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Theorem archirngz 33197
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 12655 . . 3 -1 ∈ ℤ
2 archirng.1 . . . . . . . . . 10 (𝜑𝑊 ∈ oGrp)
3 ogrpgrp 33081 . . . . . . . . . 10 (𝑊 ∈ oGrp → 𝑊 ∈ Grp)
42, 3syl 17 . . . . . . . . 9 (𝜑𝑊 ∈ Grp)
5 1zzd 12650 . . . . . . . . 9 (𝜑 → 1 ∈ ℤ)
6 archirng.3 . . . . . . . . 9 (𝜑𝑋𝐵)
7 archirng.b . . . . . . . . . 10 𝐵 = (Base‘𝑊)
8 archirng.x . . . . . . . . . 10 · = (.g𝑊)
9 eqid 2736 . . . . . . . . . 10 (invg𝑊) = (invg𝑊)
107, 8, 9mulgneg 19111 . . . . . . . . 9 ((𝑊 ∈ Grp ∧ 1 ∈ ℤ ∧ 𝑋𝐵) → (-1 · 𝑋) = ((invg𝑊)‘(1 · 𝑋)))
114, 5, 6, 10syl3anc 1372 . . . . . . . 8 (𝜑 → (-1 · 𝑋) = ((invg𝑊)‘(1 · 𝑋)))
127, 8mulg1 19100 . . . . . . . . . 10 (𝑋𝐵 → (1 · 𝑋) = 𝑋)
136, 12syl 17 . . . . . . . . 9 (𝜑 → (1 · 𝑋) = 𝑋)
1413fveq2d 6909 . . . . . . . 8 (𝜑 → ((invg𝑊)‘(1 · 𝑋)) = ((invg𝑊)‘𝑋))
1511, 14eqtrd 2776 . . . . . . 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 33100 . . . . . . . . 9 ((𝑊 ∈ oGrp ∧ 𝑋𝐵) → ( 0 < 𝑋 ↔ ((invg𝑊)‘𝑋) < 0 ))
2019biimpa 476 . . . . . . . 8 (((𝑊 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → ((invg𝑊)‘𝑋) < 0 )
212, 6, 16, 20syl21anc 837 . . . . . . 7 (𝜑 → ((invg𝑊)‘𝑋) < 0 )
2215, 21eqbrtrd 5164 . . . . . 6 (𝜑 → (-1 · 𝑋) < 0 )
2322adantr 480 . . . . 5 ((𝜑𝑌 = 0 ) → (-1 · 𝑋) < 0 )
24 simpr 484 . . . . 5 ((𝜑𝑌 = 0 ) → 𝑌 = 0 )
2523, 24breqtrrd 5170 . . . 4 ((𝜑𝑌 = 0 ) → (-1 · 𝑋) < 𝑌)
26 isogrp 33080 . . . . . . . . . 10 (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
2726simprbi 496 . . . . . . . . 9 (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
28 omndtos 33083 . . . . . . . . 9 (𝑊 ∈ oMnd → 𝑊 ∈ Toset)
292, 27, 283syl 18 . . . . . . . 8 (𝜑𝑊 ∈ Toset)
30 tospos 18466 . . . . . . . 8 (𝑊 ∈ Toset → 𝑊 ∈ Poset)
3129, 30syl 17 . . . . . . 7 (𝜑𝑊 ∈ Poset)
327, 18grpidcl 18984 . . . . . . . 8 (𝑊 ∈ Grp → 0𝐵)
332, 3, 323syl 18 . . . . . . 7 (𝜑0𝐵)
34 archirng.l . . . . . . . 8 = (le‘𝑊)
357, 34posref 18365 . . . . . . 7 ((𝑊 ∈ Poset ∧ 0𝐵) → 0 0 )
3631, 33, 35syl2anc 584 . . . . . 6 (𝜑0 0 )
3736adantr 480 . . . . 5 ((𝜑𝑌 = 0 ) → 0 0 )
38 1m1e0 12339 . . . . . . . . . 10 (1 − 1) = 0
3938negeqi 11502 . . . . . . . . 9 -(1 − 1) = -0
40 ax-1cn 11214 . . . . . . . . . 10 1 ∈ ℂ
4140, 40negsubdii 11595 . . . . . . . . 9 -(1 − 1) = (-1 + 1)
42 neg0 11556 . . . . . . . . 9 -0 = 0
4339, 41, 423eqtr3i 2772 . . . . . . . 8 (-1 + 1) = 0
4443oveq1i 7442 . . . . . . 7 ((-1 + 1) · 𝑋) = (0 · 𝑋)
457, 18, 8mulg0 19093 . . . . . . . 8 (𝑋𝐵 → (0 · 𝑋) = 0 )
466, 45syl 17 . . . . . . 7 (𝜑 → (0 · 𝑋) = 0 )
4744, 46eqtrid 2788 . . . . . 6 (𝜑 → ((-1 + 1) · 𝑋) = 0 )
4847adantr 480 . . . . 5 ((𝜑𝑌 = 0 ) → ((-1 + 1) · 𝑋) = 0 )
4937, 24, 483brtr4d 5174 . . . 4 ((𝜑𝑌 = 0 ) → 𝑌 ((-1 + 1) · 𝑋))
5025, 49jca 511 . . 3 ((𝜑𝑌 = 0 ) → ((-1 · 𝑋) < 𝑌𝑌 ((-1 + 1) · 𝑋)))
51 oveq1 7439 . . . . . 6 (𝑛 = -1 → (𝑛 · 𝑋) = (-1 · 𝑋))
5251breq1d 5152 . . . . 5 (𝑛 = -1 → ((𝑛 · 𝑋) < 𝑌 ↔ (-1 · 𝑋) < 𝑌))
53 oveq1 7439 . . . . . . 7 (𝑛 = -1 → (𝑛 + 1) = (-1 + 1))
5453oveq1d 7447 . . . . . 6 (𝑛 = -1 → ((𝑛 + 1) · 𝑋) = ((-1 + 1) · 𝑋))
