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Theorem archirngz 32025
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 12539 . . 3 -1 ∈ ℤ
2 archirng.1 . . . . . . . . . 10 (𝜑𝑊 ∈ oGrp)
3 ogrpgrp 31911 . . . . . . . . . 10 (𝑊 ∈ oGrp → 𝑊 ∈ Grp)
42, 3syl 17 . . . . . . . . 9 (𝜑𝑊 ∈ Grp)
5 1zzd 12534 . . . . . . . . 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 18894 . . . . . . . . 9 ((𝑊 ∈ Grp ∧ 1 ∈ ℤ ∧ 𝑋𝐵) → (-1 · 𝑋) = ((invg𝑊)‘(1 · 𝑋)))
114, 5, 6, 10syl3anc 1371 . . . . . . . 8 (𝜑 → (-1 · 𝑋) = ((invg𝑊)‘(1 · 𝑋)))
127, 8mulg1 18883 . . . . . . . . . 10 (𝑋𝐵 → (1 · 𝑋) = 𝑋)
136, 12syl 17 . . . . . . . . 9 (𝜑 → (1 · 𝑋) = 𝑋)
1413fveq2d 6846 . . . . . . . 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 31930 . . . . . . . . 9 ((𝑊 ∈ oGrp ∧ 𝑋𝐵) → ( 0 < 𝑋 ↔ ((invg𝑊)‘𝑋) < 0 ))
2019biimpa 477 . . . . . . . 8 (((𝑊 ∈ oGrp ∧ 𝑋𝐵) ∧ 0 < 𝑋) → ((invg𝑊)‘𝑋) < 0 )
212, 6, 16, 20syl21anc 836 . . . . . . 7 (𝜑 → ((invg𝑊)‘𝑋) < 0 )
2215, 21eqbrtrd 5127 . . . . . 6 (𝜑 → (-1 · 𝑋) < 0 )
2322adantr 481 . . . . 5 ((𝜑𝑌 = 0 ) → (-1 · 𝑋) < 0 )
24 simpr 485 . . . . 5 ((𝜑𝑌 = 0 ) → 𝑌 = 0 )
2523, 24breqtrrd 5133 . . . 4 ((𝜑𝑌 = 0 ) → (-1 · 𝑋) < 𝑌)
26 isogrp 31910 . . . . . . . . . 10 (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
2726simprbi 497 . . . . . . . . 9 (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
28 omndtos 31913 . . . . . . . . 9 (𝑊 ∈ oMnd → 𝑊 ∈ Toset)
292, 27, 283syl 18 . . . . . . . 8 (𝜑𝑊 ∈ Toset)
30 tospos 18309 . . . . . . . 8 (𝑊 ∈ Toset → 𝑊 ∈ Poset)
3129, 30syl 17 . . . . . . 7 (𝜑𝑊 ∈ Poset)
327, 18grpidcl 18778 . . . . . . . 8 (𝑊 ∈ Grp → 0𝐵)
332, 3, 323syl 18 . . . . . . 7 (𝜑0𝐵)
34 archirng.l . . . . . . . 8 = (le‘𝑊)
357, 34posref 18207 . . . . . . 7 ((𝑊 ∈ Poset ∧ 0𝐵) → 0 0 )
3631, 33, 35syl2anc 584 . . . . . 6 (𝜑0 0 )
3736adantr 481 . . . . 5 ((𝜑𝑌 = 0 ) → 0 0 )
38 1m1e0 12225 . . . . . . . . . 10 (1 − 1) = 0
3938negeqi 11394 . . . . . . . . 9 -(1 − 1) = -0
40 ax-1cn 11109 . . . . . . . . . 10 1 ∈ ℂ
4140, 40negsubdii 11486 . . . . . . . . 9 -(1 − 1) = (-1 + 1)
42 neg0 11447 . . . . . . . . 9 -0 = 0
4339, 41, 423eqtr3i 2772 . . . . . . . 8 (-1 + 1) = 0
4443oveq1i 7367 . . . . . . 7 ((-1 + 1) · 𝑋) = (0 · 𝑋)
457, 18, 8mulg0 18879 . . . . . . . 8 (𝑋𝐵 → (0 · 𝑋) = 0 )
466, 45syl 17 . . . . . . 7 (𝜑 → (0 · 𝑋) = 0 )
4744, 46eqtrid 2788 . . . . . 6 (𝜑 → ((-1 + 1) · 𝑋) = 0 )
4847adantr 481 . . . . 5 ((𝜑𝑌 = 0 ) → ((-1 + 1) · 𝑋) = 0 )
4937, 24, 483brtr4d 5137 . . . 4 ((𝜑𝑌 = 0 ) → 𝑌 ((-1 + 1) · 𝑋))
5025, 49jca 512 . . 3 ((𝜑𝑌 = 0 ) → ((-1 · 𝑋) < 𝑌𝑌 ((-1 + 1) · 𝑋)))
51 oveq1 7364 . . . . . 6 (𝑛 = -1 → (𝑛 · 𝑋) = (-1 · 𝑋))
5251breq1d 5115 . . . . 5 (𝑛 = -1 → ((𝑛 · 𝑋) < 𝑌 ↔ (-1 · 𝑋) < 𝑌))
53 oveq1 7364 . . . . . . 7 (𝑛 = -1 → (𝑛 + 1) = (-1 + 1))
5453oveq1d 7372 . . . . . 6 (𝑛 = -1 → ((𝑛 + 1) · 𝑋) = ((-1 + 1) · 𝑋))
5554breq2d 5117 . . . . 5 (𝑛 = -1 → (𝑌 ((𝑛 + 1) · 𝑋) ↔ 𝑌 ((-1 + 1) · 𝑋)))
5652, 55anbi12d 631 . . . 4 (𝑛 = -1 → (((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)) ↔ ((-1 · 𝑋) < 𝑌𝑌 ((-1 + 1) · 𝑋))))
