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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 482 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → 𝑛 = 𝑁) | |
| 2 | 1 | eqeq1d 2738 | . . 3 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑛 = 0 ↔ 𝑁 = 0)) | 
| 3 | 1 | breq2d 5154 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (0 < 𝑛 ↔ 0 < 𝑁)) | 
| 4 | simpr 484 | . . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
| 5 | 4 | sneqd 4637 | . . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → {𝑥} = {𝑋}) | 
| 6 | 5 | xpeq2d 5714 | . . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (ℕ × {𝑥}) = (ℕ × {𝑋})) | 
| 7 | 6 | seqeq3d 14051 | . . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑋}))) | 
| 8 | mulgval.s | . . . . . 6 ⊢ 𝑆 = seq1( + , (ℕ × {𝑋})) | |
| 9 | 7, 8 | eqtr4di 2794 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = 𝑆) | 
| 10 | 9, 1 | fveq12d 6912 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (𝑆‘𝑁)) | 
| 11 | 1 | negeqd 11503 | . . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → -𝑛 = -𝑁) | 
| 12 | 9, 11 | fveq12d 6912 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘-𝑛) = (𝑆‘-𝑁)) | 
| 13 | 12 | fveq2d 6909 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) = (𝐼‘(𝑆‘-𝑁))) | 
| 14 | 3, 10, 13 | ifbieq12d 4553 | . . 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 4551 | . 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 19088 | . 2 ⊢ · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) | 
| 22 | 18 | fvexi 6919 | . . 3 ⊢ 0 ∈ V | 
| 23 | fvex 6918 | . . . 4 ⊢ (𝑆‘𝑁) ∈ V | |
| 24 | fvex 6918 | . . . 4 ⊢ (𝐼‘(𝑆‘-𝑁)) ∈ V | |
| 25 | 23, 24 | ifex 4575 | . . 3 ⊢ if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁))) ∈ V | 
| 26 | 22, 25 | ifex 4575 | . 2 ⊢ if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁)))) ∈ V | 
| 27 | 15, 21, 26 | ovmpoa 7589 | 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 395 = wceq 1539 ∈ wcel 2107 ifcif 4524 {csn 4625 class class class wbr 5142 × cxp 5682 ‘cfv 6560 (class class class)co 7432 0cc0 11156 1c1 11157 < clt 11296 -cneg 11494 ℕcn 12267 ℤcz 12615 seqcseq 14043 Basecbs 17248 +gcplusg 17298 0gc0g 17485 invgcminusg 18953 .gcmg 19086 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-n0 12529 df-z 12616 df-uz 12880 df-seq 14044 df-mulg 19087 | 
| This theorem is referenced by: mulg0 19093 mulgnn 19094 mulgnegnn 19103 subgmulg 19159 | 
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