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Theorem archirngz 30745
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 12006 . . 3 -1 ∈ ℤ
2 archirng.1 . . . . . . . . . 10 (𝜑𝑊 ∈ oGrp)
3 ogrpgrp 30631 . . . . . . . . . 10 (𝑊 ∈ oGrp → 𝑊 ∈ Grp)
42, 3syl 17 . . . . . . . . 9 (𝜑𝑊 ∈ Grp)
5 1zzd 12001 . . . . . . . . 9 (𝜑 → 1 ∈ ℤ)
6 archirng.3 . . . . . . . . 9 (𝜑𝑋𝐵)
7 archirng.b . . . . . . . . . 10 𝐵 = (Base‘𝑊)
8 archirng.x . . . . . . . . . 10 · = (.g𝑊)
9 eqid 2818 . . . . . . . . . 10 (invg𝑊) = (invg𝑊)
107, 8, 9mulgneg 18184 . . . . . . . . 9 ((𝑊 ∈ Grp ∧ 1 ∈ ℤ ∧ 𝑋𝐵) → (-1 · 𝑋) = ((invg𝑊)‘(1 · 𝑋)))
114, 5, 6, 10syl3anc 1363 . . . . . . . 8 (𝜑 → (-1 · 𝑋) = ((invg𝑊)‘(1 · 𝑋)))
127, 8mulg1 18173 . . . . . . . . . 10 (𝑋𝐵 → (1 · 𝑋) = 𝑋)
136, 12syl 17 . . . . . . . . 9 (𝜑 → (1 · 𝑋) = 𝑋)
1413fveq2d 6667 . . . . . . . 8 (𝜑 → ((invg𝑊)‘(1 · 𝑋)) = ((invg𝑊)‘𝑋))
1511, 14eqtrd 2853 . . . . . . 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 30650 . . . . . . . . 9 ((𝑊 ∈ oGrp ∧ 𝑋𝐵) → ( 0 < 𝑋 ↔ ((invg𝑊)‘𝑋) < 0 ))
2019biimpa 477 . . . . . . . 8 (((𝑊 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → ((invg𝑊)‘𝑋) < 0 )
212, 6, 16, 20syl21anc 833 . . . . . . 7 (𝜑 → ((invg𝑊)‘𝑋) < 0 )
2215, 21eqbrtrd 5079 . . . . . 6 (𝜑 → (-1 · 𝑋) < 0 )
2322adantr 481 . . . . 5 ((𝜑𝑌 = 0 ) → (-1 · 𝑋) < 0 )
24 simpr 485 . . . . 5 ((𝜑𝑌 = 0 ) → 𝑌 = 0 )
2523, 24breqtrrd 5085 . . . 4 ((𝜑𝑌 = 0 ) → (-1 · 𝑋) < 𝑌)
26 isogrp 30630 . . . . . . . . . 10 (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
2726simprbi 497 . . . . . . . . 9 (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
28 omndtos 30633 . . . . . . . . 9 (𝑊 ∈ oMnd → 𝑊 ∈ Toset)
292, 27, 283syl 18 . . . . . . . 8 (𝜑𝑊 ∈ Toset)
30 tospos 30572 . . . . . . . 8 (𝑊 ∈ Toset → 𝑊 ∈ Poset)
3129, 30syl 17 . . . . . . 7 (𝜑𝑊 ∈ Poset)
327, 18grpidcl 18069 . . . . . . . 8 (𝑊 ∈ Grp → 0𝐵)
332, 3, 323syl 18 . . . . . . 7 (𝜑0𝐵)
34 archirng.l . . . . . . . 8 = (le‘𝑊)
357, 34posref 17549 . . . . . . 7 ((𝑊 ∈ Poset ∧ 0𝐵) → 0 0 )
3631, 33, 35syl2anc 584 . . . . . 6 (𝜑0 0 )
3736adantr 481 . . . . 5 ((𝜑𝑌 = 0 ) → 0 0 )
38 1m1e0 11697 . . . . . . . . . 10 (1 − 1) = 0
3938negeqi 10867 . . . . . . . . 9 -(1 − 1) = -0
40 ax-1cn 10583 . . . . . . . . . 10 1 ∈ ℂ
4140, 40negsubdii 10959 . . . . . . . . 9 -(1 − 1) = (-1 + 1)
42 neg0 10920 . . . . . . . . 9 -0 = 0
4339, 41, 423eqtr3i 2849 . . . . . . . 8 (-1 + 1) = 0
4443oveq1i 7155 . . . . . . 7 ((-1 + 1) · 𝑋) = (0 · 𝑋)
457, 18, 8mulg0 18169 . . . . . . . 8 (𝑋𝐵 → (0 · 𝑋) = 0 )
466, 45syl 17 . . . . . . 7 (𝜑 → (0 · 𝑋) = 0 )
4744, 46syl5eq 2865 . . . . . 6 (𝜑 → ((-1 + 1) · 𝑋) = 0 )
4847adantr 481 . . . . 5 ((𝜑𝑌 = 0 ) → ((-1 + 1) · 𝑋) = 0 )
4937, 24, 483brtr4d 5089 . . . 4 ((𝜑𝑌 = 0 ) → 𝑌 ((-1 + 1) · 𝑋))
5025, 49jca 512 . . 3 ((𝜑𝑌 = 0 ) → ((-1 · 𝑋) < 𝑌𝑌 ((-1 + 1) · 𝑋)))
51 oveq1 7152 . . . . . 6 (𝑛 = -1 → (𝑛 · 𝑋) = (-1 · 𝑋))
5251breq1d 5067 . . . . 5 (𝑛 = -1 → ((𝑛 · 𝑋) < 𝑌 ↔ (-1 · 𝑋) < 𝑌))
53 oveq1 7152 . . . . . . 7 (𝑛 = -1 → (𝑛 + 1) = (-1 + 1))
5453oveq1d 7160 . . . . . 6 (𝑛 = -1 → ((𝑛 + 1) · 𝑋) = ((-1 + 1) · 𝑋))
5554breq2d 5069 . . . . 5 (𝑛 = -1 → (𝑌 ((𝑛 + 1) · 𝑋) ↔ 𝑌 ((-1 + 1) · 𝑋)))