5756rspcev 3581 . . 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 12525 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℤ)
6160ad2antrr 724 . . . . . . 7 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → 𝑚 ∈ ℤ)
6261znegcld 12609 . . . . . 6 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → -𝑚 ∈ ℤ)
63 2z 12535 . . . . . . 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 12612 . . . . 5 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → (-𝑚 − 2) ∈ ℤ)
66 nn0cn 12423 . . . . . . . . . . 11 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
6766adantl 482 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℂ)
68 2cnd 12231 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 2 ∈ ℂ)
6967, 68negdi2d 11526 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -(𝑚 + 2) = (-𝑚 − 2))
7069oveq1d 7372 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) · 𝑋) = ((-𝑚 − 2) · 𝑋))
712ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ oGrp)
72 archirngz.1 . . . . . . . . . . . 12 (𝜑 → (oppg𝑊) ∈ oGrp)
7372ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (oppg𝑊) ∈ oGrp)
7471, 73jca 512 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp))
754ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ Grp)
7660peano2zd 12610 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 1) ∈ ℤ)
776ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑋𝐵)
787, 8mulgcl 18893 . . . . . . . . . . 11 ((𝑊 ∈ Grp ∧ (𝑚 + 1) ∈ ℤ ∧ 𝑋𝐵) → ((𝑚 + 1) · 𝑋) ∈ 𝐵)
7975, 76, 77, 78syl3anc 1371 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) · 𝑋) ∈ 𝐵)
8063a1i 11 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 2 ∈ ℤ)
8160, 80zaddcld 12611 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 2) ∈ ℤ)
827, 8mulgcl 18893 . . . . . . . . . . 11 ((𝑊 ∈ Grp ∧ (𝑚 + 2) ∈ ℤ ∧ 𝑋𝐵) → ((𝑚 + 2) · 𝑋) ∈ 𝐵)
8375, 81, 77, 82syl3anc 1371 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 2) · 𝑋) ∈ 𝐵)
8475, 32syl 17 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 0𝐵)
8516ad2antrr 724 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 0 < 𝑋)
86 eqid 2736 . . . . . . . . . . . . 13 (+g𝑊) = (+g𝑊)
877, 17, 86ogrpaddlt 31925 . . . . . . . . . . . 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 1383 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ( 0 (+g𝑊)((𝑚 + 1) · 𝑋)) < (𝑋(+g𝑊)((𝑚 + 1) · 𝑋)))
897, 86, 18grplid 18780 . . . . . . . . . . . 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 11150 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ0 → 1 ∈ ℂ)
9266, 91, 91addassd 11177 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) = (𝑚 + (1 + 1)))
93 1p1e2 12278 . . . . . . . . . . . . . . . . 17 (1 + 1) = 2
9493oveq2i 7368 . . . . . . . . . . . . . . . 16 (𝑚 + (1 + 1)) = (𝑚 + 2)
9592, 94eqtrdi 2792 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) = (𝑚 + 2))
9666, 91addcld 11174 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℂ)
