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Theorem mulgfng 12843
Description: Functionality of the group multiple operation. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
mulgfn.b 𝐵 = (Base‘𝐺)
mulgfn.t · = (.g𝐺)
Assertion
Ref Expression
mulgfng (𝐺𝑉· Fn (ℤ × 𝐵))

Proof of Theorem mulgfng
Dummy variables 𝑢 𝑣 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2744 . . . . . . 7 (𝐺𝑉𝐺 ∈ V)
2 fn0g 12656 . . . . . . . 8 0g Fn V
3 funfvex 5521 . . . . . . . . 9 ((Fun 0g𝐺 ∈ dom 0g) → (0g𝐺) ∈ V)
43funfni 5305 . . . . . . . 8 ((0g Fn V ∧ 𝐺 ∈ V) → (0g𝐺) ∈ V)
52, 4mpan 424 . . . . . . 7 (𝐺 ∈ V → (0g𝐺) ∈ V)
61, 5syl 14 . . . . . 6 (𝐺𝑉 → (0g𝐺) ∈ V)
76ad2antrr 488 . . . . 5 (((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ 𝑛 = 0) → (0g𝐺) ∈ V)
8 nnuz 9531 . . . . . . . . . 10 ℕ = (ℤ‘1)
9 1zzd 9248 . . . . . . . . . 10 ((𝐺𝑉𝑥𝐵) → 1 ∈ ℤ)
10 fvconst2g 5719 . . . . . . . . . . . . 13 ((𝑥𝐵𝑢 ∈ ℕ) → ((ℕ × {𝑥})‘𝑢) = 𝑥)
11 simpl 109 . . . . . . . . . . . . 13 ((𝑥𝐵𝑢 ∈ ℕ) → 𝑥𝐵)
1210, 11eqeltrd 2250 . . . . . . . . . . . 12 ((𝑥𝐵𝑢 ∈ ℕ) → ((ℕ × {𝑥})‘𝑢) ∈ 𝐵)
1312elexd 2746 . . . . . . . . . . 11 ((𝑥𝐵𝑢 ∈ ℕ) → ((ℕ × {𝑥})‘𝑢) ∈ V)
1413adantll 476 . . . . . . . . . 10 (((𝐺𝑉𝑥𝐵) ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑥})‘𝑢) ∈ V)
15 simprl 529 . . . . . . . . . . 11 (((𝐺𝑉𝑥𝐵) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → 𝑢 ∈ V)
16 plusgslid 12522 . . . . . . . . . . . . 13 (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
1716slotex 12452 . . . . . . . . . . . 12 (𝐺𝑉 → (+g𝐺) ∈ V)
1817ad2antrr 488 . . . . . . . . . . 11 (((𝐺𝑉𝑥𝐵) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (+g𝐺) ∈ V)
19 simprr 530 . . . . . . . . . . 11 (((𝐺𝑉𝑥𝐵) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → 𝑣 ∈ V)
20 ovexg 5896 . . . . . . . . . . 11 ((𝑢 ∈ V ∧ (+g𝐺) ∈ V ∧ 𝑣 ∈ V) → (𝑢(+g𝐺)𝑣) ∈ V)
2115, 18, 19, 20syl3anc 1236 . . . . . . . . . 10 (((𝐺𝑉𝑥𝐵) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (𝑢(+g𝐺)𝑣) ∈ V)
228, 9, 14, 21seqf 10426 . . . . . . . . 9 ((𝐺𝑉𝑥𝐵) → seq1((+g𝐺), (ℕ × {𝑥})):ℕ⟶V)
2322adantrl 478 . . . . . . . 8 ((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) → seq1((+g𝐺), (ℕ × {𝑥})):ℕ⟶V)
2423ad2antrr 488 . . . . . . 7 ((((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) ∧ 0 < 𝑛) → seq1((+g𝐺), (ℕ × {𝑥})):ℕ⟶V)
25 simprl 529 . . . . . . . . 9 ((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) → 𝑛 ∈ ℤ)
2625ad2antrr 488 . . . . . . . 8 ((((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) ∧ 0 < 𝑛) → 𝑛 ∈ ℤ)
27 simpr 110 . . . . . . . 8 ((((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) ∧ 0 < 𝑛) → 0 < 𝑛)
28 elnnz 9231 . . . . . . . 8 (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℤ ∧ 0 < 𝑛))
