Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  archiabllem1b Structured version   Visualization version   GIF version

Theorem archiabllem1b 29874
Description: Lemma for archiabl 29880. (Contributed by Thierry Arnoux, 13-Apr-2018.)
Hypotheses
Ref Expression
archiabllem.b 𝐵 = (Base‘𝑊)
archiabllem.0 0 = (0g𝑊)
archiabllem.e = (le‘𝑊)
archiabllem.t < = (lt‘𝑊)
archiabllem.m · = (.g𝑊)
archiabllem.g (𝜑𝑊 ∈ oGrp)
archiabllem.a (𝜑𝑊 ∈ Archi)
archiabllem1.u (𝜑𝑈𝐵)
archiabllem1.p (𝜑0 < 𝑈)
archiabllem1.s ((𝜑𝑥𝐵0 < 𝑥) → 𝑈 𝑥)
Assertion
Ref Expression
archiabllem1b ((𝜑𝑦𝐵) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈))
Distinct variable groups:   𝑥,𝑛,𝑦,𝐵   𝑈,𝑛,𝑥   𝑛,𝑊,𝑥,𝑦   𝜑,𝑛,𝑥,𝑦   · ,𝑛,𝑥   0 ,𝑛,𝑥   < ,𝑛,𝑥   𝑥,
Allowed substitution hints:   < (𝑦)   · (𝑦)   𝑈(𝑦)   (𝑦,𝑛)   0 (𝑦)

Proof of Theorem archiabllem1b
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 0zd 11427 . . 3 (((𝜑𝑦𝐵) ∧ 𝑦 = 0 ) → 0 ∈ ℤ)
2 simpr 476 . . . 4 (((𝜑𝑦𝐵) ∧ 𝑦 = 0 ) → 𝑦 = 0 )
3 archiabllem1.u . . . . . 6 (𝜑𝑈𝐵)
4 archiabllem.b . . . . . . 7 𝐵 = (Base‘𝑊)
5 archiabllem.0 . . . . . . 7 0 = (0g𝑊)
6 archiabllem.m . . . . . . 7 · = (.g𝑊)
74, 5, 6mulg0 17593 . . . . . 6 (𝑈𝐵 → (0 · 𝑈) = 0 )
83, 7syl 17 . . . . 5 (𝜑 → (0 · 𝑈) = 0 )
98ad2antrr 762 . . . 4 (((𝜑𝑦𝐵) ∧ 𝑦 = 0 ) → (0 · 𝑈) = 0 )
102, 9eqtr4d 2688 . . 3 (((𝜑𝑦𝐵) ∧ 𝑦 = 0 ) → 𝑦 = (0 · 𝑈))
11 oveq1 6697 . . . . 5 (𝑛 = 0 → (𝑛 · 𝑈) = (0 · 𝑈))
1211eqeq2d 2661 . . . 4 (𝑛 = 0 → (𝑦 = (𝑛 · 𝑈) ↔ 𝑦 = (0 · 𝑈)))
1312rspcev 3340 . . 3 ((0 ∈ ℤ ∧ 𝑦 = (0 · 𝑈)) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈))
141, 10, 13syl2anc 694 . 2 (((𝜑𝑦𝐵) ∧ 𝑦 = 0 ) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈))
15 simplr 807 . . . . . . 7 ((((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑚 ∈ ℕ) ∧ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈)) → 𝑚 ∈ ℕ)
1615nnzd 11519 . . . . . 6 ((((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑚 ∈ ℕ) ∧ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈)) → 𝑚 ∈ ℤ)
1716znegcld 11522 . . . . 5 ((((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑚 ∈ ℕ) ∧ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈)) → -𝑚 ∈ ℤ)
1833ad2ant1 1102 . . . . . . . 8 ((𝜑𝑦𝐵𝑦 < 0 ) → 𝑈𝐵)
1918ad2antrr 762 . . . . . . 7 ((((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑚 ∈ ℕ) ∧ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈)) → 𝑈𝐵)
20 eqid 2651 . . . . . . . 8 (invg𝑊) = (invg𝑊)
214, 6, 20mulgnegnn 17598 . . . . . . 7 ((𝑚 ∈ ℕ ∧ 𝑈𝐵) → (-𝑚 · 𝑈) = ((invg𝑊)‘(𝑚 · 𝑈)))
2215, 19, 21syl2anc 694 . . . . . 6 ((((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑚 ∈ ℕ) ∧ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈)) → (-𝑚 · 𝑈) = ((invg𝑊)‘(𝑚 · 𝑈)))