9796, 91addcomd 11357 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) = (1 + (𝑚 + 1)))
9895, 97eqtr3d 2778 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → (𝑚 + 2) = (1 + (𝑚 + 1)))
9998oveq1d 7372 . . . . . . . . . . . . 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 12534 . . . . . . . . . . . . 13 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 1 ∈ ℤ)
1027, 8, 86mulgdir 18908 . . . . . . . . . . . . 13 ((𝑊 ∈ Grp ∧ (1 ∈ ℤ ∧ (𝑚 + 1) ∈ ℤ ∧ 𝑋𝐵)) → ((1 + (𝑚 + 1)) · 𝑋) = ((1 · 𝑋)(+g𝑊)((𝑚 + 1) · 𝑋)))
10375, 101, 76, 77, 102syl13anc 1372 . . . . . . . . . . . 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 7372 . . . . . . . . . . . 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 5136 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) · 𝑋) < ((𝑚 + 2) · 𝑋))
1087, 17, 9ogrpinvlt 31931 . . . . . . . . . . 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 1373 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((invg𝑊)‘((𝑚 + 2) · 𝑋)) < ((invg𝑊)‘((𝑚 + 1) · 𝑋)))
1117, 8, 9mulgneg 18894 . . . . . . . . . 10 ((𝑊 ∈ Grp ∧ (𝑚 + 2) ∈ ℤ ∧ 𝑋𝐵) → (-(𝑚 + 2) · 𝑋) = ((invg𝑊)‘((𝑚 + 2) · 𝑋)))
11275, 81, 77, 111syl3anc 1371 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) · 𝑋) = ((invg𝑊)‘((𝑚 + 2) · 𝑋)))
1137, 8, 9mulgneg 18894 . . . . . . . . . 10 ((𝑊 ∈ Grp ∧ (𝑚 + 1) ∈ ℤ ∧ 𝑋𝐵) → (-(𝑚 + 1) · 𝑋) = ((invg𝑊)‘((𝑚 + 1) · 𝑋)))
11475, 76, 77, 113syl3anc 1371 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 1) · 𝑋) = ((invg𝑊)‘((𝑚 + 1) · 𝑋)))
115110, 112, 1143brtr4d 5137 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) · 𝑋) < (-(𝑚 + 1) · 𝑋))
11670, 115eqbrtrrd 5129 . . . . . . 7 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-𝑚 − 2) · 𝑋) < (-(𝑚 + 1) · 𝑋))
117116ad2antrr 724 . . . . . 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 724 . . . . . . 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 730 . . . . . . . . 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 18798 . . . . . . . . . . . 12 ((𝑊 ∈ Grp ∧ 𝑌𝐵) → ((invg𝑊)‘𝑌) ∈ 𝐵)
1224, 120, 121syl2anc 584 . . . . . . . . . . 11 (𝜑 → ((invg𝑊)‘𝑌) ∈ 𝐵)
123122ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((invg𝑊)‘𝑌) ∈ 𝐵)
124123ad2antrr 724 . . . . . . . . 9 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌)) → ((invg𝑊)‘𝑌) ∈ 𝐵)
12579ad2antrr 724 . . . . . . . . 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 776 . . . . . . . . 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 18208 . . . . . . . . . 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 1378 . . . . . . . 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 6846 . . . . . . 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 18814 . . . . . . . . 9 ((𝑊 ∈ Grp ∧ 𝑌𝐵) → ((invg𝑊)‘((invg𝑊)‘𝑌)) = 𝑌)
1334, 120, 132syl2anc 584 . . . . . . . 8 (𝜑 → ((invg𝑊)‘((invg𝑊)‘𝑌)) = 𝑌)
134133ad4antr 730 . . . . . . 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 5133 . . . . 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 11150 . . . . . . . . . . . . 13 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 1 ∈ ℂ)
