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Theorem archirngz 29973
Description: Property of Archimedean left and right ordered groups. (Contributed by Thierry Arnoux, 6-May-2018.)
Hypotheses
Ref Expression
archirng.b 𝐵 = (Base‘𝑊)
archirng.0 0 = (0g𝑊)
archirng.i < = (lt‘𝑊)
archirng.l = (le‘𝑊)
archirng.x · = (.g𝑊)
archirng.1 (𝜑𝑊 ∈ oGrp)
archirng.2 (𝜑𝑊 ∈ Archi)
archirng.3 (𝜑𝑋𝐵)
archirng.4 (𝜑𝑌𝐵)
archirng.5 (𝜑0 < 𝑋)
archirngz.1 (𝜑 → (oppg𝑊) ∈ oGrp)
Assertion
Ref Expression
archirngz (𝜑 → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
Distinct variable groups:   𝑛,𝑋   𝑛,𝑌   𝜑,𝑛   0 ,𝑛   ,𝑛   < ,𝑛   · ,𝑛
Allowed substitution hints:   𝐵(𝑛)   𝑊(𝑛)

Proof of Theorem archirngz
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 neg1z 11526 . . 3 -1 ∈ ℤ
2 archirng.1 . . . . . . . . . 10 (𝜑𝑊 ∈ oGrp)
3 ogrpgrp 29933 . . . . . . . . . 10 (𝑊 ∈ oGrp → 𝑊 ∈ Grp)
42, 3syl 17 . . . . . . . . 9 (𝜑𝑊 ∈ Grp)
5 1zzd 11521 . . . . . . . . 9 (𝜑 → 1 ∈ ℤ)
6 archirng.3 . . . . . . . . 9 (𝜑𝑋𝐵)
7 archirng.b . . . . . . . . . 10 𝐵 = (Base‘𝑊)
8 archirng.x . . . . . . . . . 10 · = (.g𝑊)
9 eqid 2724 . . . . . . . . . 10 (invg𝑊) = (invg𝑊)
107, 8, 9mulgneg 17682 . . . . . . . . 9 ((𝑊 ∈ Grp ∧ 1 ∈ ℤ ∧ 𝑋𝐵) → (-1 · 𝑋) = ((invg𝑊)‘(1 · 𝑋)))
114, 5, 6, 10syl3anc 1439 . . . . . . . 8 (𝜑 → (-1 · 𝑋) = ((invg𝑊)‘(1 · 𝑋)))
127, 8mulg1 17670 . . . . . . . . . 10 (𝑋𝐵 → (1 · 𝑋) = 𝑋)
136, 12syl 17 . . . . . . . . 9 (𝜑 → (1 · 𝑋) = 𝑋)
1413fveq2d 6308 . . . . . . . 8 (𝜑 → ((invg𝑊)‘(1 · 𝑋)) = ((invg𝑊)‘𝑋))
1511, 14eqtrd 2758 . . . . . . 7 (𝜑 → (-1 · 𝑋) = ((invg𝑊)‘𝑋))
16 archirng.5 . . . . . . . 8 (𝜑0 < 𝑋)
17 archirng.i . . . . . . . . . 10 < = (lt‘𝑊)
18 archirng.0 . . . . . . . . . 10 0 = (0g𝑊)
197, 17, 9, 18ogrpinv0lt 29953 . . . . . . . . 9 ((𝑊 ∈ oGrp ∧ 𝑋𝐵) → ( 0 < 𝑋 ↔ ((invg𝑊)‘𝑋) < 0 ))
2019biimpa 502 . . . . . . . 8 (((𝑊 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → ((invg𝑊)‘𝑋) < 0 )
212, 6, 16, 20syl21anc 1438 . . . . . . 7 (𝜑 → ((invg𝑊)‘𝑋) < 0 )
2215, 21eqbrtrd 4782 . . . . . 6 (𝜑 → (-1 · 𝑋) < 0 )
2322adantr 472 . . . . 5 ((𝜑𝑌 = 0 ) → (-1 · 𝑋) < 0 )
24 simpr 479 . . . . 5 ((𝜑𝑌 = 0 ) → 𝑌 = 0 )
2523, 24breqtrrd 4788 . . . 4 ((𝜑𝑌 = 0 ) → (-1 · 𝑋) < 𝑌)
26 isogrp 29932 . . . . . . . . . 10 (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
2726simprbi 483 . . . . . . . . 9 (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
28 omndtos 29935 . . . . . . . . 9 (𝑊 ∈ oMnd → 𝑊 ∈ Toset)
292, 27, 283syl 18 . . . . . . . 8 (𝜑𝑊 ∈ Toset)
30 tospos 29888 . . . . . . . 8 (𝑊 ∈ Toset → 𝑊 ∈ Poset)
3129, 30syl 17 . . . . . . 7 (𝜑𝑊 ∈ Poset)
327, 18grpidcl 17572 . . . . . . . 8 (𝑊 ∈ Grp → 0𝐵)
332, 3, 323syl 18 . . . . . . 7 (𝜑0𝐵)
34 archirng.l . . . . . . . 8 = (le‘𝑊)
357, 34posref 17073 . . . . . . 7 ((𝑊 ∈ Poset ∧ 0𝐵) → 0 0 )
3631, 33, 35syl2anc 696 . . . . . 6 (𝜑0 0 )
3736adantr 472 . . . . 5 ((𝜑𝑌 = 0 ) → 0 0 )
38 1m1e0 11202 . . . . . . . . . 10 (1 − 1) = 0
3938negeqi 10387 . . . . . . . . 9 -(1 − 1) = -0
40 ax-1cn 10107 . . . . . . . . . 10 1 ∈ ℂ
4140, 40negsubdii 10479 . . . . . . . . 9 -(1 − 1) = (-1 + 1)
42 neg0 10440 . . . . . . . . 9 -0 = 0
4339, 41, 423eqtr3i 2754 . . . . . . . 8 (-1 + 1) = 0
4443oveq1i 6775 . . . . . . 7 ((-1 + 1) · 𝑋) = (0 · 𝑋)
457, 18, 8mulg0 17668 . . . . . . . 8 (𝑋𝐵 → (0 · 𝑋) = 0 )
466, 45syl 17 . . . . . . 7 (𝜑 → (0 · 𝑋) = 0 )
