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Theorem mulgfvalg 12939
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
mulgfvalg (𝐺𝑉· = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))
Distinct variable groups:   𝑥, 0 ,𝑛   𝑥,𝐵,𝑛   𝑥, + ,𝑛   𝑥,𝐺,𝑛   𝑥,𝐼,𝑛
Allowed substitution hints:   · (𝑥,𝑛)   𝑉(𝑥,𝑛)

Proof of Theorem mulgfvalg
Dummy variables 𝑤 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulgval.t . 2 · = (.g𝐺)
2 df-mulg 12938 . . 3 .g = (𝑤 ∈ V ↦ (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑤) ↦ if(𝑛 = 0, (0g𝑤), seq1((+g𝑤), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑤)‘(𝑠‘-𝑛))))))
3 eqidd 2178 . . . 4 (𝑤 = 𝐺 → ℤ = ℤ)
4 fveq2 5515 . . . . 5 (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
5 mulgval.b . . . . 5 𝐵 = (Base‘𝐺)
64, 5eqtr4di 2228 . . . 4 (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
7 fveq2 5515 . . . . . 6 (𝑤 = 𝐺 → (0g𝑤) = (0g𝐺))
8 mulgval.o . . . . . 6 0 = (0g𝐺)
97, 8eqtr4di 2228 . . . . 5 (𝑤 = 𝐺 → (0g𝑤) = 0 )
10 seqex 10444 . . . . . . 7 seq1((+g𝑤), (ℕ × {𝑥})) ∈ V
1110a1i 9 . . . . . 6 (𝑤 = 𝐺 → seq1((+g𝑤), (ℕ × {𝑥})) ∈ V)
12 id 19 . . . . . . . . 9 (𝑠 = seq1((+g𝑤), (ℕ × {𝑥})) → 𝑠 = seq1((+g𝑤), (ℕ × {𝑥})))
13 fveq2 5515 . . . . . . . . . . 11 (𝑤 = 𝐺 → (+g𝑤) = (+g𝐺))
14 mulgval.p . . . . . . . . . . 11 + = (+g𝐺)
1513, 14eqtr4di 2228 . . . . . . . . . 10 (𝑤 = 𝐺 → (+g𝑤) = + )
1615seqeq2d 10449 . . . . . . . . 9 (𝑤 = 𝐺 → seq1((+g𝑤), (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑥})))
1712, 16sylan9eqr 2232 . . . . . . . 8 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → 𝑠 = seq1( + , (ℕ × {𝑥})))
1817fveq1d 5517 . . . . . . 7 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → (𝑠𝑛) = (seq1( + , (ℕ × {𝑥}))‘𝑛))
19 simpl 109 . . . . . . . . . 10 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → 𝑤 = 𝐺)
2019fveq2d 5519 . . . . . . . . 9 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → (invg𝑤) = (invg𝐺))
21 mulgval.i . . . . . . . . 9 𝐼 = (invg𝐺)
2220, 21eqtr4di 2228 . . . . . . . 8 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → (invg𝑤) = 𝐼)
2317fveq1d 5517 . . . . . . . 8 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → (𝑠‘-𝑛) = (seq1( + , (ℕ × {𝑥}))‘-𝑛))
2422, 23fveq12d 5522 . . . . . . 7 ((𝑤 = 𝐺𝑠 = seq1((+g𝑤), (ℕ × {𝑥}))) → ((invg𝑤)‘(𝑠‘-𝑛)) = (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))
2518, 24ifeq12d 3553 . . . . . 6 ((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))))
2611, 25csbied 3103 . . . . 5 (𝑤 = 𝐺seq1((+g𝑤), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑤)‘(𝑠‘-𝑛))) = if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))
279, 26ifeq12d 3553 . . . 4 (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)))))
283, 6, 27mpoeq123dv 5936 . . 3 (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))))))
29 elex 2748 . . 3 (𝐺𝑉𝐺 ∈ V)
30 zex 9260 . . . 4 ℤ ∈ V
31 basfn 12514 . . . . . 6 Base Fn V
32 funfvex 5532 . . . . . . 7 ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
3332funfni 5316 . . . . . 6 ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
3431, 29, 33sylancr 414 . . . . 5 (𝐺𝑉 → (Base‘𝐺) ∈ V)
355, 34eqeltrid 2264 . . . 4 (𝐺𝑉𝐵 ∈ V)
36 mpoexga 6212 . . . 4 ((ℤ ∈ V ∧ 𝐵 ∈ V) → (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) ∈ V)
3730, 35, 36sylancr 414 . . 3 (𝐺𝑉 → (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) ∈ V)
382, 28, 29, 37fvmptd3 5609 . 2 (𝐺𝑉 → (.g𝐺) = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))
391, 38eqtrid 2222 1 (𝐺𝑉· = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  Vcvv 2737  csb 3057  ifcif 3534  {csn 3592   class class class wbr 4003   × cxp 4624   Fn wfn 5211  cfv 5216  cmpo 5876  0cc0 7810  1c1 7811   < clt 7990  -cneg 8127  cn 8917  cz 9251  seqcseq 10442  Basecbs 12456  +gcplusg 12530  0gc0g 12695  invgcminusg 12832  .gcmg 12937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-neg 8129  df-inn 8918  df-z 9252  df-seqfrec 10443  df-ndx 12459  df-slot 12460  df-base 12462  df-mulg 12938
This theorem is referenced by:  mulgval  12940  mulgfng  12941  mulgpropdg  12978
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