MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulgneg2 Structured version   Visualization version   GIF version

Theorem mulgneg2 17515
Description: Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 13-Dec-2014.)
Hypotheses
Ref Expression
mulgneg2.b 𝐵 = (Base‘𝐺)
mulgneg2.m · = (.g𝐺)
mulgneg2.i 𝐼 = (invg𝐺)
Assertion
Ref Expression
mulgneg2 ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) → (-𝑁 · 𝑋) = (𝑁 · (𝐼𝑋)))

Proof of Theorem mulgneg2
Dummy variables 𝑥 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 negeq 10233 . . . . . . 7 (𝑥 = 0 → -𝑥 = -0)
2 neg0 10287 . . . . . . 7 -0 = 0
31, 2syl6eq 2671 . . . . . 6 (𝑥 = 0 → -𝑥 = 0)
43oveq1d 6630 . . . . 5 (𝑥 = 0 → (-𝑥 · 𝑋) = (0 · 𝑋))
5 oveq1 6622 . . . . 5 (𝑥 = 0 → (𝑥 · (𝐼𝑋)) = (0 · (𝐼𝑋)))
64, 5eqeq12d 2636 . . . 4 (𝑥 = 0 → ((-𝑥 · 𝑋) = (𝑥 · (𝐼𝑋)) ↔ (0 · 𝑋) = (0 · (𝐼𝑋))))
7 negeq 10233 . . . . . 6 (𝑥 = 𝑛 → -𝑥 = -𝑛)
87oveq1d 6630 . . . . 5 (𝑥 = 𝑛 → (-𝑥 · 𝑋) = (-𝑛 · 𝑋))
9 oveq1 6622 . . . . 5 (𝑥 = 𝑛 → (𝑥 · (𝐼𝑋)) = (𝑛 · (𝐼𝑋)))
108, 9eqeq12d 2636 . . . 4 (𝑥 = 𝑛 → ((-𝑥 · 𝑋) = (𝑥 · (𝐼𝑋)) ↔ (-𝑛 · 𝑋) = (𝑛 · (𝐼𝑋))))
11 negeq 10233 . . . . . 6 (𝑥 = (𝑛 + 1) → -𝑥 = -(𝑛 + 1))
1211oveq1d 6630 . . . . 5 (𝑥 = (𝑛 + 1) → (-𝑥 · 𝑋) = (-(𝑛 + 1) · 𝑋))
13 oveq1 6622 . . . . 5 (𝑥 = (𝑛 + 1) → (𝑥 · (𝐼𝑋)) = ((𝑛 + 1) · (𝐼𝑋)))
1412, 13eqeq12d 2636 . . . 4 (𝑥 = (𝑛 + 1) → ((-𝑥 · 𝑋) = (𝑥 · (𝐼𝑋)) ↔ (-(𝑛 + 1) · 𝑋) = ((𝑛 + 1) · (𝐼𝑋))))
15 negeq 10233 . . . . . 6 (𝑥 = -𝑛 → -𝑥 = --𝑛)
1615oveq1d 6630 . . . . 5 (𝑥 = -𝑛 → (-𝑥 · 𝑋) = (--𝑛 · 𝑋))
17 oveq1 6622 . . . . 5 (𝑥 = -𝑛 → (𝑥 · (𝐼𝑋)) = (-𝑛 · (𝐼𝑋)))
1816, 17eqeq12d 2636 . . . 4 (𝑥 = -𝑛 → ((-𝑥 · 𝑋) = (𝑥 · (𝐼𝑋)) ↔ (--𝑛 · 𝑋) = (-𝑛 · (𝐼𝑋))))
19 negeq 10233 . . . . . 6 (𝑥 = 𝑁 → -𝑥 = -𝑁)
2019oveq1d 6630 . . . . 5 (𝑥 = 𝑁 → (-𝑥 · 𝑋) = (-𝑁 · 𝑋))
21 oveq1 6622 . . . . 5 (𝑥 = 𝑁 → (𝑥 · (𝐼𝑋)) = (𝑁 · (𝐼𝑋)))
2220, 21eqeq12d 2636 . . . 4 (𝑥 = 𝑁 → ((-𝑥 · 𝑋) = (𝑥 · (𝐼𝑋)) ↔ (-𝑁 · 𝑋) = (𝑁 · (𝐼𝑋))))
23 mulgneg2.b . . . . . . 7 𝐵 = (Base‘𝐺)
24 eqid 2621 . . . . . . 7 (0g𝐺) = (0g𝐺)
25 mulgneg2.m . . . . . . 7 · = (.g𝐺)
2623, 24, 25mulg0 17486 . . . . . 6 (𝑋𝐵 → (0 · 𝑋) = (0g𝐺))
2726adantl 482 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (0 · 𝑋) = (0g𝐺))
28 mulgneg2.i . . . . . . 7 𝐼 = (invg𝐺)
2923, 28grpinvcl 17407 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝐼𝑋) ∈ 𝐵)
3023, 24, 25mulg0 17486 . . . . . 6 ((𝐼𝑋) ∈ 𝐵 → (0 · (𝐼𝑋)) = (0g𝐺))
3129, 30syl 17 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (0 · (𝐼𝑋)) = (0g𝐺))
3227, 31eqtr4d 2658 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (0 · 𝑋) = (0 · (𝐼𝑋)))