5652, 55anbi12d 630 . . . 4 (𝑛 = -1 → (((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)) ↔ ((-1 · 𝑋) < 𝑌𝑌 ((-1 + 1) · 𝑋))))
5756rspcev 3620 . . 3 ((-1 ∈ ℤ ∧ ((-1 · 𝑋) < 𝑌𝑌 ((-1 + 1) · 𝑋))) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
581, 50, 57sylancr 587 . 2 ((𝜑𝑌 = 0 ) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
59 simpr 485 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
6059nn0zd 12073 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℤ)
6160ad2antrr 722 . . . . . . 7 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → 𝑚 ∈ ℤ)
6261znegcld 12077 . . . . . 6 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → -𝑚 ∈ ℤ)
63 2z 12002 . . . . . . 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 12080 . . . . 5 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → (-𝑚 − 2) ∈ ℤ)
66 nn0cn 11895 . . . . . . . . . . 11 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
6766adantl 482 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℂ)
68 2cnd 11703 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 2 ∈ ℂ)
6967, 68negdi2d 10999 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -(𝑚 + 2) = (-𝑚 − 2))
7069oveq1d 7160 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) · 𝑋) = ((-𝑚 − 2) · 𝑋))
712ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ oGrp)
72 archirngz.1 . . . . . . . . . . . 12 (𝜑 → (oppg𝑊) ∈ oGrp)
7372ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (oppg𝑊) ∈ oGrp)
7471, 73jca 512 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp))
754ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ Grp)
7660peano2zd 12078 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 1) ∈ ℤ)
776ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑋𝐵)
787, 8mulgcl 18183 . . . . . . . . . . 11 ((𝑊 ∈ Grp ∧ (𝑚 + 1) ∈ ℤ ∧ 𝑋𝐵) → ((𝑚 + 1) · 𝑋) ∈ 𝐵)
7975, 76, 77, 78syl3anc 1363 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) · 𝑋) ∈ 𝐵)
8063a1i 11 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 2 ∈ ℤ)
8160, 80zaddcld 12079 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 2) ∈ ℤ)
827, 8mulgcl 18183 . . . . . . . . . . 11 ((𝑊 ∈ Grp ∧ (𝑚 + 2) ∈ ℤ ∧ 𝑋𝐵) → ((𝑚 + 2) · 𝑋) ∈ 𝐵)
8375, 81, 77, 82syl3anc 1363 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 2) · 𝑋) ∈ 𝐵)
8475, 32syl 17 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 0𝐵)
8516ad2antrr 722 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 0 < 𝑋)
86 eqid 2818 . . . . . . . . . . . . 13 (+g𝑊) = (+g𝑊)
877, 17, 86ogrpaddlt 30645 . . . . . . . . . . . 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 1375 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ( 0 (+g𝑊)((𝑚 + 1) · 𝑋)) < (𝑋(+g𝑊)((𝑚 + 1) · 𝑋)))
897, 86, 18grplid 18071 . . . . . . . . . . . 12 ((𝑊 ∈ Grp ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵) → ( 0 (+g𝑊)((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) · 𝑋))
9075, 79, 89syl2anc 584 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ( 0 (+g𝑊)((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) · 𝑋))
91 1cnd 10624 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ0 → 1 ∈ ℂ)
9266, 91, 91addassd 10651 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) = (𝑚 + (1 + 1)))