23 simpr 476 . . . . . . 7 ((((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑚 ∈ ℕ) ∧ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈)) → ((invg𝑊)‘𝑦) = (𝑚 · 𝑈))
2423fveq2d 6233 . . . . . 6 ((((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑚 ∈ ℕ) ∧ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈)) → ((invg𝑊)‘((invg𝑊)‘𝑦)) = ((invg𝑊)‘(𝑚 · 𝑈)))
25 archiabllem.g . . . . . . . . . 10 (𝜑𝑊 ∈ oGrp)
26253ad2ant1 1102 . . . . . . . . 9 ((𝜑𝑦𝐵𝑦 < 0 ) → 𝑊 ∈ oGrp)
27 ogrpgrp 29831 . . . . . . . . 9 (𝑊 ∈ oGrp → 𝑊 ∈ Grp)
2826, 27syl 17 . . . . . . . 8 ((𝜑𝑦𝐵𝑦 < 0 ) → 𝑊 ∈ Grp)
29 simp2 1082 . . . . . . . 8 ((𝜑𝑦𝐵𝑦 < 0 ) → 𝑦𝐵)
304, 20grpinvinv 17529 . . . . . . . 8 ((𝑊 ∈ Grp ∧ 𝑦𝐵) → ((invg𝑊)‘((invg𝑊)‘𝑦)) = 𝑦)
3128, 29, 30syl2anc 694 . . . . . . 7 ((𝜑𝑦𝐵𝑦 < 0 ) → ((invg𝑊)‘((invg𝑊)‘𝑦)) = 𝑦)
3231ad2antrr 762 . . . . . 6 ((((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑚 ∈ ℕ) ∧ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈)) → ((invg𝑊)‘((invg𝑊)‘𝑦)) = 𝑦)
3322, 24, 323eqtr2rd 2692 . . . . 5 ((((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑚 ∈ ℕ) ∧ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈)) → 𝑦 = (-𝑚 · 𝑈))
34 oveq1 6697 . . . . . . 7 (𝑛 = -𝑚 → (𝑛 · 𝑈) = (-𝑚 · 𝑈))
3534eqeq2d 2661 . . . . . 6 (𝑛 = -𝑚 → (𝑦 = (𝑛 · 𝑈) ↔ 𝑦 = (-𝑚 · 𝑈)))
3635rspcev 3340 . . . . 5 ((-𝑚 ∈ ℤ ∧ 𝑦 = (-𝑚 · 𝑈)) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈))
3717, 33, 36syl2anc 694 . . . 4 ((((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑚 ∈ ℕ) ∧ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈)) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈))
38 archiabllem.e . . . . 5 = (le‘𝑊)
39 archiabllem.t . . . . 5 < = (lt‘𝑊)
40 archiabllem.a . . . . . 6 (𝜑𝑊 ∈ Archi)
41403ad2ant1 1102 . . . . 5 ((𝜑𝑦𝐵𝑦 < 0 ) → 𝑊 ∈ Archi)
42 archiabllem1.p . . . . . 6 (𝜑0 < 𝑈)
43423ad2ant1 1102 . . . . 5 ((𝜑𝑦𝐵𝑦 < 0 ) → 0 < 𝑈)
44 simp1 1081 . . . . . 6 ((𝜑𝑦𝐵𝑦 < 0 ) → 𝜑)
45 archiabllem1.s . . . . . 6 ((𝜑𝑥𝐵0 < 𝑥) → 𝑈 𝑥)
4644, 45syl3an1 1399 . . . . 5 (((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑥𝐵0 < 𝑥) → 𝑈 𝑥)
474, 20grpinvcl 17514 . . . . . 6 ((𝑊 ∈ Grp ∧ 𝑦𝐵) → ((invg𝑊)‘𝑦) ∈ 𝐵)
4828, 29, 47syl2anc 694 . . . . 5 ((𝜑𝑦𝐵𝑦 < 0 ) → ((invg𝑊)‘𝑦) ∈ 𝐵)
494, 5grpidcl 17497 . . . . . . . 8 (𝑊 ∈ Grp → 0𝐵)
5028, 49syl 17 . . . . . . 7 ((𝜑𝑦𝐵𝑦 < 0 ) → 0𝐵)
51 simp3 1083 . . . . . . 7 ((𝜑𝑦𝐵𝑦 < 0 ) → 𝑦 < 0 )
52 eqid 2651 . . . . . . . 8 (+g𝑊) = (+g𝑊)
534, 39, 52ogrpaddlt 29846 . . . . . . 7 ((𝑊 ∈ oGrp ∧ (𝑦𝐵0𝐵 ∧ ((invg𝑊)‘𝑦) ∈ 𝐵) ∧ 𝑦 < 0 ) → (𝑦(+g𝑊)((invg𝑊)‘𝑦)) < ( 0 (+g𝑊)((invg𝑊)‘𝑦)))
5426, 29, 50, 48, 51, 53syl131anc 1379 . . . . . 6 ((𝜑𝑦𝐵𝑦 < 0 ) → (𝑦(+g𝑊)((invg𝑊)‘𝑦)) < ( 0 (+g𝑊)((invg𝑊)‘𝑦)))
554, 52, 5, 20grprinv 17516 . . . . . . 7 ((𝑊 ∈ Grp ∧ 𝑦𝐵) → (𝑦(+g𝑊)((invg𝑊)‘𝑦)) = 0 )
5628, 29, 55syl2anc 694 . . . . . 6 ((𝜑𝑦𝐵𝑦 < 0 ) → (𝑦(+g𝑊)((invg𝑊)‘𝑦)) = 0 )