5554breq2d 5154 . . . . 5 (𝑛 = -1 → (𝑌 ((𝑛 + 1) · 𝑋) ↔ 𝑌 ((-1 + 1) · 𝑋)))
5652, 55anbi12d 632 . . . 4 (𝑛 = -1 → (((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)) ↔ ((-1 · 𝑋) < 𝑌𝑌 ((-1 + 1) · 𝑋))))
5756rspcev 3621 . . 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 484 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
6059nn0zd 12641 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℤ)
6160ad2antrr 726 . . . . . . 7 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → 𝑚 ∈ ℤ)
6261znegcld 12726 . . . . . 6 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → -𝑚 ∈ ℤ)
63 2z 12651 . . . . . . 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 12729 . . . . 5 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → (-𝑚 − 2) ∈ ℤ)
66 nn0cn 12538 . . . . . . . . . . 11 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
6766adantl 481 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℂ)
68 2cnd 12345 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 2 ∈ ℂ)
6967, 68negdi2d 11635 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -(𝑚 + 2) = (-𝑚 − 2))
7069oveq1d 7447 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) · 𝑋) = ((-𝑚 − 2) · 𝑋))
712ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ oGrp)
72 archirngz.1 . . . . . . . . . . . 12 (𝜑 → (oppg𝑊) ∈ oGrp)
7372ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (oppg𝑊) ∈ oGrp)
7471, 73jca 511 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp))
754ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ Grp)
7660peano2zd 12727 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 1) ∈ ℤ)
776ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑋𝐵)
787, 8mulgcl 19110 . . . . . . . . . . 11 ((𝑊 ∈ Grp ∧ (𝑚 + 1) ∈ ℤ ∧ 𝑋𝐵) → ((𝑚 + 1) · 𝑋) ∈ 𝐵)
7975, 76, 77, 78syl3anc 1372 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) · 𝑋) ∈ 𝐵)
8063a1i 11 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 2 ∈ ℤ)
8160, 80zaddcld 12728 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 2) ∈ ℤ)
827, 8mulgcl 19110 . . . . . . . . . . 11 ((𝑊 ∈ Grp ∧ (𝑚 + 2) ∈ ℤ ∧ 𝑋𝐵) → ((𝑚 + 2) · 𝑋) ∈ 𝐵)
8375, 81, 77, 82syl3anc 1372 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 2) · 𝑋) ∈ 𝐵)
8475, 32syl 17 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 0𝐵)
8516ad2antrr 726 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 0 < 𝑋)
86 eqid 2736 . . . . . . . . . . . . 13 (+g𝑊) = (+g𝑊)
877, 17, 86ogrpaddlt 33095 . . . . . . . . . . . 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 1384 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ( 0 (+g𝑊)((𝑚 + 1) · 𝑋)) < (𝑋(+g𝑊)((𝑚 + 1) · 𝑋)))
897, 86, 18grplid 18986 . . . . . . . . . . . 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 11257 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ0 → 1 ∈ ℂ)
9266, 91, 91addassd 11284 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) = (𝑚 + (1 + 1)))
93 1p1e2 12392 . . . . . . . . . . . . . . . . 17 (1 + 1) = 2
9493oveq2i 7443 . . . . . . . . . . . . . . . 16 (𝑚 + (1 + 1)) = (𝑚 + 2)