13867, 68, 137addsubassd 11532 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 2) − 1) = (𝑚 + (2 − 1)))
139 2m1e1 12279 . . . . . . . . . . . . 13 (2 − 1) = 1
140139oveq2i 7368 . . . . . . . . . . . 12 (𝑚 + (2 − 1)) = (𝑚 + 1)
141138, 140eqtr2di 2793 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 1) = ((𝑚 + 2) − 1))
142141negeqd 11395 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -(𝑚 + 1) = -((𝑚 + 2) − 1))
14367, 68addcld 11174 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 2) ∈ ℂ)
144143, 137negsubdid 11527 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -((𝑚 + 2) − 1) = (-(𝑚 + 2) + 1))
14569oveq1d 7372 . . . . . . . . . 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 7372 . . . . . . . 8 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((-𝑚 − 2) + 1) · 𝑋) = (-(𝑚 + 1) · 𝑋))
14829ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ Toset)
149148, 30syl 17 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ Poset)
15060znegcld 12609 . . . . . . . . . . . 12 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -𝑚 ∈ ℤ)
151150, 80zsubcld 12612 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-𝑚 − 2) ∈ ℤ)
152151peano2zd 12610 . . . . . . . . . 10 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-𝑚 − 2) + 1) ∈ ℤ)
1537, 8mulgcl 18893 . . . . . . . . . 10 ((𝑊 ∈ Grp ∧ ((-𝑚 − 2) + 1) ∈ ℤ ∧ 𝑋𝐵) → (((-𝑚 − 2) + 1) · 𝑋) ∈ 𝐵)
15475, 152, 77, 153syl3anc 1371 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((-𝑚 − 2) + 1) · 𝑋) ∈ 𝐵)
1557, 34posref 18207 . . . . . . . . 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 5129 . . . . . . 7 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 1) · 𝑋) (((-𝑚 − 2) + 1) · 𝑋))
158157ad2antrr 724 . . . . . 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 5127 . . . . 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 7364 . . . . . . . 8 (𝑛 = (-𝑚 − 2) → (𝑛 · 𝑋) = ((-𝑚 − 2) · 𝑋))
161160breq1d 5115 . . . . . . 7 (𝑛 = (-𝑚 − 2) → ((𝑛 · 𝑋) < 𝑌 ↔ ((-𝑚 − 2) · 𝑋) < 𝑌))
162 oveq1 7364 . . . . . . . . 9 (𝑛 = (-𝑚 − 2) → (𝑛 + 1) = ((-𝑚 − 2) + 1))
163162oveq1d 7372 . . . . . . . 8 (𝑛 = (-𝑚 − 2) → ((𝑛 + 1) · 𝑋) = (((-𝑚 − 2) + 1) · 𝑋))
164163breq2d 5117 . . . . . . 7 (𝑛 = (-𝑚 − 2) → (𝑌 ((𝑛 + 1) · 𝑋) ↔ 𝑌 (((-𝑚 − 2) + 1) · 𝑋)))
165161, 164anbi12d 631 . . . . . 6 (𝑛 = (-𝑚 − 2) → (((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)) ↔ (((-𝑚 − 2) · 𝑋) < 𝑌𝑌 (((-𝑚 − 2) + 1) · 𝑋))))
166165rspcev 3581 . . . . 5 (((-𝑚 − 2) ∈ ℤ ∧ (((-𝑚 − 2) · 𝑋) < 𝑌𝑌 (((-𝑚 − 2) + 1) · 𝑋))) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
16765, 136, 159, 166syl12anc 835 . . . 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 724 . . . . . 6 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → (𝑚 + 1) ∈ ℤ)
169168znegcld 12609 . . . . 5 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → -(𝑚 + 1) ∈ ℤ)
1702ad2antrr 724 . . . . . . . . 9 (((𝜑𝑌 < 0 ) ∧ (𝑚 ∈ ℕ0 ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋)) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋))) → 𝑊 ∈ oGrp)