2926, 27, 28sylanbrc 417 . . . . . . 7 ((((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) ∧ 0 < 𝑛) → 𝑛 ∈ ℕ)
3024, 29ffvelrnd 5641 . . . . . 6 ((((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) ∧ 0 < 𝑛) → (seq1((+g𝐺), (ℕ × {𝑥}))‘𝑛) ∈ V)
31 mulgfn.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
32 eqid 2173 . . . . . . . . . 10 (invg𝐺) = (invg𝐺)
3331, 32grpinvfng 12774 . . . . . . . . 9 (𝐺𝑉 → (invg𝐺) Fn 𝐵)
34 basfn 12482 . . . . . . . . . . . 12 Base Fn V
35 funfvex 5521 . . . . . . . . . . . . 13 ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
3635funfni 5305 . . . . . . . . . . . 12 ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
3734, 36mpan 424 . . . . . . . . . . 11 (𝐺 ∈ V → (Base‘𝐺) ∈ V)
3831, 37eqeltrid 2260 . . . . . . . . . 10 (𝐺 ∈ V → 𝐵 ∈ V)
391, 38syl 14 . . . . . . . . 9 (𝐺𝑉𝐵 ∈ V)
40 fnex 5727 . . . . . . . . 9 (((invg𝐺) Fn 𝐵𝐵 ∈ V) → (invg𝐺) ∈ V)
4133, 39, 40syl2anc 411 . . . . . . . 8 (𝐺𝑉 → (invg𝐺) ∈ V)
4241ad3antrrr 492 . . . . . . 7 ((((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → (invg𝐺) ∈ V)
4323ad2antrr 488 . . . . . . . 8 ((((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → seq1((+g𝐺), (ℕ × {𝑥})):ℕ⟶V)
4425znegcld 9345 . . . . . . . . . 10 ((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) → -𝑛 ∈ ℤ)
4544ad2antrr 488 . . . . . . . . 9 ((((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → -𝑛 ∈ ℤ)
46 simplr 528 . . . . . . . . . . 11 ((((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → ¬ 𝑛 = 0)
47 simpr 110 . . . . . . . . . . 11 ((((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → ¬ 0 < 𝑛)
48 ztri3or0 9263 . . . . . . . . . . . . 13 (𝑛 ∈ ℤ → (𝑛 < 0 ∨ 𝑛 = 0 ∨ 0 < 𝑛))
4925, 48syl 14 . . . . . . . . . . . 12 ((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) → (𝑛 < 0 ∨ 𝑛 = 0 ∨ 0 < 𝑛))
5049ad2antrr 488 . . . . . . . . . . 11 ((((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → (𝑛 < 0 ∨ 𝑛 = 0 ∨ 0 < 𝑛))
5146, 47, 50ecase23d 1348 . . . . . . . . . 10 ((((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → 𝑛 < 0)
5225zred 9343 . . . . . . . . . . . 12 ((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) → 𝑛 ∈ ℝ)
5352ad2antrr 488 . . . . . . . . . . 11 ((((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → 𝑛 ∈ ℝ)
5453lt0neg1d 8443 . . . . . . . . . 10 ((((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → (𝑛 < 0 ↔ 0 < -𝑛))
5551, 54mpbid 148 . . . . . . . . 9 ((((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → 0 < -𝑛)
56 elnnz 9231 . . . . . . . . 9 (-𝑛 ∈ ℕ ↔ (-𝑛 ∈ ℤ ∧ 0 < -𝑛))
5745, 55, 56sylanbrc 417 . . . . . . . 8 ((((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → -𝑛 ∈ ℕ)
5843, 57ffvelrnd 5641 . . . . . . 7 ((((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → (seq1((+g𝐺), (ℕ × {𝑥}))‘-𝑛) ∈ V)
