Theorem archirngz 29074

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 = (0g‘𝑊)
archirng.i	⊢ < = (lt‘𝑊)
archirng.l	⊢ ≤ = (le‘𝑊)
archirng.x	⊢ · = (.g‘𝑊)
archirng.1	⊢ (𝜑 → 𝑊 ∈ oGrp)
archirng.2	⊢ (𝜑 → 𝑊 ∈ Archi)
archirng.3	⊢ (𝜑 → 𝑋 ∈ 𝐵)
archirng.4	⊢ (𝜑 → 𝑌 ∈ 𝐵)
archirng.5	⊢ (𝜑 → 0 < 𝑋)
archirngz.1	⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp)

Assertion

Ref	Expression
archirngz	⊢ (𝜑 → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((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.

Step	Hyp	Ref	Expression
1		neg1z 11290	. . 3 ⊢ -1 ∈ ℤ
2		archirng.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ oGrp)
3		ogrpgrp 29034	. . . . . . . . . 10 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ Grp)
4	2, 3	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ Grp)
5		1zzd 11285	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
6		archirng.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		archirng.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑊)
8		archirng.x	. . . . . . . . . 10 ⊢ · = (.g‘𝑊)
9		eqid 2610	. . . . . . . . . 10 ⊢ (invg‘𝑊) = (invg‘𝑊)
10	7, 8, 9	mulgneg 17383	. . . . . . . . 9 ⊢ ((𝑊 ∈ Grp ∧ 1 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (-1 · 𝑋) = ((invg‘𝑊)‘(1 · 𝑋)))
11	4, 5, 6, 10	syl3anc 1318	. . . . . . . 8 ⊢ (𝜑 → (-1 · 𝑋) = ((invg‘𝑊)‘(1 · 𝑋)))
12	7, 8	mulg1 17371	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = 𝑋)
13	6, 12	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (1 · 𝑋) = 𝑋)
14	13	fveq2d 6107	. . . . . . . 8 ⊢ (𝜑 → ((invg‘𝑊)‘(1 · 𝑋)) = ((invg‘𝑊)‘𝑋))
15	11, 14	eqtrd 2644	. . . . . . 7 ⊢ (𝜑 → (-1 · 𝑋) = ((invg‘𝑊)‘𝑋))
16		archirng.5	. . . . . . . 8 ⊢ (𝜑 → 0 < 𝑋)
17		archirng.i	. . . . . . . . . 10 ⊢ < = (lt‘𝑊)
18		archirng.0	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑊)
19	7, 17, 9, 18	ogrpinv0lt 29054	. . . . . . . . 9 ⊢ ((𝑊 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ ((invg‘𝑊)‘𝑋) < 0 ))
20	19	biimpa 500	. . . . . . . 8 ⊢ (((𝑊 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → ((invg‘𝑊)‘𝑋) < 0 )
21	2, 6, 16, 20	syl21anc 1317	. . . . . . 7 ⊢ (𝜑 → ((invg‘𝑊)‘𝑋) < 0 )
22	15, 21	eqbrtrd 4605	. . . . . 6 ⊢ (𝜑 → (-1 · 𝑋) < 0 )
23	22	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (-1 · 𝑋) < 0 )
24		simpr 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝑌 = 0 )
25	23, 24	breqtrrd 4611	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (-1 · 𝑋) < 𝑌)
26		isogrp 29033	. . . . . . . . . 10 ⊢ (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
27	26	simprbi 479	. . . . . . . . 9 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
28		omndtos 29036	. . . . . . . . 9 ⊢ (𝑊 ∈ oMnd → 𝑊 ∈ Toset)
29	2, 27, 28	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ Toset)
30		tospos 28989	. . . . . . . 8 ⊢ (𝑊 ∈ Toset → 𝑊 ∈ Poset)
31	29, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Poset)
32	7, 18	grpidcl 17273	. . . . . . . 8 ⊢ (𝑊 ∈ Grp → 0 ∈ 𝐵)
33	2, 3, 32	3syl 18	. . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝐵)
34		archirng.l	. . . . . . . 8 ⊢ ≤ = (le‘𝑊)
35	7, 34	posref 16774	. . . . . . 7 ⊢ ((𝑊 ∈ Poset ∧ 0 ∈ 𝐵) → 0 ≤ 0 )
36	31, 33, 35	syl2anc 691	. . . . . 6 ⊢ (𝜑 → 0 ≤ 0 )
37	36	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 0 ≤ 0 )
38		1m1e0 10966	. . . . . . . . . 10 ⊢ (1 − 1) = 0
39	38	negeqi 10153	. . . . . . . . 9 ⊢ -(1 − 1) = -0
40		ax-1cn 9873	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
41	40, 40	negsubdii 10245	. . . . . . . . 9 ⊢ -(1 − 1) = (-1 + 1)
42		neg0 10206	. . . . . . . . 9 ⊢ -0 = 0
43	39, 41, 42	3eqtr3i 2640	. . . . . . . 8 ⊢ (-1 + 1) = 0
44	43	oveq1i 6559	. . . . . . 7 ⊢ ((-1 + 1) · 𝑋) = (0 · 𝑋)
45	7, 18, 8	mulg0 17369	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = 0 )
46	6, 45	syl 17	. . . . . . 7 ⊢ (𝜑 → (0 · 𝑋) = 0 )
47	44, 46	syl5eq 2656	. . . . . 6 ⊢ (𝜑 → ((-1 + 1) · 𝑋) = 0 )
48	47	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → ((-1 + 1) · 𝑋) = 0 )
49	37, 24, 48	3brtr4d 4615	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝑌 ≤ ((-1 + 1) · 𝑋))
50	25, 49	jca 553	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → ((-1 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((-1 + 1) · 𝑋)))
51		oveq1 6556	. . . . . 6 ⊢ (𝑛 = -1 → (𝑛 · 𝑋) = (-1 · 𝑋))
52	51	breq1d 4593	. . . . 5 ⊢ (𝑛 = -1 → ((𝑛 · 𝑋) < 𝑌 ↔ (-1 · 𝑋) < 𝑌))
53		oveq1 6556	. . . . . . 7 ⊢ (𝑛 = -1 → (𝑛 + 1) = (-1 + 1))
54	53	oveq1d 6564	. . . . . 6 ⊢ (𝑛 = -1 → ((𝑛 + 1) · 𝑋) = ((-1 + 1) · 𝑋))
55	54	breq2d 4595	. . . . 5 ⊢ (𝑛 = -1 → (𝑌 ≤ ((𝑛 + 1) · 𝑋) ↔ 𝑌 ≤ ((-1 + 1) · 𝑋)))
