ × {𝑋})) |
7 | 6 | seqeq3d 13372 | . . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑋}))) |
8 | mulgval.s | . . . . . 6 ⊢ 𝑆 = seq1( + , (ℕ × {𝑋})) | |
9 | 7, 8 | eqtr4di 2851 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = 𝑆) |
10 | 9, 1 | fveq12d 6652 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (𝑆‘𝑁)) |
11 | 1 | negeqd 10869 | . . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → -𝑛 = -𝑁) |
12 | 9, 11 | fveq12d 6652 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘-𝑛) = (𝑆‘-𝑁)) |
13 | 12 | fveq2d 6649 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) = (𝐼‘(𝑆‘-𝑁))) |
14 | 3, 10, 13 | ifbieq12d 4452 | . . 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 4450 | . 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 18218 | . 2 ⊢ · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) |
22 | 18 | fvexi 6659 | . . 3 ⊢ 0 ∈ V |
23 | fvex 6658 | . . . 4 ⊢ (𝑆‘𝑁) ∈ V | |
24 | fvex 6658 | . . . 4 ⊢ (𝐼‘(𝑆‘-𝑁)) ∈ V | |
25 | 23, 24 | ifex 4473 | . . 3 ⊢ if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁))) ∈ V |
26 | 22, 25 | ifex 4473 | . 2 ⊢ if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁)))) ∈ V |
27 | 15, 21, 26 | ovmpoa 7284 | 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 399 = wceq 1538 ∈ wcel 2111 ifcif 4425 {csn 4525 class class class wbr 5030 × cxp 5517 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 < clt 10664 -cneg 10860 ℕcn 11625 ℤcz 11969 seqcseq 13364 Basecbs 16475 +gcplusg 16557 0gc0g 16705 invgcminusg 18096 .gcmg 18216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-seq 13365 df-mulg 18217 |
This theorem is referenced by: mulg0 18223 mulgnn 18224 mulgnegnn 18230 subgmulg 18285 |
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