9592, 94eqtrdi 2792 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) = (𝑚 + 2))
9666, 91addcld 11281 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℂ)
9796, 91addcomd 11464 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) = (1 + (𝑚 + 1)))
9895, 97eqtr3d 2778 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → (𝑚 + 2) = (1 + (𝑚 + 1)))
9998oveq1d 7447 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0 → ((𝑚 + 2) · 𝑋) = ((1 + (𝑚 + 1)) · 𝑋))
10099adantl 481 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 2) · 𝑋) = ((1 + (𝑚 + 1)) · 𝑋))
101 1zzd 12650 . . . . . . . . . . . . 13 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 1 ∈ ℤ)
1027, 8, 86mulgdir 19125 . . . . . . . . . . . . 13 ((𝑊 ∈ Grp ∧ (1 ∈ ℤ ∧ (𝑚 + 1) ∈ ℤ ∧ 𝑋𝐵)) → ((1 + (𝑚 + 1)) · 𝑋) = ((1 · 𝑋)(+g𝑊)((𝑚 + 1) · 𝑋)))
10375, 101, 76, 77, 102syl13anc 1373 . . . . . . . . . . . 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 7447 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((1 · 𝑋)(+g𝑊)((𝑚 + 1) · 𝑋)) = (𝑋(+g𝑊)((𝑚 + 1) · 𝑋)))
106100, 103, 1053eqtrrd 2781 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑋(+g𝑊)((𝑚 + 1) · 𝑋)) = ((𝑚 + 2) · 𝑋))
10788, 90, 1063brtr3d 5173 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) · 𝑋) < ((𝑚 + 2) · 𝑋))
1087, 17, 9ogrpinvlt 33101 . . . . . . . . . . 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 476 . . . . . . . . . 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 1374 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((invg𝑊)‘((𝑚 + 2) · 𝑋)) < ((invg𝑊)‘((𝑚 + 1) · 𝑋)))
1117, 8, 9mulgneg 19111 . . . . . . . . . 10 ((𝑊 ∈ Grp ∧ (𝑚 + 2) ∈ ℤ ∧ 𝑋𝐵) → (-(𝑚 + 2) · 𝑋) = ((invg𝑊)‘((𝑚 + 2) · 𝑋)))
11275, 81, 77, 111syl3anc 1372 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) · 𝑋) = ((invg𝑊)‘((𝑚 + 2) · 𝑋)))
1137, 8, 9mulgneg 19111 . . . . . . . . . 10 ((𝑊 ∈ Grp ∧ (𝑚 + 1) ∈ ℤ ∧ 𝑋𝐵) → (-(𝑚 + 1) · 𝑋) = ((invg𝑊)‘((𝑚 + 1) · 𝑋)))
11475, 76, 77, 113syl3anc 1372 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 1) · 𝑋) = ((invg𝑊)‘((𝑚 + 1) · 𝑋)))
115110, 112, 1143brtr4d 5174 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) · 𝑋) < (-(𝑚 + 1) · 𝑋))
11670, 115eqbrtrrd 5166 . . . . . . 7 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-𝑚 − 2) · 𝑋) < (-(𝑚 + 1) · 𝑋))
117116ad2antrr 726 . . . . . 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 726 . . . . . . 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 732 . . . . . . . . 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 19006 . . . . . . . . . . . 12 ((𝑊 ∈ Grp ∧ 𝑌𝐵) → ((invg𝑊)‘𝑌) ∈ 𝐵)
1224, 120, 121syl2anc 584 . . . . . . . . . . 11 (𝜑 → ((invg𝑊)‘𝑌) ∈ 𝐵)
123122ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((invg𝑊)‘𝑌) ∈ 𝐵)
124123ad2antrr 726 . . . . . . . . 9 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → ((invg𝑊)‘𝑌) ∈ 𝐵)
