MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulgfval Structured version   Visualization version   GIF version

Theorem mulgfval 17940
Description: 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)
Assertion
Ref Expression
mulgfval · = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
Distinct variable groups:   𝑥,𝑛, 0   𝑛,𝐺,𝑥   𝑛,𝐼,𝑥   𝐵,𝑛,𝑥
Allowed substitution hints:   + (𝑥,𝑛)   · (𝑥,𝑛)

Proof of Theorem mulgfval
Dummy variables 𝑤 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulgval.t . 2 · = (.g𝐺)
2 eqidd 2779 . . . . 5 (𝑤 = 𝐺 → ℤ = ℤ)
3 fveq2 6448 . . . . . 6 (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
4 mulgval.b . . . . . 6 𝐵 = (Base‘𝐺)
53, 4syl6eqr 2832 . . . . 5 (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
6 fveq2 6448 . . . . . . 7 (𝑤 = 𝐺 → (0g𝑤) = (0g𝐺))
7 mulgval.o . . . . . . 7 0 = (0g𝐺)
86, 7syl6eqr 2832 . . . . . 6 (𝑤 = 𝐺 → (0g𝑤) = 0 )
9 seqex 13126 . . . . . . . 8 seq1((+g𝑤), (ℕ × {𝑥})) ∈ V
109a1i 11 . . . . . . 7 (𝑤 = 𝐺 → seq1((+g𝑤), (ℕ × {𝑥})) ∈ V)
11 id 22 . . . . . . . . . 10 (𝑠 = seq1((+g𝑤), (ℕ × {𝑥})) → 𝑠 = seq1((+g𝑤), (ℕ × {𝑥})))
12 fveq2 6448 . . . . . . . . . . . 12 (𝑤 = 𝐺 → (+g𝑤) = (+g𝐺))
13 mulgval.p . . . . . . . . . . . 12 + = (+g𝐺)
1412, 13syl6eqr 2832 . . . . . . . . . . 11 (𝑤 = 𝐺 → (+g𝑤) = + )
1514seqeq2d 13131 . . . . . . . . . 10 (𝑤 = 𝐺 → seq1((+g𝑤), (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑥})))
1611, 15sylan9eqr 2836 . . . . . . . . 9 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → 𝑠 = seq1( + , (ℕ × {𝑥})))
1716fveq1d 6450 . . . . . . . 8 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → (𝑠𝑛) = (seq1( + , (ℕ × {𝑥}))‘𝑛))
18 simpl 476 . . . . . . . . . . 11 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → 𝑤 = 𝐺)
1918fveq2d 6452 . . . . . . . . . 10 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → (invg𝑤) = (invg𝐺))
20 mulgval.i . . . . . . . . . 10 𝐼 = (invg𝐺)
2119, 20syl6eqr 2832 . . . . . . . . 9 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → (invg𝑤) = 𝐼)
2216fveq1d 6450 . . . . . . . . 9 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → (𝑠‘-𝑛) = (seq1( + , (ℕ × {𝑥}))‘-𝑛))
2321, 22fveq12d 6455 . . . . . . . 8 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → ((invg𝑤)‘(𝑠‘-𝑛)) = (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))
2417, 23ifeq12d 4327 . . . . . . 7 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → if(0 < 𝑛, (𝑠𝑛), ((invg𝑤)‘(𝑠‘-𝑛))) = if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))
2510, 24csbied 3778 . . . . . 6 (𝑤 = 𝐺seq1((+g𝑤), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑤)‘(𝑠‘-𝑛))) = if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))
268, 25ifeq12d 4327 . . . . 5 (𝑤 = 𝐺 → if(𝑛 = 0, (0g𝑤), seq1((+g𝑤), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑤)‘(𝑠‘-𝑛)))) = if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
272, 5, 26mpt2eq123dv 6996 . . . 4 (𝑤 = 𝐺 → (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑤) ↦ if(𝑛 = 0, (0g𝑤), seq1((+g𝑤), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑤)‘(𝑠‘-𝑛))))) = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))
28 df-mulg 17939 . . . 4 .g = (𝑤 ∈ V ↦ (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑤) ↦ if(𝑛 = 0, (0g𝑤), seq1((+g𝑤), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑤)‘(𝑠‘-𝑛))))))
29 zex 11742 . . . . 5 ℤ ∈ V
304fvexi 6462 . . . . 5 𝐵 ∈ V
3129, 30mpt2ex 7529 . . . 4 (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) ∈ V
3227, 28, 31fvmpt 6544 . . 3 (𝐺 ∈ V → (.g𝐺) = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))
33 fvprc 6441 . . . 4 𝐺 ∈ V → (.g𝐺) = ∅)
34 eqid 2778 . . . . . . 7 (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
357fvexi 6462 . . . . . . . 8 0 ∈ V
36 fvex 6461 . . . . . . . . 9 (seq1( + , (ℕ × {𝑥}))‘𝑛) ∈ V
37 fvex 6461 . . . . . . . . 9 (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)) ∈ V
3836, 37ifex 4355 . . . . . . . 8 if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))) ∈ V
3935, 38ifex 4355 . . . . . . 7 if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))) ∈ V
4034, 39fnmpt2i 7521 . . . . . 6 (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) Fn (ℤ × 𝐵)
41 fvprc 6441 . . . . . . . . . 10 𝐺 ∈ V → (Base‘𝐺) = ∅)
424, 41syl5eq 2826 . . . . . . . . 9 𝐺 ∈ V → 𝐵 = ∅)
4342xpeq2d 5387 . . . . . . . 8 𝐺 ∈ V → (ℤ × 𝐵) = (ℤ × ∅))
44 xp0 5808 . . . . . . . 8 (ℤ × ∅) = ∅
4543, 44syl6eq 2830 . . . . . . 7 𝐺 ∈ V → (ℤ × 𝐵) = ∅)
4645fneq2d 6229 . . . . . 6 𝐺 ∈ V → ((𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) Fn (ℤ × 𝐵) ↔ (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) Fn ∅))
4740, 46mpbii 225 . . . . 5 𝐺 ∈ V → (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) Fn ∅)
48 fn0 6259 . . . . 5 ((𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) Fn ∅ ↔ (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) = ∅)
4947, 48sylib 210 . . . 4 𝐺 ∈ V → (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) = ∅)
5033, 49eqtr4d 2817 . . 3 𝐺 ∈ V → (.g𝐺) = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))
5132, 50pm2.61i 177 . 2 (.g𝐺) = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
521, 51eqtri 2802 1 · = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 386   = wceq 1601  wcel 2107  Vcvv 3398  csb 3751  c0 4141  ifcif 4307  {csn 4398   class class class wbr 4888   × cxp 5355   Fn wfn 6132  cfv 6137  cmpt2 6926  0cc0 10274  1c1 10275   < clt 10413  -cneg 10609  cn 11379  cz 11733  seqcseq 13124  Basecbs 16266  +gcplusg 16349  0gc0g 16497  invgcminusg 17821  .gcmg 17938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-inf2 8837  ax-cnex 10330  ax-resscn 10331
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-neg 10611  df-z 11734  df-seq 13125  df-mulg 17939
This theorem is referenced by:  mulgval  17941  mulgfn  17942  mulgpropd  17979
  Copyright terms: Public domain W3C validator