93 1p1e2 11750 . . . . . . . . . . . . . . . . 17 (1 + 1) = 2
9493oveq2i 7156 . . . . . . . . . . . . . . . 16 (𝑚 + (1 + 1)) = (𝑚 + 2)
9592, 94syl6eq 2869 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) = (𝑚 + 2))
9666, 91addcld 10648 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℂ)
9796, 91addcomd 10830 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) = (1 + (𝑚 + 1)))
9895, 97eqtr3d 2855 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → (𝑚 + 2) = (1 + (𝑚 + 1)))
9998oveq1d 7160 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0 → ((𝑚 + 2) · 𝑋) = ((1 + (𝑚 + 1)) · 𝑋))
10099adantl 482 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 2) · 𝑋) = ((1 + (𝑚 + 1)) · 𝑋))
101 1zzd 12001 . . . . . . . . . . . . 13 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 1 ∈ ℤ)
1027, 8, 86mulgdir 18197 . . . . . . . . . . . . 13 ((𝑊 ∈ Grp ∧ (1 ∈ ℤ ∧ (𝑚 + 1) ∈ ℤ ∧ 𝑋𝐵)) → ((1 + (𝑚 + 1)) · 𝑋) = ((1 · 𝑋)(+g𝑊)((𝑚 + 1) · 𝑋)))
10375, 101, 76, 77, 102syl13anc 1364 . . . . . . . . . . . 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 7160 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((1 · 𝑋)(+g𝑊)((𝑚 + 1) · 𝑋)) = (𝑋(+g𝑊)((𝑚 + 1) · 𝑋)))
106100, 103, 1053eqtrrd 2858 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑋(+g𝑊)((𝑚 + 1) · 𝑋)) = ((𝑚 + 2) · 𝑋))
10788, 90, 1063brtr3d 5088 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) · 𝑋) < ((𝑚 + 2) · 𝑋))
1087, 17, 9ogrpinvlt 30651 . . . . . . . . . . 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 477 . . . . . . . . . 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 1365 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((invg𝑊)‘((𝑚 + 2) · 𝑋)) < ((invg𝑊)‘((𝑚 + 1) · 𝑋)))
1117, 8, 9mulgneg 18184 . . . . . . . . . 10 ((𝑊 ∈ Grp ∧ (𝑚 + 2) ∈ ℤ ∧ 𝑋𝐵) → (-(𝑚 + 2) · 𝑋) = ((invg𝑊)‘((𝑚 + 2) · 𝑋)))
11275, 81, 77, 111syl3anc 1363 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) · 𝑋) = ((invg𝑊)‘((𝑚 + 2) · 𝑋)))
1137, 8, 9mulgneg 18184 . . . . . . . . . 10 ((𝑊 ∈ Grp ∧ (𝑚 + 1) ∈ ℤ ∧ 𝑋𝐵) → (-(𝑚 + 1) · 𝑋) = ((invg𝑊)‘((𝑚 + 1) · 𝑋)))
11475, 76, 77, 113syl3anc 1363 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 1) · 𝑋) = ((invg𝑊)‘((𝑚 + 1) · 𝑋)))
115110, 112, 1143brtr4d 5089 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) · 𝑋) < (-(𝑚 + 1) · 𝑋))
11670, 115eqbrtrrd 5081 . . . . . . 7 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-𝑚 − 2) · 𝑋) < (-(𝑚 + 1) · 𝑋))
117116ad2antrr 722 . . . . . 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 722 . . . . . . 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 728 . . . . . . . . 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 18089 . . . . . . . . . . . 12 ((𝑊 ∈ Grp ∧ 𝑌𝐵) → ((invg𝑊)‘𝑌) ∈ 𝐵)
1224, 120, 121syl2anc 584 . . . . . . . . . . 11 (𝜑 → ((invg𝑊)‘𝑌) ∈ 𝐵)
123122ad2antrr 722 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((invg𝑊)‘𝑌) ∈ 𝐵)
124123ad2antrr 722 . . . . . . . . 9 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → ((invg𝑊)‘𝑌) ∈ 𝐵)
