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Theorem archiabllem1b 29523
Description: Lemma for archiabl 29529. (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 11334 . . 3 (((𝜑𝑦𝐵) ∧ 𝑦 = 0 ) → 0 ∈ ℤ)
2 simpr 477 . . . 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 17462 . . . . . 6 (𝑈𝐵 → (0 · 𝑈) = 0 )
83, 7syl 17 . . . . 5 (𝜑 → (0 · 𝑈) = 0 )
98ad2antrr 761 . . . 4 (((𝜑𝑦𝐵) ∧ 𝑦 = 0 ) → (0 · 𝑈) = 0 )
102, 9eqtr4d 2663 . . 3 (((𝜑𝑦𝐵) ∧ 𝑦 = 0 ) → 𝑦 = (0 · 𝑈))
11 oveq1 6612 . . . . 5 (𝑛 = 0 → (𝑛 · 𝑈) = (0 · 𝑈))
1211eqeq2d 2636 . . . 4 (𝑛 = 0 → (𝑦 = (𝑛 · 𝑈) ↔ 𝑦 = (0 · 𝑈)))
1312rspcev 3300 . . 3 ((0 ∈ ℤ ∧ 𝑦 = (0 · 𝑈)) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈))
141, 10, 13syl2anc 692 . 2 (((𝜑𝑦𝐵) ∧ 𝑦 = 0 ) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈))
15 simplr 791 . . . . . . 7 ((((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑚 ∈ ℕ) ∧ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈)) → 𝑚 ∈ ℕ)
1615nnzd 11425 . . . . . 6 ((((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑚 ∈ ℕ) ∧ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈)) → 𝑚 ∈ ℤ)
1716znegcld 11428 . . . . 5 ((((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑚 ∈ ℕ) ∧ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈)) → -𝑚 ∈ ℤ)
1833ad2ant1 1080 . . . . . . . 8 ((𝜑𝑦𝐵𝑦 < 0 ) → 𝑈𝐵)
1918ad2antrr 761 . . . . . . 7 ((((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑚 ∈ ℕ) ∧ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈)) → 𝑈𝐵)
20 eqid 2626 . . . . . . . 8 (invg𝑊) = (invg𝑊)
214, 6, 20mulgnegnn 17467 . . . . . . 7 ((𝑚 ∈ ℕ ∧ 𝑈𝐵) → (-𝑚 · 𝑈) = ((invg𝑊)‘(𝑚 · 𝑈)))
2215, 19, 21syl2anc 692 . . . . . 6 ((((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑚 ∈ ℕ) ∧ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈)) → (-𝑚 · 𝑈) = ((invg𝑊)‘(𝑚 · 𝑈)))
23 simpr 477 . . . . . . 7 ((((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑚 ∈ ℕ) ∧ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈)) → ((invg𝑊)‘𝑦) = (𝑚 · 𝑈))
2423fveq2d 6154 . . . . . 6 ((((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑚 ∈ ℕ) ∧ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈)) → ((invg𝑊)‘((invg𝑊)‘𝑦)) = ((invg𝑊)‘(𝑚 · 𝑈)))
25 archiabllem.g . . . . . . . . . 10 (𝜑𝑊 ∈ oGrp)
26253ad2ant1 1080 . . . . . . . . 9 ((𝜑𝑦𝐵𝑦 < 0 ) → 𝑊 ∈ oGrp)