12579ad2antrr 726 . . . . . . . . 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 777 . . . . . . . . 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 484 . . . . . . . . 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 18366 . . . . . . . . . 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 476 . . . . . . . . 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 1379 . . . . . . . 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 6909 . . . . . . 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 19024 . . . . . . . . 9 ((𝑊 ∈ Grp ∧ 𝑌𝐵) → ((invg𝑊)‘((invg𝑊)‘𝑌)) = 𝑌)
1334, 120, 132syl2anc 584 . . . . . . . 8 (𝜑 → ((invg𝑊)‘((invg𝑊)‘𝑌)) = 𝑌)
134133ad4antr 732 . . . . . . 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 2783 . . . . . 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 5170 . . . . 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 11257 . . . . . . . . . . . . 13 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 1 ∈ ℂ)
13867, 68, 137addsubassd 11641 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 2) − 1) = (𝑚 + (2 − 1)))
139 2m1e1 12393 . . . . . . . . . . . . 13 (2 − 1) = 1
140139oveq2i 7443 . . . . . . . . . . . 12 (𝑚 + (2 − 1)) = (𝑚 + 1)
141138, 140eqtr2di 2793 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 1) = ((𝑚 + 2) − 1))
142141negeqd 11503 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -(𝑚 + 1) = -((𝑚 + 2) − 1))
14367, 68addcld 11281 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 2) ∈ ℂ)
144143, 137negsubdid 11636 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -((𝑚 + 2) − 1) = (-(𝑚 + 2) + 1))
14569oveq1d 7447 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) + 1) = ((-𝑚 − 2) + 1))
146142, 144, 1453eqtrrd 2781 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-𝑚 − 2) + 1) = -(𝑚 + 1))
147146oveq1d 7447 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((-𝑚 − 2) + 1) · 𝑋) = (-(𝑚 + 1) · 𝑋))
14829ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ Toset)
149148, 30syl 17 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ Poset)
15060znegcld 12726 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -𝑚 ∈ ℤ)
151150, 80zsubcld 12729 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-𝑚 − 2) ∈ ℤ)
152151peano2zd 12727 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-𝑚 − 2) + 1) ∈ ℤ)
1537, 8mulgcl 19110 . . . . . . . . . 10 ((𝑊 ∈ Grp ∧ ((-𝑚 − 2) + 1) ∈ ℤ ∧ 𝑋𝐵) → (((-𝑚 − 2) + 1) · 𝑋) ∈ 𝐵)
15475, 152, 77, 153syl3anc 1372 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((-𝑚 − 2) + 1) · 𝑋) ∈ 𝐵)
1557, 34posref 18365 . . . . . . . . 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 5166 . . . . . . 7 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 1) · 𝑋) (((-𝑚 − 2) + 1) · 𝑋))
158157ad2antrr 726 . . . . . 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 5164 . . . . 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 7439 . . . . . . . 8 (𝑛 = (-𝑚 − 2) → (𝑛 · 𝑋) = ((-𝑚 − 2) · 𝑋))
161160breq1d 5152 . . . . . . 7 (𝑛 = (-𝑚 − 2) → ((𝑛 · 𝑋) < 𝑌 ↔ ((-𝑚 − 2) · 𝑋) < 𝑌))
162 oveq1 7439 . . . . . . . . 9 (𝑛 = (-𝑚 − 2) → (𝑛 + 1) = ((-𝑚 − 2) + 1))
163162oveq1d 7447 . . . . . . . 8 (𝑛 = (-𝑚 − 2) → ((𝑛 + 1) · 𝑋) = (((-𝑚 − 2) + 1) · 𝑋))