4744, 46syl5eq 2770 . . . . . 6 (𝜑 → ((-1 + 1) · 𝑋) = 0 )
4847adantr 472 . . . . 5 ((𝜑𝑌 = 0 ) → ((-1 + 1) · 𝑋) = 0 )
4937, 24, 483brtr4d 4792 . . . 4 ((𝜑𝑌 = 0 ) → 𝑌 ((-1 + 1) · 𝑋))
5025, 49jca 555 . . 3 ((𝜑𝑌 = 0 ) → ((-1 · 𝑋) < 𝑌𝑌 ((-1 + 1) · 𝑋)))
51 oveq1 6772 . . . . . 6 (𝑛 = -1 → (𝑛 · 𝑋) = (-1 · 𝑋))
5251breq1d 4770 . . . . 5 (𝑛 = -1 → ((𝑛 · 𝑋) < 𝑌 ↔ (-1 · 𝑋) < 𝑌))
53 oveq1 6772 . . . . . . 7 (𝑛 = -1 → (𝑛 + 1) = (-1 + 1))
5453oveq1d 6780 . . . . . 6 (𝑛 = -1 → ((𝑛 + 1) · 𝑋) = ((-1 + 1) · 𝑋))
5554breq2d 4772 . . . . 5 (𝑛 = -1 → (𝑌 ((𝑛 + 1) · 𝑋) ↔ 𝑌 ((-1 + 1) · 𝑋)))
5652, 55anbi12d 749 . . . 4 (𝑛 = -1 → (((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)) ↔ ((-1 · 𝑋) < 𝑌𝑌 ((-1 + 1) · 𝑋))))
5756rspcev 3413 . . 3 ((-1 ∈ ℤ ∧ ((-1 · 𝑋) < 𝑌𝑌 ((-1 + 1) · 𝑋))) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
581, 50, 57sylancr 698 . 2 ((𝜑𝑌 = 0 ) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
59 simpr 479 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
6059nn0zd 11593 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℤ)
6160ad2antrr 764 . . . . . . 7 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → 𝑚 ∈ ℤ)
6261znegcld 11597 . . . . . 6 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → -𝑚 ∈ ℤ)
63 2z 11522 . . . . . . 7 2 ∈ ℤ
6463a1i 11 . . . . . 6 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → 2 ∈ ℤ)
6562, 64zsubcld 11600 . . . . 5 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → (-𝑚 − 2) ∈ ℤ)
66 nn0cn 11415 . . . . . . . . . . 11 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
6766adantl 473 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℂ)
68 2cnd 11206 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 2 ∈ ℂ)
6967, 68negdi2d 10519 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -(𝑚 + 2) = (-𝑚 − 2))
7069oveq1d 6780 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) · 𝑋) = ((-𝑚 − 2) · 𝑋))
712ad2antrr 764 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ oGrp)
72 archirngz.1 . . . . . . . . . . . 12 (𝜑 → (oppg𝑊) ∈ oGrp)
7372ad2antrr 764 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (oppg𝑊) ∈ oGrp)
7471, 73jca 555 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp))
754ad2antrr 764 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ Grp)
7660peano2zd 11598 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 1) ∈ ℤ)
776ad2antrr 764 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑋𝐵)
787, 8mulgcl 17681 . . . . . . . . . . 11 ((𝑊 ∈ Grp ∧ (𝑚 + 1) ∈ ℤ ∧ 𝑋𝐵) → ((𝑚 + 1) · 𝑋) ∈ 𝐵)
7975, 76, 77, 78syl3anc 1439 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) · 𝑋) ∈ 𝐵)
8063a1i 11 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 2 ∈ ℤ)
8160, 80zaddcld 11599 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 2) ∈ ℤ)
827, 8mulgcl 17681 . . . . . . . . . . 11 ((𝑊 ∈ Grp ∧ (𝑚 + 2) ∈ ℤ ∧ 𝑋𝐵) → ((𝑚 + 2) · 𝑋) ∈ 𝐵)
8375, 81, 77, 82syl3anc 1439 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 2) · 𝑋) ∈ 𝐵)
8475, 32syl 17 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 0𝐵)
8516ad2antrr 764 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 0 < 𝑋)
86 eqid 2724 . . . . . . . . . . . . 13 (+g𝑊) = (+g𝑊)
877, 17, 86ogrpaddlt 29948 . . . . . . . . . . . 12 ((𝑊 ∈ oGrp ∧ ( 0𝐵𝑋𝐵 ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵) ∧ 0 < 𝑋) → ( 0 (+g𝑊)((𝑚 + 1) · 𝑋)) < (𝑋(+g𝑊)((𝑚 + 1) · 𝑋)))
