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Theorem mulgval 17751
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 468 . . . 4 ((𝑛 = 𝑁𝑥 = 𝑋) → 𝑛 = 𝑁)
21eqeq1d 2773 . . 3 ((𝑛 = 𝑁𝑥 = 𝑋) → (𝑛 = 0 ↔ 𝑁 = 0))
31breq2d 4798 . . . 4 ((𝑛 = 𝑁𝑥 = 𝑋) → (0 < 𝑛 ↔ 0 < 𝑁))
4 simpr 471 . . . . . . . . 9 ((𝑛 = 𝑁𝑥 = 𝑋) → 𝑥 = 𝑋)
54sneqd 4328 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → {𝑥} = {𝑋})
65xpeq2d 5279 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → (ℕ × {𝑥}) = (ℕ × {𝑋}))
76seqeq3d 13016 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑋})))
8 mulgval.s . . . . . 6 𝑆 = seq1( + , (ℕ × {𝑋}))
97, 8syl6eqr 2823 . . . . 5 ((𝑛 = 𝑁𝑥 = 𝑋) → seq1( + , (ℕ × {𝑥})) = 𝑆)
109, 1fveq12d 6338 . . . 4 ((𝑛 = 𝑁𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘𝑛) = (𝑆𝑁))
111negeqd 10477 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → -𝑛 = -𝑁)
129, 11fveq12d 6338 . . . . 5 ((𝑛 = 𝑁𝑥 = 𝑋) → (seq1( + , (ℕ × {𝑥}))‘-𝑛) = (𝑆‘-𝑁))
1312fveq2d 6336 . . . 4 ((𝑛 = 𝑁𝑥 = 𝑋) → (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) = (𝐼‘(𝑆‘-𝑁)))
143, 10, 13ifbieq12d 4252 . . 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 4250 . 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 17750 . 2 · = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
22 fvex 6342 . . . 4 (0g𝐺) ∈ V
2318, 22eqeltri 2846 . . 3 0 ∈ V
24 fvex 6342 . . . 4 (𝑆𝑁) ∈ V
25 fvex 6342 . . . 4 (𝐼‘(𝑆‘-𝑁)) ∈ V
2624, 25ifex 4295 . . 3 if(0 < 𝑁, (𝑆𝑁), (𝐼‘(𝑆‘-𝑁))) ∈ V
2723, 26ifex 4295 . 2 if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆𝑁), (𝐼‘(𝑆‘-𝑁)))) ∈ V
2815, 21, 27ovmpt2a 6938 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 382   = wceq 1631  wcel 2145  Vcvv 3351  ifcif 4225  {csn 4316   class class class wbr 4786   × cxp 5247  cfv 6031  (class class class)co 6793  0cc0 10138  1c1 10139   < clt 10276  -cneg 10469  cn 11222  cz 11579  seqcseq 13008  Basecbs 16064  +gcplusg 16149  0gc0g 16308  invgcminusg 17631  .gcmg 17748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-neg 10471  df-z 11580  df-seq 13009  df-mulg 17749
This theorem is referenced by:  mulg0  17754  mulgnn  17755  mulgnegnn  17759  subgmulg  17816
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