17172ad2antrr 724 . . . . . . . . 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 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 724 . . . . . . 7 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((invg𝑊)‘𝑌) ∈ 𝐵)
17579ad2antrr 724 . . . . . . 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 31931 . . . . . . . 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 1373 . . . . . 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 724 . . . . . . 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 730 . . . . . 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 5136 . . . . 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 781 . . . . . 6 (((((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋))) ∧ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → 𝜑)
1857, 8mulgcl 18893 . . . . . . . . . . . 12 ((𝑊 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝑋𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
18675, 60, 77, 185syl3anc 1371 . . . . . . . . . . 11 (((𝜑𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 · 𝑋) ∈ 𝐵)
1877, 17, 9ogrpinvlt 31931 . . . . . . . . . . 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 1371 . . . . . . . . . 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 715 . . . . . . . 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 11458 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝑚 + 1) = (-𝑚 + -1))
19366, 40, 192sylancl 586 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → -(𝑚 + 1) = (-𝑚 + -1))
194193oveq1d 7372 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0 → (-(𝑚 + 1) + 1) = ((-𝑚 + -1) + 1))
19566negcld 11499 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → -𝑚 ∈ ℂ)
19691negcld 11499 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → -1 ∈ ℂ)
197195, 196, 91addassd 11177 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → ((-𝑚 + -1) + 1) = (-𝑚 + (-1 + 1)))
19843oveq2i 7368 . . . . . . . . . . . . . . 15 (-𝑚 + (-1 + 1)) = (-𝑚 + 0)
199198a1i 11 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → (-𝑚 + (-1 + 1)) = (-𝑚 + 0))
200195addid1d 11355 . . . . . . . . . . . . . 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 7372 . . . . . . . . . . 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 18894 . . . . . . . . . . 11 ((𝑊 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝑋𝐵) → (-𝑚 · 𝑋) = ((invg𝑊)‘(𝑚 · 𝑋)))
20675, 60, 77, 205syl3anc 1371 . . . . . . . . . 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 724 . . . . . . . 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 5136 . . . . . 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 7392 . . . . . . 7 (𝜑 → ((-(𝑚 + 1) + 1) · 𝑋) ∈ V)
21234, 17pltle 18222 . . . . . . 7 ((𝑊 ∈ oGrp ∧ 𝑌𝐵 ∧ ((-(𝑚 + 1) + 1) · 𝑋) ∈ V) → (𝑌 < ((-(𝑚 + 1) + 1) · 𝑋) → 𝑌 ((-(𝑚 + 1) + 1) · 𝑋)))