59 fvexg 5523 . . . . . . 7 (((invg𝐺) ∈ V ∧ (seq1((+g𝐺), (ℕ × {𝑥}))‘-𝑛) ∈ V) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑥}))‘-𝑛)) ∈ V)
6042, 58, 59syl2anc 411 . . . . . 6 ((((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) ∧ ¬ 0 < 𝑛) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑥}))‘-𝑛)) ∈ V)
61 0zd 9233 . . . . . . 7 (((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) → 0 ∈ ℤ)
62 simplrl 533 . . . . . . 7 (((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℤ)
63 zdclt 9298 . . . . . . 7 ((0 ∈ ℤ ∧ 𝑛 ∈ ℤ) → DECID 0 < 𝑛)
6461, 62, 63syl2anc 411 . . . . . 6 (((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) → DECID 0 < 𝑛)
6530, 60, 64ifcldadc 3558 . . . . 5 (((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) ∧ ¬ 𝑛 = 0) → if(0 < 𝑛, (seq1((+g𝐺), (ℕ × {𝑥}))‘𝑛), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑥}))‘-𝑛))) ∈ V)
66 0zd 9233 . . . . . 6 ((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) → 0 ∈ ℤ)
67 zdceq 9296 . . . . . 6 ((𝑛 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑛 = 0)
6825, 66, 67syl2anc 411 . . . . 5 ((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) → DECID 𝑛 = 0)
697, 65, 68ifcldadc 3558 . . . 4 ((𝐺𝑉 ∧ (𝑛 ∈ ℤ ∧ 𝑥𝐵)) → if(𝑛 = 0, (0g𝐺), if(0 < 𝑛, (seq1((+g𝐺), (ℕ × {𝑥}))‘𝑛), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑥}))‘-𝑛)))) ∈ V)
7069ralrimivva 2555 . . 3 (𝐺𝑉 → ∀𝑛 ∈ ℤ ∀𝑥𝐵 if(𝑛 = 0, (0g𝐺), if(0 < 𝑛, (seq1((+g𝐺), (ℕ × {𝑥}))‘𝑛), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑥}))‘-𝑛)))) ∈ V)
71 eqid 2173 . . . 4 (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, (0g𝐺), if(0 < 𝑛, (seq1((+g𝐺), (ℕ × {𝑥}))‘𝑛), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑥}))‘-𝑛))))) = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, (0g𝐺), if(0 < 𝑛, (seq1((+g𝐺), (ℕ × {𝑥}))‘𝑛), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑥}))‘-𝑛)))))
7271fnmpo 6190 . . 3 (∀𝑛 ∈ ℤ ∀𝑥𝐵 if(𝑛 = 0, (0g𝐺), if(0 < 𝑛, (seq1((+g𝐺), (ℕ × {𝑥}))‘𝑛), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑥}))‘-𝑛)))) ∈ V → (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, (0g𝐺), if(0 < 𝑛, (seq1((+g𝐺), (ℕ × {𝑥}))‘𝑛), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑥}))‘-𝑛))))) Fn (ℤ × 𝐵))
7370, 72syl 14 . 2 (𝐺𝑉 → (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, (0g𝐺), if(0 < 𝑛, (seq1((+g𝐺), (ℕ × {𝑥}))‘𝑛), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑥}))‘-𝑛))))) Fn (ℤ × 𝐵))
74 eqid 2173 . . . 4 (+g𝐺) = (+g𝐺)
75 eqid 2173 . . . 4 (0g𝐺) = (0g𝐺)
76 mulgfn.t . . . 4 · = (.g𝐺)
7731, 74, 75, 32, 76mulgfvalg 12841 . . 3 (𝐺𝑉· = (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, (0g𝐺), if(0 < 𝑛, (seq1((+g𝐺), (ℕ × {𝑥}))‘𝑛), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑥}))‘-𝑛))))))
