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	⊢ + = (+g‘𝐺)
mulgval.o	⊢ 0 = (0g‘𝐺)
mulgval.i	⊢ 𝐼 = (invg‘𝐺)
mulgval.t	⊢ · = (.g‘𝐺)

Assertion

Ref	Expression
mulgfvalg	⊢ (𝐺 ∈ 𝑉 → · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ 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.

Step	Hyp	Ref	Expression
1		mulgval.t	. 2 ⊢ · = (.g‘𝐺)
2		df-mulg 12938	. . 3 ⊢ .g = (𝑤 ∈ V ↦ (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑤) ↦ if(𝑛 = 0, (0g‘𝑤), ⦋seq1((+g‘𝑤), (ℕ × {𝑥})) / 𝑠⦌if(0 < 𝑛, (𝑠‘𝑛), ((invg‘𝑤)‘(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)
6	4, 5	eqtr4di 2228	. . . 4 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵)
7		fveq2 5515	. . . . . 6 ⊢ (𝑤 = 𝐺 → (0g‘𝑤) = (0g‘𝐺))
8		mulgval.o	. . . . . 6 ⊢ 0 = (0g‘𝐺)
9	7, 8	eqtr4di 2228	. . . . 5 ⊢ (𝑤 = 𝐺 → (0g‘𝑤) = 0 )
10		seqex 10444	. . . . . . 7 ⊢ seq1((+g‘𝑤), (ℕ × {𝑥})) ∈ V
11	10	a1i 9	. . . . . 6 ⊢ (𝑤 = 𝐺 → seq1((+g‘𝑤), (ℕ × {𝑥})) ∈ V)
12		id 19	. . . . . . . . 9 ⊢ (𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥})) → 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥})))
13		fveq2 5515	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = (+g‘𝐺))
14		mulgval.p	. . . . . . . . . . 11 ⊢ + = (+g‘𝐺)
15	13, 14	eqtr4di 2228	. . . . . . . . . 10 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = + )
16	15	seqeq2d 10449	. . . . . . . . 9 ⊢ (𝑤 = 𝐺 → seq1((+g‘𝑤), (ℕ × {𝑥})) = seq1( + , (ℕ × {𝑥})))
17	12, 16	sylan9eqr 2232	. . . . . . . 8 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → 𝑠 = seq1( + , (ℕ × {𝑥})))
18	17	fveq1d 5517	. . . . . . 7 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (𝑠‘𝑛) = (seq1( + , (ℕ × {𝑥}))‘𝑛))
19		simpl 109	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → 𝑤 = 𝐺)
20	19	fveq2d 5519	. . . . . . . . 9 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (invg‘𝑤) = (invg‘𝐺))
21		mulgval.i	. . . . . . . . 9 ⊢ 𝐼 = (invg‘𝐺)
22	20, 21	eqtr4di 2228	. . . . . . . 8 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (invg‘𝑤) = 𝐼)
23	17	fveq1d 5517	. . . . . . . 8 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → (𝑠‘-𝑛) = (seq1( + , (ℕ × {𝑥}))‘-𝑛))
24	22, 23	fveq12d 5522	. . . . . . 7 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → ((invg‘𝑤)‘(𝑠‘-𝑛)) = (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))
25	18, 24	ifeq12d 3553	. . . . . 6 ⊢ ((𝑤 = 𝐺 ∧ 𝑠 = seq1((+g‘𝑤), (ℕ × {𝑥}))) → if(0 < 𝑛, (𝑠‘𝑛), ((invg‘𝑤)‘(𝑠‘-𝑛))) = if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))
26	11, 25	csbied 3103	. . . . 5 ⊢ (𝑤 = 𝐺 → ⦋seq1((+g‘𝑤), (ℕ × {𝑥})) / 𝑠⦌if(0 < 𝑛, (𝑠‘𝑛), ((invg‘𝑤)‘(𝑠‘-𝑛))) = if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))
27	9, 26	ifeq12d 3553	. . . 4 ⊢ (𝑤 = 𝐺 → if(𝑛 = 0, (0g‘𝑤), ⦋seq1((+g‘𝑤), (ℕ × {𝑥})) / 𝑠⦌if(0 < 𝑛, (𝑠‘𝑛), ((invg‘𝑤)‘(𝑠‘-𝑛)))) = if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
28	3, 6, 27	mpoeq123dv 5936	. . 3 ⊢ (𝑤 = 𝐺 → (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑤) ↦ if(𝑛 = 0, (0g‘𝑤), ⦋seq1((+g‘𝑤), (ℕ × {𝑥})) / 𝑠⦌if(0 < 𝑛, (𝑠‘𝑛), ((invg‘𝑤)‘(𝑠‘-𝑛))))) = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))
29		elex 2748	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ 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)
33	32	funfni 5316	. . . . . 6 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
34	31, 29, 33	sylancr 414	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → (Base‘𝐺) ∈ V)
35	5, 34	eqeltrid 2264	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝐵 ∈ V)
36		mpoexga 6212	. . . 4 ⊢ ((ℤ ∈ V ∧ 𝐵 ∈ V) → (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) ∈ V)
37	30, 35, 36	sylancr 414	. . 3 ⊢ (𝐺 ∈ 𝑉 → (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))) ∈ V)
38	2, 28, 29, 37	fvmptd3 5609	. 2 ⊢ (𝐺 ∈ 𝑉 → (.g‘𝐺) = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))
39	1, 38	eqtrid 2222	1 ⊢ (𝐺 ∈ 𝑉 → · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛))))))

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  ℕcn 8917  ℤcz 9251  seqcseq 10442  Basecbs 12456  +_gcplusg 12530  0_gc0g 12695  inv_gcminusg 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