8871, 84, 77, 79, 85, 87syl131anc 1452 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ( 0 (+g𝑊)((𝑚 + 1) · 𝑋)) < (𝑋(+g𝑊)((𝑚 + 1) · 𝑋)))
897, 86, 18grplid 17574 . . . . . . . . . . . 12 ((𝑊 ∈ Grp ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵) → ( 0 (+g𝑊)((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) · 𝑋))
9075, 79, 89syl2anc 696 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ( 0 (+g𝑊)((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) · 𝑋))
91 1cnd 10169 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ0 → 1 ∈ ℂ)
9266, 91, 91addassd 10175 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) = (𝑚 + (1 + 1)))
93 1p1e2 11247 . . . . . . . . . . . . . . . . 17 (1 + 1) = 2
9493oveq2i 6776 . . . . . . . . . . . . . . . 16 (𝑚 + (1 + 1)) = (𝑚 + 2)
9592, 94syl6eq 2774 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) = (𝑚 + 2))
9666, 91addcld 10172 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℂ)
9796, 91addcomd 10351 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) = (1 + (𝑚 + 1)))
9895, 97eqtr3d 2760 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → (𝑚 + 2) = (1 + (𝑚 + 1)))
9998oveq1d 6780 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0 → ((𝑚 + 2) · 𝑋) = ((1 + (𝑚 + 1)) · 𝑋))
10099adantl 473 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 2) · 𝑋) = ((1 + (𝑚 + 1)) · 𝑋))
101 1zzd 11521 . . . . . . . . . . . . 13 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 1 ∈ ℤ)
1027, 8, 86mulgdir 17695 . . . . . . . . . . . . 13 ((𝑊 ∈ Grp ∧ (1 ∈ ℤ ∧ (𝑚 + 1) ∈ ℤ ∧ 𝑋𝐵)) → ((1 + (𝑚 + 1)) · 𝑋) = ((1 · 𝑋)(+g𝑊)((𝑚 + 1) · 𝑋)))
10375, 101, 76, 77, 102syl13anc 1441 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((1 + (𝑚 + 1)) · 𝑋) = ((1 · 𝑋)(+g𝑊)((𝑚 + 1) · 𝑋)))
10477, 12syl 17 . . . . . . . . . . . . 13 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (1 · 𝑋) = 𝑋)
105104oveq1d 6780 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((1 · 𝑋)(+g𝑊)((𝑚 + 1) · 𝑋)) = (𝑋(+g𝑊)((𝑚 + 1) · 𝑋)))
106100, 103, 1053eqtrrd 2763 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑋(+g𝑊)((𝑚 + 1) · 𝑋)) = ((𝑚 + 2) · 𝑋))
10788, 90, 1063brtr3d 4791 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) · 𝑋) < ((𝑚 + 2) · 𝑋))
1087, 17, 9ogrpinvlt 29954 . . . . . . . . . . 11 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp) ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵 ∧ ((𝑚 + 2) · 𝑋) ∈ 𝐵) → (((𝑚 + 1) · 𝑋) < ((𝑚 + 2) · 𝑋) ↔ ((invg𝑊)‘((𝑚 + 2) · 𝑋)) < ((invg𝑊)‘((𝑚 + 1) · 𝑋))))
109108biimpa 502 . . . . . . . . . 10 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp) ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵 ∧ ((𝑚 + 2) · 𝑋) ∈ 𝐵) ∧ ((𝑚 + 1) · 𝑋) < ((𝑚 + 2) · 𝑋)) → ((invg𝑊)‘((𝑚 + 2) · 𝑋)) < ((invg𝑊)‘((𝑚 + 1) · 𝑋)))
11074, 79, 83, 107, 109syl31anc 1442 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((invg𝑊)‘((𝑚 + 2) · 𝑋)) < ((invg𝑊)‘((𝑚 + 1) · 𝑋)))
1117, 8, 9mulgneg 17682 . . . . . . . . . 10 ((𝑊 ∈ Grp ∧ (𝑚 + 2) ∈ ℤ ∧ 𝑋𝐵) → (-(𝑚 + 2) · 𝑋) = ((invg𝑊)‘((𝑚 + 2) · 𝑋)))
11275, 81, 77, 111syl3anc 1439 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) · 𝑋) = ((invg𝑊)‘((𝑚 + 2) · 𝑋)))
1137, 8, 9mulgneg 17682 . . . . . . . . . 10 ((𝑊 ∈ Grp ∧ (𝑚 + 1) ∈ ℤ ∧ 𝑋𝐵) → (-(𝑚 + 1) · 𝑋) = ((invg𝑊)‘((𝑚 + 1) · 𝑋)))