164163breq2d 5154 . . . . . . 7 (𝑛 = (-𝑚 − 2) → (𝑌 ((𝑛 + 1) · 𝑋) ↔ 𝑌 (((-𝑚 − 2) + 1) · 𝑋)))
165161, 164anbi12d 632 . . . . . 6 (𝑛 = (-𝑚 − 2) → (((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)) ↔ (((-𝑚 − 2) · 𝑋) < 𝑌𝑌 (((-𝑚 − 2) + 1) · 𝑋))))
166165rspcev 3621 . . . . 5 (((-𝑚 − 2) ∈ ℤ ∧ (((-𝑚 − 2) · 𝑋) < 𝑌𝑌 (((-𝑚 − 2) + 1) · 𝑋))) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
16765, 136, 159, 166syl12anc 836 . . . 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 726 . . . . . 6 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → (𝑚 + 1) ∈ ℤ)
169168znegcld 12726 . . . . 5 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → -(𝑚 + 1) ∈ ℤ)
1702ad2antrr 726 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ (𝑚 ∈ ℕ0 ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋)) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋))) → 𝑊 ∈ oGrp)
17172ad2antrr 726 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ (𝑚 ∈ ℕ0 ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋)) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋))) → (oppg𝑊) ∈ oGrp)
172170, 171jca 511 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ (𝑚 ∈ ℕ0 ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋)) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋))) → (𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp))
1731723anassrs 1360 . . . . . . 7 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → (𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp))
174123ad2antrr 726 . . . . . . 7 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((invg𝑊)‘𝑌) ∈ 𝐵)
17579ad2antrr 726 . . . . . . 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 484 . . . . . . 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 33101 . . . . . . . 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 476 . . . . . . 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 1374 . . . . . 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 726 . . . . . . 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 2742 . . . . . 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 732 . . . . . 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 5173 . . . . 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 782 . . . . . 6 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → 𝜑)
1857, 8mulgcl 19110 . . . . . . . . . . . 12 ((𝑊 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝑋𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
18675, 60, 77, 185syl3anc 1372 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 · 𝑋) ∈ 𝐵)
1877, 17, 9ogrpinvlt 33101 . . . . . . . . . . 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 1372 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ↔ ((invg𝑊)‘((invg𝑊)‘𝑌)) < ((invg𝑊)‘(𝑚 · 𝑋))))
189188biimpa 476 . . . . . . . . 9 ((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 · 𝑋) < ((invg𝑊)‘𝑌)) → ((invg𝑊)‘((invg𝑊)‘𝑌)) < ((invg𝑊)‘(𝑚 · 𝑋)))
190189adantrr 717 . . . . . . . 8 ((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) → ((invg𝑊)‘((invg𝑊)‘𝑌)) < ((invg𝑊)‘(𝑚 · 𝑋)))