12579ad2antrr 722 . . . . . . . . 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 774 . . . . . . . . 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 485 . . . . . . . . 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 17550 . . . . . . . . . 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 477 . . . . . . . . 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 1370 . . . . . . . 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 6667 . . . . . . 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 18104 . . . . . . . . 9 ((𝑊 ∈ Grp ∧ 𝑌𝐵) → ((invg𝑊)‘((invg𝑊)‘𝑌)) = 𝑌)
1334, 120, 132syl2anc 584 . . . . . . . 8 (𝜑 → ((invg𝑊)‘((invg𝑊)‘𝑌)) = 𝑌)
134133ad4antr 728 . . . . . . 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 2860 . . . . . 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 5085 . . . . 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 10624 . . . . . . . . . . . . 13 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 1 ∈ ℂ)
13867, 68, 137addsubassd 11005 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 2) − 1) = (𝑚 + (2 − 1)))
139 2m1e1 11751 . . . . . . . . . . . . 13 (2 − 1) = 1
140139oveq2i 7156 . . . . . . . . . . . 12 (𝑚 + (2 − 1)) = (𝑚 + 1)
141138, 140syl6req 2870 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 1) = ((𝑚 + 2) − 1))
142141negeqd 10868 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -(𝑚 + 1) = -((𝑚 + 2) − 1))
14367, 68addcld 10648 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 2) ∈ ℂ)
144143, 137negsubdid 11000 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -((𝑚 + 2) − 1) = (-(𝑚 + 2) + 1))
14569oveq1d 7160 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) + 1) = ((-𝑚 − 2) + 1))
146142, 144, 1453eqtrrd 2858 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-𝑚 − 2) + 1) = -(𝑚 + 1))
147146oveq1d 7160 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((-𝑚 − 2) + 1) · 𝑋) = (-(𝑚 + 1) · 𝑋))
14829ad2antrr 722 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ Toset)
149148, 30syl 17 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ Poset)
15060znegcld 12077 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -𝑚 ∈ ℤ)
151150, 80zsubcld 12080 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-𝑚 − 2) ∈ ℤ)
152151peano2zd 12078 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-𝑚 − 2) + 1) ∈ ℤ)
1537, 8mulgcl 18183 . . . . . . . . . 10 ((𝑊 ∈ Grp ∧ ((-𝑚 − 2) + 1) ∈ ℤ ∧ 𝑋𝐵) → (((-𝑚 − 2) + 1) · 𝑋) ∈ 𝐵)
15475, 152, 77, 153syl3anc 1363 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((-𝑚 − 2) + 1) · 𝑋) ∈ 𝐵)
1557, 34posref 17549 . . . . . . . . 9 ((𝑊 ∈ Poset ∧ (((-𝑚 − 2) + 1) · 𝑋) ∈ 𝐵) → (((-𝑚 − 2) + 1) · 𝑋) (((-𝑚 − 2) + 1) · 𝑋))
156149, 154, 155syl2anc 584 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((-𝑚 − 2) + 1) · 𝑋) (((-𝑚 − 2) + 1) · 𝑋))
157147, 156eqbrtrrd 5081 . . . . . . 7 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 1) · 𝑋) (((-𝑚 − 2) + 1) · 𝑋))
158157ad2antrr 722 . . . . . 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 5079 . . . . 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 7152 . . . . . . . 8 (𝑛 = (-𝑚 − 2) → (𝑛 · 𝑋) = ((-𝑚 − 2) · 𝑋))
161160breq1d 5067 . . . . . . 7 (𝑛 = (-𝑚 − 2) → ((𝑛 · 𝑋) < 𝑌 ↔ ((-𝑚 − 2) · 𝑋) < 𝑌))
162 oveq1 7152 . . . . . . . . 9 (𝑛 = (-𝑚 − 2) → (𝑛 + 1) = ((-𝑚 − 2) + 1))