11475, 76, 77, 113syl3anc 1439 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 1) · 𝑋) = ((invg𝑊)‘((𝑚 + 1) · 𝑋)))
115110, 112, 1143brtr4d 4792 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) · 𝑋) < (-(𝑚 + 1) · 𝑋))
11670, 115eqbrtrrd 4784 . . . . . . 7 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-𝑚 − 2) · 𝑋) < (-(𝑚 + 1) · 𝑋))
117116ad2antrr 764 . . . . . 6 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → ((-𝑚 − 2) · 𝑋) < (-(𝑚 + 1) · 𝑋))
118114ad2antrr 764 . . . . . . 7 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → (-(𝑚 + 1) · 𝑋) = ((invg𝑊)‘((𝑚 + 1) · 𝑋)))
11931ad4antr 771 . . . . . . . . 9 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → 𝑊 ∈ Poset)
120 archirng.4 . . . . . . . . . . . 12 (𝜑𝑌𝐵)
1217, 9grpinvcl 17589 . . . . . . . . . . . 12 ((𝑊 ∈ Grp ∧ 𝑌𝐵) → ((invg𝑊)‘𝑌) ∈ 𝐵)
1224, 120, 121syl2anc 696 . . . . . . . . . . 11 (𝜑 → ((invg𝑊)‘𝑌) ∈ 𝐵)
123122ad2antrr 764 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((invg𝑊)‘𝑌) ∈ 𝐵)
124123ad2antrr 764 . . . . . . . . 9 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → ((invg𝑊)‘𝑌) ∈ 𝐵)
12579ad2antrr 764 . . . . . . . . 9 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → ((𝑚 + 1) · 𝑋) ∈ 𝐵)
126 simplrr 820 . . . . . . . . 9 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))
127 simpr 479 . . . . . . . . 9 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌))
1287, 34posasymb 17074 . . . . . . . . . 10 ((𝑊 ∈ Poset ∧ ((invg𝑊)‘𝑌) ∈ 𝐵 ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵) → ((((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) ↔ ((invg𝑊)‘𝑌) = ((𝑚 + 1) · 𝑋)))
129128biimpa 502 . . . . . . . . 9 (((𝑊 ∈ Poset ∧ ((invg𝑊)‘𝑌) ∈ 𝐵 ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵) ∧ (((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌))) → ((invg𝑊)‘𝑌) = ((𝑚 + 1) · 𝑋))
130119, 124, 125, 126, 127, 129syl32anc 1447 . . . . . . . 8 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → ((invg𝑊)‘𝑌) = ((𝑚 + 1) · 𝑋))
131130fveq2d 6308 . . . . . . 7 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → ((invg𝑊)‘((invg𝑊)‘𝑌)) = ((invg𝑊)‘((𝑚 + 1) · 𝑋)))
1327, 9grpinvinv 17604 . . . . . . . . 9 ((𝑊 ∈ Grp ∧ 𝑌𝐵) → ((invg𝑊)‘((invg𝑊)‘𝑌)) = 𝑌)
1334, 120, 132syl2anc 696 . . . . . . . 8 (𝜑 → ((invg𝑊)‘((invg𝑊)‘𝑌)) = 𝑌)
134133ad4antr 771 . . . . . . 7 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → ((invg𝑊)‘((invg𝑊)‘𝑌)) = 𝑌)
135118, 131, 1343eqtr2rd 2765 . . . . . 6 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → 𝑌 = (-(𝑚 + 1) · 𝑋))
136117, 135breqtrrd 4788 . . . . 5 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → ((-𝑚 − 2) · 𝑋) < 𝑌)
137 1cnd 10169 . . . . . . . . . . . . 13 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 1 ∈ ℂ)
13867, 68, 137addsubassd 10525 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 2) − 1) = (𝑚 + (2 − 1)))
139 2m1e1 11248 . . . . . . . . . . . . 13 (2 − 1) = 1
140139oveq2i 6776 . . . . . . . . . . . 12 (𝑚 + (2 − 1)) = (𝑚 + 1)
141138, 140syl6req 2775 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 1) = ((𝑚 + 2) − 1))
142141negeqd 10388 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -(𝑚 + 1) = -((𝑚 + 2) − 1))
14367, 68addcld 10172 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 2) ∈ ℂ)