191190adantr 480 . . . . . . 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 11567 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝑚 + 1) = (-𝑚 + -1))
19366, 40, 192sylancl 586 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → -(𝑚 + 1) = (-𝑚 + -1))
194193oveq1d 7447 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0 → (-(𝑚 + 1) + 1) = ((-𝑚 + -1) + 1))
19566negcld 11608 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → -𝑚 ∈ ℂ)
19691negcld 11608 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → -1 ∈ ℂ)
197195, 196, 91addassd 11284 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → ((-𝑚 + -1) + 1) = (-𝑚 + (-1 + 1)))
19843oveq2i 7443 . . . . . . . . . . . . . . 15 (-𝑚 + (-1 + 1)) = (-𝑚 + 0)
199198a1i 11 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → (-𝑚 + (-1 + 1)) = (-𝑚 + 0))
200195addridd 11462 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → (-𝑚 + 0) = -𝑚)
201197, 199, 2003eqtrd 2780 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0 → ((-𝑚 + -1) + 1) = -𝑚)
202194, 201eqtrd 2776 . . . . . . . . . . . 12 (𝑚 ∈ ℕ0 → (-(𝑚 + 1) + 1) = -𝑚)
203202oveq1d 7447 . . . . . . . . . . 11 (𝑚 ∈ ℕ0 → ((-(𝑚 + 1) + 1) · 𝑋) = (-𝑚 · 𝑋))
204203adantl 481 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-(𝑚 + 1) + 1) · 𝑋) = (-𝑚 · 𝑋))
2057, 8, 9mulgneg 19111 . . . . . . . . . . 11 ((𝑊 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝑋𝐵) → (-𝑚 · 𝑋) = ((invg𝑊)‘(𝑚 · 𝑋)))
20675, 60, 77, 205syl3anc 1372 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-𝑚 · 𝑋) = ((invg𝑊)‘(𝑚 · 𝑋)))
207204, 206eqtrd 2776 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-(𝑚 + 1) + 1) · 𝑋) = ((invg𝑊)‘(𝑚 · 𝑋)))
208207ad2antrr 726 . . . . . . . 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 2742 . . . . . . 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 5173 . . . . . 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 7467 . . . . . . 7 (𝜑 → ((-(𝑚 + 1) + 1) · 𝑋) ∈ V)
21234, 17pltle 18379 . . . . . . 7 ((𝑊 ∈ oGrp ∧ 𝑌𝐵 ∧ ((-(𝑚 + 1) + 1) · 𝑋) ∈ V) → (𝑌 < ((-(𝑚 + 1) + 1) · 𝑋) → 𝑌 ((-(𝑚 + 1) + 1) · 𝑋)))
2132, 120, 211, 212syl3anc 1372 . . . . . 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 7439 . . . . . . . 8 (𝑛 = -(𝑚 + 1) → (𝑛 · 𝑋) = (-(𝑚 + 1) · 𝑋))
216215breq1d 5152 . . . . . . 7 (𝑛 = -(𝑚 + 1) → ((𝑛 · 𝑋) < 𝑌 ↔ (-(𝑚 + 1) · 𝑋) < 𝑌))
217 oveq1 7439 . . . . . . . . 9 (𝑛 = -(𝑚 + 1) → (𝑛 + 1) = (-(𝑚 + 1) + 1))
218217oveq1d 7447 . . . . . . . 8 (𝑛 = -(𝑚 + 1) → ((𝑛 + 1) · 𝑋) = ((-(𝑚 + 1) + 1) · 𝑋))
219218breq2d 5154 . . . . . . 7 (𝑛 = -(𝑚 + 1) → (𝑌 ((𝑛 + 1) · 𝑋) ↔ 𝑌 ((-(𝑚 + 1) + 1) · 𝑋)))
220216, 219anbi12d 632 . . . . . 6 (𝑛 = -(𝑚 + 1) → (((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)) ↔ ((-(𝑚 + 1) · 𝑋) < 𝑌𝑌 ((-(𝑚 + 1) + 1) · 𝑋))))
221220rspcev 3621 . . . . 5 ((-(𝑚 + 1) ∈ ℤ ∧ ((-(𝑚 + 1) · 𝑋) < 𝑌𝑌 ((-(𝑚 + 1) + 1) · 𝑋))) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