163162oveq1d 7160 . . . . . . . 8 (𝑛 = (-𝑚 − 2) → ((𝑛 + 1) · 𝑋) = (((-𝑚 − 2) + 1) · 𝑋))
164163breq2d 5069 . . . . . . 7 (𝑛 = (-𝑚 − 2) → (𝑌 ((𝑛 + 1) · 𝑋) ↔ 𝑌 (((-𝑚 − 2) + 1) · 𝑋)))
165161, 164anbi12d 630 . . . . . 6 (𝑛 = (-𝑚 − 2) → (((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)) ↔ (((-𝑚 − 2) · 𝑋) < 𝑌𝑌 (((-𝑚 − 2) + 1) · 𝑋))))
166165rspcev 3620 . . . . 5 (((-𝑚 − 2) ∈ ℤ ∧ (((-𝑚 − 2) · 𝑋) < 𝑌𝑌 (((-𝑚 − 2) + 1) · 𝑋))) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
16765, 136, 159, 166syl12anc 832 . . . 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 722 . . . . . 6 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → (𝑚 + 1) ∈ ℤ)
169168znegcld 12077 . . . . 5 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → -(𝑚 + 1) ∈ ℤ)
1702ad2antrr 722 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ (𝑚 ∈ ℕ0 ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋)) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋))) → 𝑊 ∈ oGrp)
17172ad2antrr 722 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ (𝑚 ∈ ℕ0 ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋)) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋))) → (oppg𝑊) ∈ oGrp)
172170, 171jca 512 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ (𝑚 ∈ ℕ0 ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋)) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋))) → (𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp))
1731723anassrs 1352 . . . . . . 7 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → (𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp))
174123ad2antrr 722 . . . . . . 7 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((invg𝑊)‘𝑌) ∈ 𝐵)
17579ad2antrr 722 . . . . . . 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 485 . . . . . . 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 30651 . . . . . . . 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 477 . . . . . . 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 1365 . . . . . 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 722 . . . . . . 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 2824 . . . . . 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 728 . . . . . 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 5088 . . . . 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 779 . . . . . 6 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → 𝜑)
1857, 8mulgcl 18183 . . . . . . . . . . . 12 ((𝑊 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝑋𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
18675, 60, 77, 185syl3anc 1363 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 · 𝑋) ∈ 𝐵)
1877, 17, 9ogrpinvlt 30651 . . . . . . . . . . 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 1363 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ↔ ((invg𝑊)‘((invg𝑊)‘𝑌)) < ((invg𝑊)‘(𝑚 · 𝑋))))
189188biimpa 477 . . . . . . . . 9 ((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 · 𝑋) < ((invg𝑊)‘𝑌)) → ((invg𝑊)‘((invg𝑊)‘𝑌)) < ((invg𝑊)‘(𝑚 · 𝑋)))
190189adantrr 713 . . . . . . . 8 ((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) → ((invg𝑊)‘((invg𝑊)‘𝑌)) < ((invg𝑊)‘(𝑚 · 𝑋)))