2132, 120, 211, 212syl3anc 1371 . . . . . 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 7364 . . . . . . . 8 (𝑛 = -(𝑚 + 1) → (𝑛 · 𝑋) = (-(𝑚 + 1) · 𝑋))
216215breq1d 5115 . . . . . . 7 (𝑛 = -(𝑚 + 1) → ((𝑛 · 𝑋) < 𝑌 ↔ (-(𝑚 + 1) · 𝑋) < 𝑌))
217 oveq1 7364 . . . . . . . . 9 (𝑛 = -(𝑚 + 1) → (𝑛 + 1) = (-(𝑚 + 1) + 1))
218217oveq1d 7372 . . . . . . . 8 (𝑛 = -(𝑚 + 1) → ((𝑛 + 1) · 𝑋) = ((-(𝑚 + 1) + 1) · 𝑋))
219218breq2d 5117 . . . . . . 7 (𝑛 = -(𝑚 + 1) → (𝑌 ((𝑛 + 1) · 𝑋) ↔ 𝑌 ((-(𝑚 + 1) + 1) · 𝑋)))
220216, 219anbi12d 631 . . . . . 6 (𝑛 = -(𝑚 + 1) → (((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)) ↔ ((-(𝑚 + 1) · 𝑋) < 𝑌𝑌 ((-(𝑚 + 1) + 1) · 𝑋))))
221220rspcev 3581 . . . . 5 ((-(𝑚 + 1) ∈ ℤ ∧ ((-(𝑚 + 1) · 𝑋) < 𝑌𝑌 ((-(𝑚 + 1) + 1) · 𝑋))) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
222169, 183, 214, 221syl12anc 835 . . . 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 31829 . . . . . 6 ((𝑊 ∈ Toset ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵 ∧ ((invg𝑊)‘𝑌) ∈ 𝐵) → (((𝑚 + 1) · 𝑋) ((invg𝑊)‘𝑌) ∨ ((invg𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)))
224148, 79, 123, 223syl3anc 1371 . . . . 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 957 . . 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 5115 . . . . . 6 (𝜑 → (((invg𝑊)‘((invg𝑊)‘𝑌)) < 0𝑌 < 0 ))
234233biimpar 478 . . . . 5 ((𝜑𝑌 < 0 ) → ((invg𝑊)‘((invg𝑊)‘𝑌)) < 0 )
2357, 17, 9, 18ogrpinv0lt 31930 . . . . . . 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 32024 . . 3 ((𝜑𝑌 < 0 ) → ∃𝑚 ∈ ℕ0 ((𝑚 · 𝑋) < ((invg𝑊)‘𝑌) ∧ ((invg𝑊)‘𝑌) ((𝑚 + 1) · 𝑋)))
240226, 239r19.29a 3159 . 2 ((𝜑𝑌 < 0 ) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
241 nn0ssz 12522 . . 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 32024 . . 3 ((𝜑0 < 𝑌) → ∃𝑛 ∈ ℕ0 ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
249 ssrexv 4011 . . 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 31830 . . 3 ((𝑊 ∈ Toset ∧ 𝑌𝐵0𝐵) → (𝑌 = 0𝑌 < 00 < 𝑌))
25229, 120, 33, 251syl3anc 1371 . 2 (𝜑 → (𝑌 = 0𝑌 < 00 < 𝑌))
25358, 240, 250, 252mpjao3dan 1431 1 (𝜑 → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845  w3o 1086  w3a 1087   = wceq 1541  wcel 2106  wrex 3073  Vcvv 3445  wss 3910   class class class wbr 5105  cfv 6496  (class class class)co 7357  cc 11049  0cc0 11051  1c1 11052   + caddc 11054  cmin 11385  -cneg 11386  2c2 12208  0cn0 12413  cz 12499  Basecbs 17083  +gcplusg 17133  lecple 17140  0gc0g 17321  Posetcpo 18196  ltcplt 18197  Tosetctos 18305  Grpcgrp 18748  invgcminusg 18749  .gcmg 18872  oppgcoppg 19123  oMndcomnd 31905  oGrpcogrp 31906  Archicarchi 32013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-seq 13907  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-ple 17153  df-0g 17323  df-proset 18184  df-poset 18202  df-plt 18219  df-toset 18306  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-mulg 18873  df-oppg 19124  df-omnd 31907  df-ogrp 31908  df-inftm 32014  df-archi 32015
This theorem is referenced by:  archiabllem2c  32031
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