33 oveq1 6622 . . . . . 6 ((-𝑛 · 𝑋) = (𝑛 · (𝐼𝑋)) → ((-𝑛 · 𝑋)(+g𝐺)(𝐼𝑋)) = ((𝑛 · (𝐼𝑋))(+g𝐺)(𝐼𝑋)))
34 nn0cn 11262 . . . . . . . . . . 11 (𝑛 ∈ ℕ0𝑛 ∈ ℂ)
3534adantl 482 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℂ)
36 ax-1cn 9954 . . . . . . . . . 10 1 ∈ ℂ
37 negdi 10298 . . . . . . . . . 10 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝑛 + 1) = (-𝑛 + -1))
3835, 36, 37sylancl 693 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ0) → -(𝑛 + 1) = (-𝑛 + -1))
3938oveq1d 6630 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ0) → (-(𝑛 + 1) · 𝑋) = ((-𝑛 + -1) · 𝑋))
40 simpll 789 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ0) → 𝐺 ∈ Grp)
41 nn0negz 11375 . . . . . . . . . 10 (𝑛 ∈ ℕ0 → -𝑛 ∈ ℤ)
4241adantl 482 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ0) → -𝑛 ∈ ℤ)
43 1z 11367 . . . . . . . . . 10 1 ∈ ℤ
44 znegcl 11372 . . . . . . . . . 10 (1 ∈ ℤ → -1 ∈ ℤ)
4543, 44mp1i 13 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ0) → -1 ∈ ℤ)
46 simplr 791 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑋𝐵)
47 eqid 2621 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
4823, 25, 47mulgdir 17513 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (-𝑛 ∈ ℤ ∧ -1 ∈ ℤ ∧ 𝑋𝐵)) → ((-𝑛 + -1) · 𝑋) = ((-𝑛 · 𝑋)(+g𝐺)(-1 · 𝑋)))
4940, 42, 45, 46, 48syl13anc 1325 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ0) → ((-𝑛 + -1) · 𝑋) = ((-𝑛 · 𝑋)(+g𝐺)(-1 · 𝑋)))
5023, 25, 28mulgm1 17502 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (-1 · 𝑋) = (𝐼𝑋))
5150adantr 481 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ0) → (-1 · 𝑋) = (𝐼𝑋))
5251oveq2d 6631 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ0) → ((-𝑛 · 𝑋)(+g𝐺)(-1 · 𝑋)) = ((-𝑛 · 𝑋)(+g𝐺)(𝐼𝑋)))
5339, 49, 523eqtrd 2659 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ0) → (-(𝑛 + 1) · 𝑋) = ((-𝑛 · 𝑋)(+g𝐺)(𝐼𝑋)))
54 grpmnd 17369 . . . . . . . . 9 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
5554ad2antrr 761 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ0) → 𝐺 ∈ Mnd)
56 simpr 477 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
5729adantr 481 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ0) → (𝐼𝑋) ∈ 𝐵)
5823, 25, 47mulgnn0p1 17492 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝑛 ∈ ℕ0 ∧ (𝐼𝑋) ∈ 𝐵) → ((𝑛 + 1) · (𝐼𝑋)) = ((𝑛 · (𝐼𝑋))(+g𝐺)(𝐼𝑋)))
5955, 56, 57, 58syl3anc 1323 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑛 + 1) · (𝐼𝑋)) = ((𝑛 · (𝐼𝑋))(+g𝐺)(𝐼𝑋)))
6053, 59eqeq12d 2636 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ0) → ((-(𝑛 + 1) · 𝑋) = ((𝑛 + 1) · (𝐼𝑋)) ↔ ((-𝑛 · 𝑋)(+g𝐺)(𝐼𝑋)) = ((𝑛 · (𝐼𝑋))(+g𝐺)(𝐼𝑋))))
