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Theorem mulgval 17742
 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 474 . . . 4 ((𝑛 = 𝑁𝑥 = 𝑋) → 𝑛 = 𝑁)
21eqeq1d 2760 . . 3 ((𝑛 = 𝑁𝑥 = 𝑋) → (𝑛 = 0 ↔ 𝑁 = 0))
31breq2d 4814 . . . 4 ((𝑛 = 𝑁𝑥 = 𝑋) → (0 < 𝑛 ↔ 0 < 𝑁))
4 simpr 479 . . . . . . . . 9 ((𝑛 = 𝑁𝑥 = 𝑋) → 𝑥 = 𝑋)
54sneqd 4331 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → {𝑥} = {𝑋})
65xpeq2d 5294 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → (ℕ × {𝑥}) = (ℕ × {𝑋}))
76seqeq3d 13001 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑋})))
8 mulgval.s . . . . . 6 𝑆 = seq1( + , (ℕ × {𝑋}))
97, 8syl6eqr 2810 . . . . 5 ((𝑛 = 𝑁𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = 𝑆)
109, 1fveq12d 6356 . . . 4 ((𝑛 = 𝑁𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (𝑆𝑁))
111negeqd 10465 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → -𝑛 = -𝑁)
129, 11fveq12d 6356 . . . . 5 ((𝑛 = 𝑁𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘-𝑛) = (𝑆‘-𝑁))
1312fveq2d 6354 . . . 4 ((𝑛 = 𝑁𝑥 = 𝑋) → (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) = (𝐼‘(𝑆‘-𝑁)))
143, 10, 13ifbieq12d 4255 . . 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 4253 . 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 17741 . 2 · = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
22 fvex 6360 . . . 4 (0g𝐺) ∈ V
2318, 22eqeltri 2833 . . 3 0 ∈ V
24 fvex 6360 . . . 4 (𝑆𝑁) ∈ V
25 fvex 6360 . . . 4 (𝐼‘(𝑆‘-𝑁)) ∈ V
2624, 25ifex 4298 . . 3 if(0 < 𝑁, (𝑆𝑁), (𝐼‘(𝑆‘-𝑁))) ∈ V
2723, 26ifex 4298 . 2 if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆𝑁), (𝐼‘(𝑆‘-𝑁)))) ∈ V
2815, 21, 27ovmpt2a 6954 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 383   = wceq 1630   ∈ wcel 2137  Vcvv 3338  ifcif 4228  {csn 4319   class class class wbr 4802   × cxp 5262  ‘cfv 6047  (class class class)co 6811  0cc0 10126  1c1 10127   < clt 10264  -cneg 10457  ℕcn 11210  ℤcz 11567  seqcseq 12993  Basecbs 16057  +gcplusg 16141  0gc0g 16300  invgcminusg 17622  .gcmg 17739 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-inf2 8709  ax-cnex 10182  ax-resscn 10183 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-1st 7331  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-neg 10459  df-z 11568  df-seq 12994  df-mulg 17740 This theorem is referenced by:  mulg0  17745  mulgnn  17746  mulgnegnn  17750  subgmulg  17807
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