144143, 137negsubdid 10520 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -((𝑚 + 2) − 1) = (-(𝑚 + 2) + 1))
14569oveq1d 6780 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) + 1) = ((-𝑚 − 2) + 1))
146142, 144, 1453eqtrrd 2763 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-𝑚 − 2) + 1) = -(𝑚 + 1))
147146oveq1d 6780 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((-𝑚 − 2) + 1) · 𝑋) = (-(𝑚 + 1) · 𝑋))
14829ad2antrr 764 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ Toset)
149148, 30syl 17 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ Poset)
15060znegcld 11597 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -𝑚 ∈ ℤ)
151150, 80zsubcld 11600 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-𝑚 − 2) ∈ ℤ)
152151peano2zd 11598 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-𝑚 − 2) + 1) ∈ ℤ)
1537, 8mulgcl 17681 . . . . . . . . . 10 ((𝑊 ∈ Grp ∧ ((-𝑚 − 2) + 1) ∈ ℤ ∧ 𝑋𝐵) → (((-𝑚 − 2) + 1) · 𝑋) ∈ 𝐵)
15475, 152, 77, 153syl3anc 1439 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((-𝑚 − 2) + 1) · 𝑋) ∈ 𝐵)
1557, 34posref 17073 . . . . . . . . 9 ((𝑊 ∈ Poset ∧ (((-𝑚 − 2) + 1) · 𝑋) ∈ 𝐵) → (((-𝑚 − 2) + 1) · 𝑋) (((-𝑚 − 2) + 1) · 𝑋))
156149, 154, 155syl2anc 696 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((-𝑚 − 2) + 1) · 𝑋) (((-𝑚 − 2) + 1) · 𝑋))
157147, 156eqbrtrrd 4784 . . . . . . 7 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 1) · 𝑋) (((-𝑚 − 2) + 1) · 𝑋))
158157ad2antrr 764 . . . . . 6 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → (-(𝑚 + 1) · 𝑋) (((-𝑚 − 2) + 1) · 𝑋))
159135, 158eqbrtrd 4782 . . . . 5 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → 𝑌 (((-𝑚 − 2) + 1) · 𝑋))
160 oveq1 6772 . . . . . . . 8 (𝑛 = (-𝑚 − 2) → (𝑛 · 𝑋) = ((-𝑚 − 2) · 𝑋))
161160breq1d 4770 . . . . . . 7 (𝑛 = (-𝑚 − 2) → ((𝑛 · 𝑋) < 𝑌 ↔ ((-𝑚 − 2) · 𝑋) < 𝑌))
162 oveq1 6772 . . . . . . . . 9 (𝑛 = (-𝑚 − 2) → (𝑛 + 1) = ((-𝑚 − 2) + 1))
163162oveq1d 6780 . . . . . . . 8 (𝑛 = (-𝑚 − 2) → ((𝑛 + 1) · 𝑋) = (((-𝑚 − 2) + 1) · 𝑋))
164163breq2d 4772 . . . . . . 7 (𝑛 = (-𝑚 − 2) → (𝑌 ((𝑛 + 1) · 𝑋) ↔ 𝑌 (((-𝑚 − 2) + 1) · 𝑋)))
165161, 164anbi12d 749 . . . . . 6 (𝑛 = (-𝑚 − 2) → (((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)) ↔ (((-𝑚 − 2) · 𝑋) < 𝑌𝑌 (((-𝑚 − 2) + 1) · 𝑋))))
166165rspcev 3413 . . . . 5 (((-𝑚 − 2) ∈ ℤ ∧ (((-𝑚 − 2) · 𝑋) < 𝑌𝑌 (((-𝑚 − 2) + 1) · 𝑋))) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
16765, 136, 159, 166syl12anc 1437 . . . 4 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
16876ad2antrr 764 . . . . . 6 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → (𝑚 + 1) ∈ ℤ)
169168znegcld 11597 . . . . 5 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → -(𝑚 + 1) ∈ ℤ)
1702ad2antrr 764 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ (𝑚 ∈ ℕ0 ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋)) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋))) → 𝑊 ∈ oGrp)
17172ad2antrr 764 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ (𝑚 ∈ ℕ0 ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋)) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋))) → (oppg𝑊) ∈ oGrp)
172170, 171jca 555 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ (𝑚 ∈ ℕ0 ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋)) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋))) → (𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp))