27 ogrpgrp 29480 . . . . . . . . 9 (𝑊 ∈ oGrp → 𝑊 ∈ Grp)
2826, 27syl 17 . . . . . . . 8 ((𝜑𝑦𝐵𝑦 < 0 ) → 𝑊 ∈ Grp)
29 simp2 1060 . . . . . . . 8 ((𝜑𝑦𝐵𝑦 < 0 ) → 𝑦𝐵)
304, 20grpinvinv 17398 . . . . . . . 8 ((𝑊 ∈ Grp ∧ 𝑦𝐵) → ((invg𝑊)‘((invg𝑊)‘𝑦)) = 𝑦)
3128, 29, 30syl2anc 692 . . . . . . 7 ((𝜑𝑦𝐵𝑦 < 0 ) → ((invg𝑊)‘((invg𝑊)‘𝑦)) = 𝑦)
3231ad2antrr 761 . . . . . 6 ((((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑚 ∈ ℕ) ∧ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈)) → ((invg𝑊)‘((invg𝑊)‘𝑦)) = 𝑦)
3322, 24, 323eqtr2rd 2667 . . . . 5 ((((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑚 ∈ ℕ) ∧ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈)) → 𝑦 = (-𝑚 · 𝑈))
34 oveq1 6612 . . . . . . 7 (𝑛 = -𝑚 → (𝑛 · 𝑈) = (-𝑚 · 𝑈))
3534eqeq2d 2636 . . . . . 6 (𝑛 = -𝑚 → (𝑦 = (𝑛 · 𝑈) ↔ 𝑦 = (-𝑚 · 𝑈)))
3635rspcev 3300 . . . . 5 ((-𝑚 ∈ ℤ ∧ 𝑦 = (-𝑚 · 𝑈)) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈))
3717, 33, 36syl2anc 692 . . . 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 1080 . . . . 5 ((𝜑𝑦𝐵𝑦 < 0 ) → 𝑊 ∈ Archi)
42 archiabllem1.p . . . . . 6 (𝜑0 < 𝑈)
43423ad2ant1 1080 . . . . 5 ((𝜑𝑦𝐵𝑦 < 0 ) → 0 < 𝑈)
44 simp1 1059 . . . . . 6 ((𝜑𝑦𝐵𝑦 < 0 ) → 𝜑)
45 archiabllem1.s . . . . . 6 ((𝜑𝑥𝐵0 < 𝑥) → 𝑈 𝑥)
4644, 45syl3an1 1356 . . . . 5 (((𝜑𝑦𝐵𝑦 < 0 ) ∧ 𝑥𝐵0 < 𝑥) → 𝑈 𝑥)
474, 20grpinvcl 17383 . . . . . 6 ((𝑊 ∈ Grp ∧ 𝑦𝐵) → ((invg𝑊)‘𝑦) ∈ 𝐵)
4828, 29, 47syl2anc 692 . . . . 5 ((𝜑𝑦𝐵𝑦 < 0 ) → ((invg𝑊)‘𝑦) ∈ 𝐵)
494, 5grpidcl 17366 . . . . . . . 8 (𝑊 ∈ Grp → 0𝐵)
5028, 49syl 17 . . . . . . 7 ((𝜑𝑦𝐵𝑦 < 0 ) → 0𝐵)
51 simp3 1061 . . . . . . 7 ((𝜑𝑦𝐵𝑦 < 0 ) → 𝑦 < 0 )
52 eqid 2626 . . . . . . . 8 (+g𝑊) = (+g𝑊)
534, 39, 52ogrpaddlt 29495 . . . . . . 7 ((𝑊 ∈ oGrp ∧ (𝑦𝐵0𝐵 ∧ ((invg𝑊)‘𝑦) ∈ 𝐵) ∧ 𝑦 < 0 ) → (𝑦(+g𝑊)((invg𝑊)‘𝑦)) < ( 0 (+g𝑊)((invg𝑊)‘𝑦)))
5426, 29, 50, 48, 51, 53syl131anc 1336 . . . . . 6 ((𝜑𝑦𝐵𝑦 < 0 ) → (𝑦(+g𝑊)((invg𝑊)‘𝑦)) < ( 0 (+g𝑊)((invg𝑊)‘𝑦)))
554, 52, 5, 20grprinv 17385 . . . . . . 7 ((𝑊 ∈ Grp ∧ 𝑦𝐵) → (𝑦(+g𝑊)((invg𝑊)‘𝑦)) = 0 )
5628, 29, 55syl2anc 692 . . . . . 6 ((𝜑𝑦𝐵𝑦 < 0 ) → (𝑦(+g𝑊)((invg𝑊)‘𝑦)) = 0 )
574, 52, 5grplid 17368 . . . . . . 7 ((𝑊 ∈ Grp ∧ ((invg𝑊)‘𝑦) ∈ 𝐵) → ( 0 (+g𝑊)((invg𝑊)‘𝑦)) = ((invg𝑊)‘𝑦))
5828, 48, 57syl2anc 692 . . . . . 6 ((𝜑𝑦𝐵𝑦 < 0 ) → ( 0 (+g𝑊)((invg𝑊)‘𝑦)) = ((invg𝑊)‘𝑦))
5954, 56, 583brtr3d 4649 . . . . 5 ((𝜑𝑦𝐵𝑦 < 0 ) → 0 < ((invg𝑊)‘𝑦))