6133, 60syl5ibr 236 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ0) → ((-𝑛 · 𝑋) = (𝑛 · (𝐼𝑋)) → (-(𝑛 + 1) · 𝑋) = ((𝑛 + 1) · (𝐼𝑋))))
6261ex 450 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑛 ∈ ℕ0 → ((-𝑛 · 𝑋) = (𝑛 · (𝐼𝑋)) → (-(𝑛 + 1) · 𝑋) = ((𝑛 + 1) · (𝐼𝑋)))))
63 fveq2 6158 . . . . . 6 ((-𝑛 · 𝑋) = (𝑛 · (𝐼𝑋)) → (𝐼‘(-𝑛 · 𝑋)) = (𝐼‘(𝑛 · (𝐼𝑋))))
64 simpll 789 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ) → 𝐺 ∈ Grp)
65 nnnegz 11340 . . . . . . . . 9 (𝑛 ∈ ℕ → -𝑛 ∈ ℤ)
6665adantl 482 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ) → -𝑛 ∈ ℤ)
67 simplr 791 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ) → 𝑋𝐵)
6823, 25, 28mulgneg 17500 . . . . . . . 8 ((𝐺 ∈ Grp ∧ -𝑛 ∈ ℤ ∧ 𝑋𝐵) → (--𝑛 · 𝑋) = (𝐼‘(-𝑛 · 𝑋)))
6964, 66, 67, 68syl3anc 1323 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ) → (--𝑛 · 𝑋) = (𝐼‘(-𝑛 · 𝑋)))
70 id 22 . . . . . . . 8 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ)
7123, 25, 28mulgnegnn 17491 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ (𝐼𝑋) ∈ 𝐵) → (-𝑛 · (𝐼𝑋)) = (𝐼‘(𝑛 · (𝐼𝑋))))
7270, 29, 71syl2anr 495 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ) → (-𝑛 · (𝐼𝑋)) = (𝐼‘(𝑛 · (𝐼𝑋))))
7369, 72eqeq12d 2636 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ) → ((--𝑛 · 𝑋) = (-𝑛 · (𝐼𝑋)) ↔ (𝐼‘(-𝑛 · 𝑋)) = (𝐼‘(𝑛 · (𝐼𝑋)))))
7463, 73syl5ibr 236 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑛 ∈ ℕ) → ((-𝑛 · 𝑋) = (𝑛 · (𝐼𝑋)) → (--𝑛 · 𝑋) = (-𝑛 · (𝐼𝑋))))
7574ex 450 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑛 ∈ ℕ → ((-𝑛 · 𝑋) = (𝑛 · (𝐼𝑋)) → (--𝑛 · 𝑋) = (-𝑛 · (𝐼𝑋)))))
766, 10, 14, 18, 22, 32, 62, 75zindd 11438 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁 ∈ ℤ → (-𝑁 · 𝑋) = (𝑁 · (𝐼𝑋))))
77763impia 1258 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑁 ∈ ℤ) → (-𝑁 · 𝑋) = (𝑁 · (𝐼𝑋)))
78773com23 1268 1 ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) → (-𝑁 · 𝑋) = (𝑁 · (𝐼𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  cfv 5857  (class class class)co 6615  cc 9894  0cc0 9896  1c1 9897   + caddc 9899  -cneg 10227  cn 10980  0cn0 11252  cz 11337  Basecbs 15800  +gcplusg 15881  0gc0g 16040  Mndcmnd 17234  Grpcgrp 17362  invgcminusg 17363  .gcmg 17480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-seq 12758  df-0g 16042  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-grp 17365  df-minusg 17366  df-mulg 17481
This theorem is referenced by:  mulgass  17519  mulgsubdi  18175  cyggeninv  18225
  Copyright terms: Public domain W3C validator