1731723anassrs 1415 . . . . . . 7 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → (𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp))
174123ad2antrr 764 . . . . . . 7 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((invg𝑊)‘𝑌) ∈ 𝐵)
17579ad2antrr 764 . . . . . . 7 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((𝑚 + 1) · 𝑋) ∈ 𝐵)
176 simpr 479 . . . . . . 7 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋))
1777, 17, 9ogrpinvlt 29954 . . . . . . . 8 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp) ∧ ((invg𝑊)‘𝑌) ∈ 𝐵 ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵) → (((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋) ↔ ((invg𝑊)‘((𝑚 + 1) · 𝑋)) < ((invg𝑊)‘((invg𝑊)‘𝑌))))
178177biimpa 502 . . . . . . 7 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp) ∧ ((invg𝑊)‘𝑌) ∈ 𝐵 ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((invg𝑊)‘((𝑚 + 1) · 𝑋)) < ((invg𝑊)‘((invg𝑊)‘𝑌)))
179173, 174, 175, 176, 178syl31anc 1442 . . . . . 6 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((invg𝑊)‘((𝑚 + 1) · 𝑋)) < ((invg𝑊)‘((invg𝑊)‘𝑌)))
180114ad2antrr 764 . . . . . . 7 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → (-(𝑚 + 1) · 𝑋) = ((invg𝑊)‘((𝑚 + 1) · 𝑋)))
181180eqcomd 2730 . . . . . 6 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((invg𝑊)‘((𝑚 + 1) · 𝑋)) = (-(𝑚 + 1) · 𝑋))
182133ad4antr 771 . . . . . 6 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((invg𝑊)‘((invg𝑊)‘𝑌)) = 𝑌)
183179, 181, 1823brtr3d 4791 . . . . 5 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → (-(𝑚 + 1) · 𝑋) < 𝑌)
184 simp-4l 825 . . . . . 6 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → 𝜑)
1857, 8mulgcl 17681 . . . . . . . . . . . 12 ((𝑊 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝑋𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
18675, 60, 77, 185syl3anc 1439 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 · 𝑋) ∈ 𝐵)
1877, 17, 9ogrpinvlt 29954 . . . . . . . . . . 11 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp) ∧ (𝑚 · 𝑋) ∈ 𝐵 ∧ ((invg𝑊)‘𝑌) ∈ 𝐵) → ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ↔ ((invg𝑊)‘((invg𝑊)‘𝑌)) < ((invg𝑊)‘(𝑚 · 𝑋))))
18874, 186, 123, 187syl3anc 1439 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ↔ ((invg𝑊)‘((invg𝑊)‘𝑌)) < ((invg𝑊)‘(𝑚 · 𝑋))))
189188biimpa 502 . . . . . . . . 9 ((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 · 𝑋) < ((invg𝑊)‘𝑌)) → ((invg𝑊)‘((invg𝑊)‘𝑌)) < ((invg𝑊)‘(𝑚 · 𝑋)))
190189adantrr 755 . . . . . . . 8 ((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) → ((invg𝑊)‘((invg𝑊)‘𝑌)) < ((invg𝑊)‘(𝑚 · 𝑋)))
191190adantr 472 . . . . . . 7 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((invg𝑊)‘((invg𝑊)‘𝑌)) < ((invg𝑊)‘(𝑚 · 𝑋)))
192 negdi 10451 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝑚 + 1) = (-𝑚 + -1))
19366, 40, 192sylancl 697 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → -(𝑚 + 1) = (-𝑚 + -1))
194193oveq1d 6780 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0 → (-(𝑚 + 1) + 1) = ((-𝑚 + -1) + 1))
19566negcld 10492 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → -𝑚 ∈ ℂ)
19691negcld 10492 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → -1 ∈ ℂ)