574, 52, 5grplid 17499 . . . . . . 7 ((𝑊 ∈ Grp ∧ ((invg𝑊)‘𝑦) ∈ 𝐵) → ( 0 (+g𝑊)((invg𝑊)‘𝑦)) = ((invg𝑊)‘𝑦))
5828, 48, 57syl2anc 694 . . . . . 6 ((𝜑𝑦𝐵𝑦 < 0 ) → ( 0 (+g𝑊)((invg𝑊)‘𝑦)) = ((invg𝑊)‘𝑦))
5954, 56, 583brtr3d 4716 . . . . 5 ((𝜑𝑦𝐵𝑦 < 0 ) → 0 < ((invg𝑊)‘𝑦))
604, 5, 38, 39, 6, 26, 41, 18, 43, 46, 48, 59archiabllem1a 29873 . . . 4 ((𝜑𝑦𝐵𝑦 < 0 ) → ∃𝑚 ∈ ℕ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈))
6137, 60r19.29a 3107 . . 3 ((𝜑𝑦𝐵𝑦 < 0 ) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈))
62613expa 1284 . 2 (((𝜑𝑦𝐵) ∧ 𝑦 < 0 ) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈))
63 nnssz 11435 . . 3 ℕ ⊆ ℤ
64253ad2ant1 1102 . . . . 5 ((𝜑𝑦𝐵0 < 𝑦) → 𝑊 ∈ oGrp)
65403ad2ant1 1102 . . . . 5 ((𝜑𝑦𝐵0 < 𝑦) → 𝑊 ∈ Archi)
6633ad2ant1 1102 . . . . 5 ((𝜑𝑦𝐵0 < 𝑦) → 𝑈𝐵)
67423ad2ant1 1102 . . . . 5 ((𝜑𝑦𝐵0 < 𝑦) → 0 < 𝑈)
68 simp1 1081 . . . . . 6 ((𝜑𝑦𝐵0 < 𝑦) → 𝜑)
6968, 45syl3an1 1399 . . . . 5 (((𝜑𝑦𝐵0 < 𝑦) ∧ 𝑥𝐵0 < 𝑥) → 𝑈 𝑥)
70 simp2 1082 . . . . 5 ((𝜑𝑦𝐵0 < 𝑦) → 𝑦𝐵)
71 simp3 1083 . . . . 5 ((𝜑𝑦𝐵0 < 𝑦) → 0 < 𝑦)
724, 5, 38, 39, 6, 64, 65, 66, 67, 69, 70, 71archiabllem1a 29873 . . . 4 ((𝜑𝑦𝐵0 < 𝑦) → ∃𝑛 ∈ ℕ 𝑦 = (𝑛 · 𝑈))
73723expa 1284 . . 3 (((𝜑𝑦𝐵) ∧ 0 < 𝑦) → ∃𝑛 ∈ ℕ 𝑦 = (𝑛 · 𝑈))
74 ssrexv 3700 . . 3 (ℕ ⊆ ℤ → (∃𝑛 ∈ ℕ 𝑦 = (𝑛 · 𝑈) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈)))
7563, 73, 74mpsyl 68 . 2 (((𝜑𝑦𝐵) ∧ 0 < 𝑦) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈))
76 isogrp 29830 . . . . . 6 (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
7776simprbi 479 . . . . 5 (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
78 omndtos 29833 . . . . 5 (𝑊 ∈ oMnd → 𝑊 ∈ Toset)
7925, 77, 783syl 18 . . . 4 (𝜑𝑊 ∈ Toset)
8079adantr 480 . . 3 ((𝜑𝑦𝐵) → 𝑊 ∈ Toset)
81 simpr 476 . . 3 ((𝜑𝑦𝐵) → 𝑦𝐵)
8225, 27, 493syl 18 . . . 4 (𝜑0𝐵)
8382adantr 480 . . 3 ((𝜑𝑦𝐵) → 0𝐵)
844, 39tlt3 29793 . . 3 ((𝑊 ∈ Toset ∧ 𝑦𝐵0𝐵) → (𝑦 = 0𝑦 < 00 < 𝑦))
8580, 81, 83, 84syl3anc 1366 . 2 ((𝜑𝑦𝐵) → (𝑦 = 0𝑦 < 00 < 𝑦))
8614, 62, 75, 85mpjao3dan 1435 1 ((𝜑𝑦𝐵) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3o 1053  w3a 1054   = wceq 1523  wcel 2030  wrex 2942  wss 3607   class class class wbr 4685  cfv 5926  (class class class)co 6690  0cc0 9974  -cneg 10305  cn 11058  cz 11415  Basecbs 15904  +gcplusg 15988  lecple 15995  0gc0g 16147  ltcplt 16988  Tosetctos 17080  Grpcgrp 17469  invgcminusg 17470  .gcmg 17587  oMndcomnd 29825  oGrpcogrp 29826  Archicarchi 29859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-seq 12842  df-0g 16149  df-preset 16975  df-poset 16993  df-plt 17005  df-toset 17081  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-omnd 29827  df-ogrp 29828  df-inftm 29860  df-archi 29861
This theorem is referenced by:  archiabllem1  29875
  Copyright terms: Public domain W3C validator