604, 5, 38, 39, 6, 26, 41, 18, 43, 46, 48, 59archiabllem1a 29522 . . . 4 ((𝜑𝑦𝐵𝑦 < 0 ) → ∃𝑚 ∈ ℕ ((invg𝑊)‘𝑦) = (𝑚 · 𝑈))
6137, 60r19.29a 3076 . . 3 ((𝜑𝑦𝐵𝑦 < 0 ) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈))
62613expa 1262 . 2 (((𝜑𝑦𝐵) ∧ 𝑦 < 0 ) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈))
63 nnssz 11342 . . 3 ℕ ⊆ ℤ
64253ad2ant1 1080 . . . . 5 ((𝜑𝑦𝐵0 < 𝑦) → 𝑊 ∈ oGrp)
65403ad2ant1 1080 . . . . 5 ((𝜑𝑦𝐵0 < 𝑦) → 𝑊 ∈ Archi)
6633ad2ant1 1080 . . . . 5 ((𝜑𝑦𝐵0 < 𝑦) → 𝑈𝐵)
67423ad2ant1 1080 . . . . 5 ((𝜑𝑦𝐵0 < 𝑦) → 0 < 𝑈)
68 simp1 1059 . . . . . 6 ((𝜑𝑦𝐵0 < 𝑦) → 𝜑)
6968, 45syl3an1 1356 . . . . 5 (((𝜑𝑦𝐵0 < 𝑦) ∧ 𝑥𝐵0 < 𝑥) → 𝑈 𝑥)
70 simp2 1060 . . . . 5 ((𝜑𝑦𝐵0 < 𝑦) → 𝑦𝐵)
71 simp3 1061 . . . . 5 ((𝜑𝑦𝐵0 < 𝑦) → 0 < 𝑦)
724, 5, 38, 39, 6, 64, 65, 66, 67, 69, 70, 71archiabllem1a 29522 . . . 4 ((𝜑𝑦𝐵0 < 𝑦) → ∃𝑛 ∈ ℕ 𝑦 = (𝑛 · 𝑈))
73723expa 1262 . . 3 (((𝜑𝑦𝐵) ∧ 0 < 𝑦) → ∃𝑛 ∈ ℕ 𝑦 = (𝑛 · 𝑈))
74 ssrexv 3651 . . 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 29479 . . . . . 6 (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
7776simprbi 480 . . . . 5 (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
78 omndtos 29482 . . . . 5 (𝑊 ∈ oMnd → 𝑊 ∈ Toset)
7925, 77, 783syl 18 . . . 4 (𝜑𝑊 ∈ Toset)
8079adantr 481 . . 3 ((𝜑𝑦𝐵) → 𝑊 ∈ Toset)
81 simpr 477 . . 3 ((𝜑𝑦𝐵) → 𝑦𝐵)
8225, 27, 493syl 18 . . . 4 (𝜑0𝐵)
8382adantr 481 . . 3 ((𝜑𝑦𝐵) → 0𝐵)
844, 39tlt3 29442 . . 3 ((𝑊 ∈ Toset ∧ 𝑦𝐵0𝐵) → (𝑦 = 0𝑦 < 00 < 𝑦))
8580, 81, 83, 84syl3anc 1323 . 2 ((𝜑𝑦𝐵) → (𝑦 = 0𝑦 < 00 < 𝑦))
8614, 62, 75, 85mpjao3dan 1392 1 ((𝜑𝑦𝐵) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3o 1035  w3a 1036   = wceq 1480  wcel 1992  wrex 2913  wss 3560   class class class wbr 4618  cfv 5850  (class class class)co 6605  0cc0 9881  -cneg 10212  cn 10965  cz 11322  Basecbs 15776  +gcplusg 15857  lecple 15864  0gc0g 16016  ltcplt 16857  Tosetctos 16949  Grpcgrp 17338  invgcminusg 17339  .gcmg 17456  oMndcomnd 29474  oGrpcogrp 29475  Archicarchi 29508
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-seq 12739  df-0g 16018  df-preset 16844  df-poset 16862  df-plt 16874  df-toset 16950  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-grp 17341  df-minusg 17342  df-sbg 17343  df-mulg 17457  df-omnd 29476  df-ogrp 29477  df-inftm 29509  df-archi 29510
This theorem is referenced by:  archiabllem1  29524
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