197195, 196, 91addassd 10175 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → ((-𝑚 + -1) + 1) = (-𝑚 + (-1 + 1)))
19843oveq2i 6776 . . . . . . . . . . . . . . 15 (-𝑚 + (-1 + 1)) = (-𝑚 + 0)
199198a1i 11 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → (-𝑚 + (-1 + 1)) = (-𝑚 + 0))
200195addid1d 10349 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → (-𝑚 + 0) = -𝑚)
201197, 199, 2003eqtrd 2762 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0 → ((-𝑚 + -1) + 1) = -𝑚)
202194, 201eqtrd 2758 . . . . . . . . . . . 12 (𝑚 ∈ ℕ0 → (-(𝑚 + 1) + 1) = -𝑚)
203202oveq1d 6780 . . . . . . . . . . 11 (𝑚 ∈ ℕ0 → ((-(𝑚 + 1) + 1) · 𝑋) = (-𝑚 · 𝑋))
204203adantl 473 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-(𝑚 + 1) + 1) · 𝑋) = (-𝑚 · 𝑋))
2057, 8, 9mulgneg 17682 . . . . . . . . . . 11 ((𝑊 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝑋𝐵) → (-𝑚 · 𝑋) = ((invg𝑊)‘(𝑚 · 𝑋)))
20675, 60, 77, 205syl3anc 1439 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-𝑚 · 𝑋) = ((invg𝑊)‘(𝑚 · 𝑋)))
207204, 206eqtrd 2758 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-(𝑚 + 1) + 1) · 𝑋) = ((invg𝑊)‘(𝑚 · 𝑋)))
208207ad2antrr 764 . . . . . . . 8 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((-(𝑚 + 1) + 1) · 𝑋) = ((invg𝑊)‘(𝑚 · 𝑋)))
209208eqcomd 2730 . . . . . . 7 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((invg𝑊)‘(𝑚 · 𝑋)) = ((-(𝑚 + 1) + 1) · 𝑋))
210191, 182, 2093brtr3d 4791 . . . . . 6 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → 𝑌 < ((-(𝑚 + 1) + 1) · 𝑋))
211 ovexd 6795 . . . . . . 7 (𝜑 → ((-(𝑚 + 1) + 1) · 𝑋) ∈ V)
21234, 17pltle 17083 . . . . . . 7 ((𝑊 ∈ oGrp ∧ 𝑌𝐵 ∧ ((-(𝑚 + 1) + 1) · 𝑋) ∈ V) → (𝑌 < ((-(𝑚 + 1) + 1) · 𝑋) → 𝑌 ((-(𝑚 + 1) + 1) · 𝑋)))
2132, 120, 211, 212syl3anc 1439 . . . . . 6 (𝜑 → (𝑌 < ((-(𝑚 + 1) + 1) · 𝑋) → 𝑌 ((-(𝑚 + 1) + 1) · 𝑋)))
214184, 210, 213sylc 65 . . . . 5 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → 𝑌 ((-(𝑚 + 1) + 1) · 𝑋))
215 oveq1 6772 . . . . . . . 8 (𝑛 = -(𝑚 + 1) → (𝑛 · 𝑋) = (-(𝑚 + 1) · 𝑋))
216215breq1d 4770 . . . . . . 7 (𝑛 = -(𝑚 + 1) → ((𝑛 · 𝑋) < 𝑌 ↔ (-(𝑚 + 1) · 𝑋) < 𝑌))
217 oveq1 6772 . . . . . . . . 9 (𝑛 = -(𝑚 + 1) → (𝑛 + 1) = (-(𝑚 + 1) + 1))
218217oveq1d 6780 . . . . . . . 8 (𝑛 = -(𝑚 + 1) → ((𝑛 + 1) · 𝑋) = ((-(𝑚 + 1) + 1) · 𝑋))
219218breq2d 4772 . . . . . . 7 (𝑛 = -(𝑚 + 1) → (𝑌 ((𝑛 + 1) · 𝑋) ↔ 𝑌 ((-(𝑚 + 1) + 1) · 𝑋)))
220216, 219anbi12d 749 . . . . . 6 (𝑛 = -(𝑚 + 1) → (((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)) ↔ ((-(𝑚 + 1) · 𝑋) < 𝑌𝑌 ((-(𝑚 + 1) + 1) · 𝑋))))
221220rspcev 3413 . . . . 5 ((-(𝑚 + 1) ∈ ℤ ∧ ((-(𝑚 + 1) · 𝑋) < 𝑌𝑌 ((-(𝑚 + 1) + 1) · 𝑋))) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
222169, 183, 214, 221syl12anc 1437 . . . 4 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
2237, 34, 17tlt2 29894 . . . . . 6 ((𝑊 ∈ Toset ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵 ∧ ((invg𝑊)‘𝑌) ∈ 𝐵) → (((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌) ∨ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)))
224148, 79, 123, 223syl3anc 1439 . . . . 5 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌) ∨ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)))
225224adantr 472 . . . 4 ((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) → (((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌) ∨ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)))