191190adantr 481 . . . . . . 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 10931 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝑚 + 1) = (-𝑚 + -1))
19366, 40, 192sylancl 586 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → -(𝑚 + 1) = (-𝑚 + -1))
194193oveq1d 7160 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0 → (-(𝑚 + 1) + 1) = ((-𝑚 + -1) + 1))
19566negcld 10972 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → -𝑚 ∈ ℂ)
19691negcld 10972 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → -1 ∈ ℂ)
197195, 196, 91addassd 10651 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → ((-𝑚 + -1) + 1) = (-𝑚 + (-1 + 1)))
19843oveq2i 7156 . . . . . . . . . . . . . . 15 (-𝑚 + (-1 + 1)) = (-𝑚 + 0)
199198a1i 11 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → (-𝑚 + (-1 + 1)) = (-𝑚 + 0))
200195addid1d 10828 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → (-𝑚 + 0) = -𝑚)
201197, 199, 2003eqtrd 2857 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0 → ((-𝑚 + -1) + 1) = -𝑚)
202194, 201eqtrd 2853 . . . . . . . . . . . 12 (𝑚 ∈ ℕ0 → (-(𝑚 + 1) + 1) = -𝑚)
203202oveq1d 7160 . . . . . . . . . . 11 (𝑚 ∈ ℕ0 → ((-(𝑚 + 1) + 1) · 𝑋) = (-𝑚 · 𝑋))
204203adantl 482 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-(𝑚 + 1) + 1) · 𝑋) = (-𝑚 · 𝑋))
2057, 8, 9mulgneg 18184 . . . . . . . . . . 11 ((𝑊 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝑋𝐵) → (-𝑚 · 𝑋) = ((invg𝑊)‘(𝑚 · 𝑋)))
20675, 60, 77, 205syl3anc 1363 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-𝑚 · 𝑋) = ((invg𝑊)‘(𝑚 · 𝑋)))
207204, 206eqtrd 2853 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-(𝑚 + 1) + 1) · 𝑋) = ((invg𝑊)‘(𝑚 · 𝑋)))
208207ad2antrr 722 . . . . . . . 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 2824 . . . . . . 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 5088 . . . . . 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 7180 . . . . . . 7 (𝜑 → ((-(𝑚 + 1) + 1) · 𝑋) ∈ V)
21234, 17pltle 17559 . . . . . . 7 ((𝑊 ∈ oGrp ∧ 𝑌𝐵 ∧ ((-(𝑚 + 1) + 1) · 𝑋) ∈ V) → (𝑌 < ((-(𝑚 + 1) + 1) · 𝑋) → 𝑌 ((-(𝑚 + 1) + 1) · 𝑋)))
2132, 120, 211, 212syl3anc 1363 . . . . . 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 7152 . . . . . . . 8 (𝑛 = -(𝑚 + 1) → (𝑛 · 𝑋) = (-(𝑚 + 1) · 𝑋))
216215breq1d 5067 . . . . . . 7 (𝑛 = -(𝑚 + 1) → ((𝑛 · 𝑋) < 𝑌 ↔ (-(𝑚 + 1) · 𝑋) < 𝑌))
217 oveq1 7152 . . . . . . . . 9 (𝑛 = -(𝑚 + 1) → (𝑛 + 1) = (-(𝑚 + 1) + 1))
218217oveq1d 7160 . . . . . . . 8 (𝑛 = -(𝑚 + 1) → ((𝑛 + 1) · 𝑋) = ((-(𝑚 + 1) + 1) · 𝑋))
219218breq2d 5069 . . . . . . 7 (𝑛 = -(𝑚 + 1) → (𝑌 ((𝑛 + 1) · 𝑋) ↔ 𝑌 ((-(𝑚 + 1) + 1) · 𝑋)))
220216, 219anbi12d 630 . . . . . 6 (𝑛 = -(𝑚 + 1) → (((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)) ↔ ((-(𝑚 + 1) · 𝑋) < 𝑌𝑌 ((-(𝑚 + 1) + 1) · 𝑋))))
221220rspcev 3620 . . . . 5 ((-(𝑚 + 1) ∈ ℤ ∧ ((-(𝑚 + 1) · 𝑋) < 𝑌𝑌 ((-(𝑚 + 1) + 1) · 𝑋))) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
222169, 183, 214, 221syl12anc 832 . . . 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 30578 . . . . . 6 ((𝑊 ∈ Toset ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵 ∧ ((invg𝑊)‘𝑌) ∈ 𝐵) → (((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌) ∨ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)))