7877fneq1d 5295 . 2 (𝐺𝑉 → ( · Fn (ℤ × 𝐵) ↔ (𝑛 ∈ ℤ, 𝑥𝐵 ↦ if(𝑛 = 0, (0g𝐺), if(0 < 𝑛, (seq1((+g𝐺), (ℕ × {𝑥}))‘𝑛), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑥}))‘-𝑛))))) Fn (ℤ × 𝐵)))
7973, 78mpbird 168 1 (𝐺𝑉· Fn (ℤ × 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  DECID wdc 832  w3o 975   = wceq 1351  wcel 2144  wral 2451  Vcvv 2733  ifcif 3529  {csn 3586   class class class wbr 3995   × cxp 4615   Fn wfn 5200  wf 5201  cfv 5205  (class class class)co 5862  cmpo 5864  cr 7782  0cc0 7783  1c1 7784   < clt 7963  -cneg 8100  cn 8887  cz 9221  seqcseq 10410  Basecbs 12425  +gcplusg 12489  0gc0g 12623  invgcminusg 12736  .gcmg 12839
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-in1 612  ax-in2 613  ax-io 707  ax-5 1443  ax-7 1444  ax-gen 1445  ax-ie1 1489  ax-ie2 1490  ax-8 1500  ax-10 1501  ax-11 1502  ax-i12 1503  ax-bndl 1505  ax-4 1506  ax-17 1522  ax-i9 1526  ax-ial 1530  ax-i5r 1531  ax-13 2146  ax-14 2147  ax-ext 2155  ax-coll 4110  ax-sep 4113  ax-nul 4121  ax-pow 4166  ax-pr 4200  ax-un 4424  ax-setind 4527  ax-iinf 4578  ax-cnex 7874  ax-resscn 7875  ax-1cn 7876  ax-1re 7877  ax-icn 7878  ax-addcl 7879  ax-addrcl 7880  ax-mulcl 7881  ax-addcom 7883  ax-addass 7885  ax-distr 7887  ax-i2m1 7888  ax-0lt1 7889  ax-0id 7891  ax-rnegex 7892  ax-cnre 7894  ax-pre-ltirr 7895  ax-pre-ltwlin 7896  ax-pre-lttrn 7897  ax-pre-ltadd 7899
This theorem depends on definitions:  df-bi 117  df-dc 833  df-3or 977  df-3an 978  df-tru 1354  df-fal 1357  df-nf 1457  df-sb 1759  df-eu 2025  df-mo 2026  df-clab 2160  df-cleq 2166  df-clel 2169  df-nfc 2304  df-ne 2344  df-nel 2439  df-ral 2456  df-rex 2457  df-reu 2458  df-rab 2460  df-v 2735  df-sbc 2959  df-csb 3053  df-dif 3126  df-un 3128  df-in 3130  df-ss 3137  df-nul 3418  df-if 3530  df-pw 3571  df-sn 3592  df-pr 3593  df-op 3595  df-uni 3803  df-int 3838  df-iun 3881  df-br 3996  df-opab 4057  df-mpt 4058  df-tr 4094  df-id 4284  df-iord 4357  df-on 4359  df-ilim 4360  df-suc 4362  df-iom 4581  df-xp 4623  df-rel 4624  df-cnv 4625  df-co 4626  df-dm 4627  df-rn 4628  df-res 4629  df-ima 4630  df-iota 5167  df-fun 5207  df-fn 5208  df-f 5209  df-f1 5210  df-fo 5211  df-f1o 5212  df-fv 5213  df-riota 5818  df-ov 5865  df-oprab 5866  df-mpo 5867  df-1st 6128  df-2nd 6129  df-recs 6293  df-frec 6379  df-pnf 7965  df-mnf 7966  df-xr 7967  df-ltxr 7968  df-le 7969  df-sub 8101  df-neg 8102  df-inn 8888  df-2 8946  df-n0 9145  df-z 9222  df-uz 9497  df-seqfrec 10411  df-ndx 12428  df-slot 12429  df-base 12431  df-plusg 12502  df-0g 12625  df-minusg 12739  df-mulg 12840
This theorem is referenced by: (None)
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