226167, 222, 225mpjaodan 862 . . 3 ((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
2272adantr 472 . . . 4 ((𝜑𝑌 < 0 ) → 𝑊 ∈ oGrp)
228 archirng.2 . . . . 5 (𝜑𝑊 ∈ Archi)
229228adantr 472 . . . 4 ((𝜑𝑌 < 0 ) → 𝑊 ∈ Archi)
2306adantr 472 . . . 4 ((𝜑𝑌 < 0 ) → 𝑋𝐵)
231122adantr 472 . . . 4 ((𝜑𝑌 < 0 ) → ((invg𝑊)‘𝑌) ∈ 𝐵)
23216adantr 472 . . . 4 ((𝜑𝑌 < 0 ) → 0 < 𝑋)
233133breq1d 4770 . . . . . 6 (𝜑 → (((invg𝑊)‘((invg𝑊)‘𝑌)) < 0𝑌 < 0 ))
234233biimpar 503 . . . . 5 ((𝜑𝑌 < 0 ) → ((invg𝑊)‘((invg𝑊)‘𝑌)) < 0 )
2357, 17, 9, 18ogrpinv0lt 29953 . . . . . . 7 ((𝑊 ∈ oGrp ∧ ((invg𝑊)‘𝑌) ∈ 𝐵) → ( 0 < ((invg𝑊)‘𝑌) ↔ ((invg𝑊)‘((invg𝑊)‘𝑌)) < 0 ))
2362, 122, 235syl2anc 696 . . . . . 6 (𝜑 → ( 0 < ((invg𝑊)‘𝑌) ↔ ((invg𝑊)‘((invg𝑊)‘𝑌)) < 0 ))
237236biimpar 503 . . . . 5 ((𝜑 ∧ ((invg𝑊)‘((invg𝑊)‘𝑌)) < 0 ) → 0 < ((invg𝑊)‘𝑌))
238234, 237syldan 488 . . . 4 ((𝜑𝑌 < 0 ) → 0 < ((invg𝑊)‘𝑌))
2397, 18, 17, 34, 8, 227, 229, 230, 231, 232, 238archirng 29972 . . 3 ((𝜑𝑌 < 0 ) → ∃𝑚 ∈ ℕ0 ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋)))
240226, 239r19.29a 3180 . 2 ((𝜑𝑌 < 0 ) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
241 nn0ssz 11511 . . 3 0 ⊆ ℤ
2422adantr 472 . . . 4 ((𝜑0 < 𝑌) → 𝑊 ∈ oGrp)
243228adantr 472 . . . 4 ((𝜑0 < 𝑌) → 𝑊 ∈ Archi)
2446adantr 472 . . . 4 ((𝜑0 < 𝑌) → 𝑋𝐵)
245120adantr 472 . . . 4 ((𝜑0 < 𝑌) → 𝑌𝐵)
24616adantr 472 . . . 4 ((𝜑0 < 𝑌) → 0 < 𝑋)
247 simpr 479 . . . 4 ((𝜑0 < 𝑌) → 0 < 𝑌)
2487, 18, 17, 34, 8, 242, 243, 244, 245, 246, 247archirng 29972 . . 3 ((𝜑0 < 𝑌) → ∃𝑛 ∈ ℕ0 ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
249 ssrexv 3773 . . 3 (ℕ0 ⊆ ℤ → (∃𝑛 ∈ ℕ0 ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋))))
250241, 248, 249mpsyl 68 . 2 ((𝜑0 < 𝑌) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
2517, 17tlt3 29895 . . 3 ((𝑊 ∈ Toset ∧ 𝑌𝐵0𝐵) → (𝑌 = 0𝑌 < 00 < 𝑌))
25229, 120, 33, 251syl3anc 1439 . 2 (𝜑 → (𝑌 = 0𝑌 < 00 < 𝑌))
25358, 240, 250, 252mpjao3dan 1508 1 (𝜑 → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3o 1071  w3a 1072   = wceq 1596  wcel 2103  wrex 3015  Vcvv 3304  wss 3680   class class class wbr 4760  cfv 6001  (class class class)co 6765  cc 10047  0cc0 10049  1c1 10050   + caddc 10052  cmin 10379  -cneg 10380  2c2 11183  0cn0 11405  cz 11490  Basecbs 15980  +gcplusg 16064  lecple 16071  0gc0g 16223  Posetcpo 17062  ltcplt 17063  Tosetctos 17155  Grpcgrp 17544  invgcminusg 17545  .gcmg 17662  oppgcoppg 17896  oMndcomnd 29927  oGrpcogrp 29928  Archicarchi 29961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-inf2 8651  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-tpos 7472  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-2 11192  df-3 11193  df-4 11194  df-5 11195  df-6 11196  df-7 11197  df-8 11198  df-9 11199  df-n0 11406  df-z 11491  df-dec 11607  df-uz 11801  df-fz 12441  df-seq 12917  df-ndx 15983  df-slot 15984  df-base 15986  df-sets 15987  df-plusg 16077  df-ple 16084  df-0g 16225  df-preset 17050  df-poset 17068  df-plt 17080  df-toset 17156  df-mgm 17364  df-sgrp 17406  df-mnd 17417  df-grp 17547  df-minusg 17548  df-mulg 17663  df-oppg 17897  df-omnd 29929  df-ogrp 29930  df-inftm 29962  df-archi 29963
This theorem is referenced by:  archiabllem2c  29979
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