224148, 79, 123, 223syl3anc 1363 . . . . 5 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌) ∨ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)))
225224adantr 481 . . . 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 952 . . 3 ((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
2272adantr 481 . . . 4 ((𝜑𝑌 < 0 ) → 𝑊 ∈ oGrp)
228 archirng.2 . . . . 5 (𝜑𝑊 ∈ Archi)
229228adantr 481 . . . 4 ((𝜑𝑌 < 0 ) → 𝑊 ∈ Archi)
2306adantr 481 . . . 4 ((𝜑𝑌 < 0 ) → 𝑋𝐵)
231122adantr 481 . . . 4 ((𝜑𝑌 < 0 ) → ((invg𝑊)‘𝑌) ∈ 𝐵)
23216adantr 481 . . . 4 ((𝜑𝑌 < 0 ) → 0 < 𝑋)
233133breq1d 5067 . . . . . 6 (𝜑 → (((invg𝑊)‘((invg𝑊)‘𝑌)) < 0𝑌 < 0 ))
234233biimpar 478 . . . . 5 ((𝜑𝑌 < 0 ) → ((invg𝑊)‘((invg𝑊)‘𝑌)) < 0 )
2357, 17, 9, 18ogrpinv0lt 30650 . . . . . . 7 ((𝑊 ∈ oGrp ∧ ((invg𝑊)‘𝑌) ∈ 𝐵) → ( 0 < ((invg𝑊)‘𝑌) ↔ ((invg𝑊)‘((invg𝑊)‘𝑌)) < 0 ))
2362, 122, 235syl2anc 584 . . . . . 6 (𝜑 → ( 0 < ((invg𝑊)‘𝑌) ↔ ((invg𝑊)‘((invg𝑊)‘𝑌)) < 0 ))
237236biimpar 478 . . . . 5 ((𝜑 ∧ ((invg𝑊)‘((invg𝑊)‘𝑌)) < 0 ) → 0 < ((invg𝑊)‘𝑌))
238234, 237syldan 591 . . . 4 ((𝜑𝑌 < 0 ) → 0 < ((invg𝑊)‘𝑌))
2397, 18, 17, 34, 8, 227, 229, 230, 231, 232, 238archirng 30744 . . 3 ((𝜑𝑌 < 0 ) → ∃𝑚 ∈ ℕ0 ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋)))
240226, 239r19.29a 3286 . 2 ((𝜑𝑌 < 0 ) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
241 nn0ssz 11991 . . 3 0 ⊆ ℤ
2422adantr 481 . . . 4 ((𝜑0 < 𝑌) → 𝑊 ∈ oGrp)
243228adantr 481 . . . 4 ((𝜑0 < 𝑌) → 𝑊 ∈ Archi)
2446adantr 481 . . . 4 ((𝜑0 < 𝑌) → 𝑋𝐵)
245120adantr 481 . . . 4 ((𝜑0 < 𝑌) → 𝑌𝐵)
24616adantr 481 . . . 4 ((𝜑0 < 𝑌) → 0 < 𝑋)
247 simpr 485 . . . 4 ((𝜑0 < 𝑌) → 0 < 𝑌)
2487, 18, 17, 34, 8, 242, 243, 244, 245, 246, 247archirng 30744 . . 3 ((𝜑0 < 𝑌) → ∃𝑛 ∈ ℕ0 ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
249 ssrexv 4031 . . 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 30579 . . 3 ((𝑊 ∈ Toset ∧ 𝑌𝐵0𝐵) → (𝑌 = 0𝑌 < 00 < 𝑌))
25229, 120, 33, 251syl3anc 1363 . 2 (𝜑 → (𝑌 = 0𝑌 < 00 < 𝑌))
25358, 240, 250, 252mpjao3dan 1423 1 (𝜑 → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841  w3o 1078  w3a 1079   = wceq 1528  wcel 2105  wrex 3136  Vcvv 3492  wss 3933   class class class wbr 5057  cfv 6348  (class class class)co 7145  cc 10523  0cc0 10525  1c1 10526   + caddc 10528  cmin 10858  -cneg 10859  2c2 11680  0cn0 11885  cz 11969  Basecbs 16471  +gcplusg 16553  lecple 16560  0gc0g 16701  Posetcpo 17538  ltcplt 17539  Tosetctos 17631  Grpcgrp 18041  invgcminusg 18042  .gcmg 18162  oppgcoppg 18411  oMndcomnd 30625  oGrpcogrp 30626  Archicarchi 30733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-tpos 7881  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12881  df-seq 13358  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-plusg 16566  df-ple 16573  df-0g 16703  df-proset 17526  df-poset 17544  df-plt 17556  df-toset 17632  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-minusg 18045  df-mulg 18163  df-oppg 18412  df-omnd 30627  df-ogrp 30628  df-inftm 30734  df-archi 30735
This theorem is referenced by:  archiabllem2c  30751
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