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Theorem mulgval 17904
Description: Value of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
Hypotheses
Ref Expression
mulgval.b 𝐵 = (Base‘𝐺)
mulgval.p + = (+g𝐺)
mulgval.o 0 = (0g𝐺)
mulgval.i 𝐼 = (invg𝐺)
mulgval.t · = (.g𝐺)
mulgval.s 𝑆 = seq1( + , (ℕ × {𝑋}))
Assertion
Ref Expression
mulgval ((𝑁 ∈ ℤ ∧ 𝑋𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆𝑁), (𝐼‘(𝑆‘-𝑁)))))

Proof of Theorem mulgval
Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 476 . . . 4 ((𝑛 = 𝑁𝑥 = 𝑋) → 𝑛 = 𝑁)
21eqeq1d 2827 . . 3 ((𝑛 = 𝑁𝑥 = 𝑋) → (𝑛 = 0 ↔ 𝑁 = 0))
31breq2d 4887 . . . 4 ((𝑛 = 𝑁𝑥 = 𝑋) → (0 < 𝑛 ↔ 0 < 𝑁))
4 simpr 479 . . . . . . . . 9 ((𝑛 = 𝑁𝑥 = 𝑋) → 𝑥 = 𝑋)
54sneqd 4411 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → {𝑥} = {𝑋})
65xpeq2d 5376 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → (ℕ × {𝑥}) = (ℕ × {𝑋}))
76seqeq3d 13110 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑋})))
8 mulgval.s . . . . . 6 𝑆 = seq1( + , (ℕ × {𝑋}))
97, 8syl6eqr 2879 . . . . 5 ((𝑛 = 𝑁𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = 𝑆)
109, 1fveq12d 6444 . . . 4 ((𝑛 = 𝑁𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (𝑆𝑁))
111negeqd 10602 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → -𝑛 = -𝑁)
129, 11fveq12d 6444 . . . . 5 ((𝑛 = 𝑁𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘-𝑛) = (𝑆‘-𝑁))
1312fveq2d 6441 . . . 4 ((𝑛 = 𝑁𝑥 = 𝑋) → (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) = (𝐼‘(𝑆‘-𝑁)))
143, 10, 13ifbieq12d 4335 . . 3 ((𝑛 = 𝑁𝑥 = 𝑋) → if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))) = if(0 < 𝑁, (𝑆𝑁), (𝐼‘(𝑆‘-𝑁))))
152, 14ifbieq2d 4333 . 2 ((𝑛 = 𝑁𝑥 = 𝑋) → if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆𝑁), (𝐼‘(𝑆‘-𝑁)))))
16 mulgval.b . . 3 𝐵 = (Base‘𝐺)
17 mulgval.p . . 3 + = (+g𝐺)
18 mulgval.o . . 3 0 = (0g𝐺)
19 mulgval.i . . 3 𝐼 = (invg𝐺)
20 mulgval.t . . 3 · = (.g𝐺)
2116, 17, 18, 19, 20mulgfval 17903 . 2 · = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
2218fvexi 6451 . . 3 0 ∈ V
23 fvex 6450 . . . 4 (𝑆𝑁) ∈ V
24 fvex 6450 . . . 4 (𝐼‘(𝑆‘-𝑁)) ∈ V
2523, 24ifex 4356 . . 3 if(0 < 𝑁, (𝑆𝑁), (𝐼‘(𝑆‘-𝑁))) ∈ V
2622, 25ifex 4356 . 2 if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆𝑁), (𝐼‘(𝑆‘-𝑁)))) ∈ V
2715, 21, 26ovmpt2a 7056 1 ((𝑁 ∈ ℤ ∧ 𝑋𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆𝑁), (𝐼‘(𝑆‘-𝑁)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  ifcif 4308  {csn 4399   class class class wbr 4875   × cxp 5344  cfv 6127  (class class class)co 6910  0cc0 10259  1c1 10260   < clt 10398  -cneg 10593  cn 11357  cz 11711  seqcseq 13102  Basecbs 16229  +gcplusg 16312  0gc0g 16460  invgcminusg 17784  .gcmg 17901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-inf2 8822  ax-cnex 10315  ax-resscn 10316
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-neg 10595  df-z 11712  df-seq 13103  df-mulg 17902
This theorem is referenced by:  mulg0  17907  mulgnn  17908  mulgnegnn  17912  subgmulg  17966
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