222169, 183, 214, 221syl12anc 836 . . . 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 32960 . . . . . 6 ((𝑊 ∈ Toset ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵 ∧ ((invg𝑊)‘𝑌) ∈ 𝐵) → (((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌) ∨ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)))
224148, 79, 123, 223syl3anc 1372 . . . . 5 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌) ∨ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)))
225224adantr 480 . . . 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 960 . . 3 ((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
2272adantr 480 . . . 4 ((𝜑𝑌 < 0 ) → 𝑊 ∈ oGrp)
228 archirng.2 . . . . 5 (𝜑𝑊 ∈ Archi)
229228adantr 480 . . . 4 ((𝜑𝑌 < 0 ) → 𝑊 ∈ Archi)
2306adantr 480 . . . 4 ((𝜑𝑌 < 0 ) → 𝑋𝐵)
231122adantr 480 . . . 4 ((𝜑𝑌 < 0 ) → ((invg𝑊)‘𝑌) ∈ 𝐵)
23216adantr 480 . . . 4 ((𝜑𝑌 < 0 ) → 0 < 𝑋)
233133breq1d 5152 . . . . . 6 (𝜑 → (((invg𝑊)‘((invg𝑊)‘𝑌)) < 0𝑌 < 0 ))
234233biimpar 477 . . . . 5 ((𝜑𝑌 < 0 ) → ((invg𝑊)‘((invg𝑊)‘𝑌)) < 0 )
2357, 17, 9, 18ogrpinv0lt 33100 . . . . . . 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 477 . . . . 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 33196 . . 3 ((𝜑𝑌 < 0 ) → ∃𝑚 ∈ ℕ0 ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋)))
240226, 239r19.29a 3161 . 2 ((𝜑𝑌 < 0 ) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
241 nn0ssz 12638 . . 3 0 ⊆ ℤ
2422adantr 480 . . . 4 ((𝜑0 < 𝑌) → 𝑊 ∈ oGrp)
243228adantr 480 . . . 4 ((𝜑0 < 𝑌) → 𝑊 ∈ Archi)
2446adantr 480 . . . 4 ((𝜑0 < 𝑌) → 𝑋𝐵)
245120adantr 480 . . . 4 ((𝜑0 < 𝑌) → 𝑌𝐵)
24616adantr 480 . . . 4 ((𝜑0 < 𝑌) → 0 < 𝑋)
247 simpr 484 . . . 4 ((𝜑0 < 𝑌) → 0 < 𝑌)
2487, 18, 17, 34, 8, 242, 243, 244, 245, 246, 247archirng 33196 . . 3 ((𝜑0 < 𝑌) → ∃𝑛 ∈ ℕ0 ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
249 ssrexv 4052 . . 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 32961 . . 3 ((𝑊 ∈ Toset ∧ 𝑌𝐵0𝐵) → (𝑌 = 0𝑌 < 00 < 𝑌))
25229, 120, 33, 251syl3anc 1372 . 2 (𝜑 → (𝑌 = 0𝑌 < 00 < 𝑌))
25358, 240, 250, 252mpjao3dan 1433 1 (𝜑 → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3o 1085  w3a 1086   = wceq 1539  wcel 2107  wrex 3069  Vcvv 3479  wss 3950   class class class wbr 5142  cfv 6560  (class class class)co 7432  cc 11154  0cc0 11156  1c1 11157   + caddc 11159  cmin 11493  -cneg 11494  2c2 12322  0cn0 12528  cz 12615  Basecbs 17248  +gcplusg 17298  lecple 17305  0gc0g 17485  Posetcpo 18354  ltcplt 18355  Tosetctos 18462  Grpcgrp 18952  invgcminusg 18953  .gcmg 19086  oppgcoppg 19364  oMndcomnd 33075  oGrpcogrp 33076  Archicarchi 33185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-tpos 8252  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-seq 14044  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-plusg 17311  df-ple 17318  df-0g 17487  df-proset 18341  df-poset 18360  df-plt 18376  df-toset 18463  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-grp 18955  df-minusg 18956  df-mulg 19087  df-oppg 19365  df-omnd 33077  df-ogrp 33078  df-inftm 33186  df-archi 33187
This theorem is referenced by:  archiabllem2c  33203
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