56	52, 55	anbi12d 743	. . . 4 ⊢ (𝑛 = -1 → (((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)) ↔ ((-1 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((-1 + 1) · 𝑋))))
57	56	rspcev 3282	. . 3 ⊢ ((-1 ∈ ℤ ∧ ((-1 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((-1 + 1) · 𝑋))) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))
58	1, 50, 57	sylancr 694	. 2 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))
59		simpr 476	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
60	59	nn0zd 11356	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℤ)
61	60	ad2antrr 758	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌)) → 𝑚 ∈ ℤ)
62	61	znegcld 11360	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌)) → -𝑚 ∈ ℤ)
63		2z 11286	. . . . . . 7 ⊢ 2 ∈ ℤ
64	63	a1i 11	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌)) → 2 ∈ ℤ)
65	62, 64	zsubcld 11363	. . . . 5 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌)) → (-𝑚 − 2) ∈ ℤ)
66		nn0cn 11179	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℂ)
67	66	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℂ)
68		2cnd 10970	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 2 ∈ ℂ)
69	67, 68	negdi2d 10285	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -(𝑚 + 2) = (-𝑚 − 2))
70	69	oveq1d 6564	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) · 𝑋) = ((-𝑚 − 2) · 𝑋))
71	2	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ oGrp)
72		archirngz.1	. . . . . . . . . . . 12 ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp)
73	72	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (oppg‘𝑊) ∈ oGrp)
74	71, 73	jca 553	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp))
75	4	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ Grp)
76	60	peano2zd 11361	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 1) ∈ ℤ)
77	6	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑋 ∈ 𝐵)
78	7, 8	mulgcl 17382	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Grp ∧ (𝑚 + 1) ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((𝑚 + 1) · 𝑋) ∈ 𝐵)
79	75, 76, 77, 78	syl3anc 1318	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) · 𝑋) ∈ 𝐵)
80	63	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 2 ∈ ℤ)
81	60, 80	zaddcld 11362	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 2) ∈ ℤ)
82	7, 8	mulgcl 17382	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Grp ∧ (𝑚 + 2) ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((𝑚 + 2) · 𝑋) ∈ 𝐵)
83	75, 81, 77, 82	syl3anc 1318	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 2) · 𝑋) ∈ 𝐵)
84	75, 32	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 0 ∈ 𝐵)
85	16	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 0 < 𝑋)
86		eqid 2610	. . . . . . . . . . . . 13 ⊢ (+g‘𝑊) = (+g‘𝑊)
87	7, 17, 86	ogrpaddlt 29049	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ oGrp ∧ ( 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵) ∧ 0 < 𝑋) → ( 0 (+g‘𝑊)((𝑚 + 1) · 𝑋)) < (𝑋(+g‘𝑊)((𝑚 + 1) · 𝑋)))
88	71, 84, 77, 79, 85, 87	syl131anc 1331	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ( 0 (+g‘𝑊)((𝑚 + 1) · 𝑋)) < (𝑋(+g‘𝑊)((𝑚 + 1) · 𝑋)))
89	7, 86, 18	grplid 17275	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Grp ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵) → ( 0 (+g‘𝑊)((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) · 𝑋))
90	75, 79, 89	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ( 0 (+g‘𝑊)((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) · 𝑋))
91		1cnd 9935	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ0 → 1 ∈ ℂ)
92	66, 91, 91	addassd 9941	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) = (𝑚 + (1 + 1)))
93		1p1e2 11011	. . . . . . . . . . . . . . . . 17 ⊢ (1 + 1) = 2
94	93	oveq2i 6560	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 + (1 + 1)) = (𝑚 + 2)
95	92, 94	syl6eq 2660	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) = (𝑚 + 2))
96	66, 91	addcld 9938	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℂ)
97	96, 91	addcomd 10117	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) = (1 + (𝑚 + 1)))
98	95, 97	eqtr3d 2646	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 2) = (1 + (𝑚 + 1)))
99	98	oveq1d 6564	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ0 → ((𝑚 + 2) · 𝑋) = ((1 + (𝑚 + 1)) · 𝑋))
100	99	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 2) · 𝑋) = ((1 + (𝑚 + 1)) · 𝑋))
101		1zzd 11285	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 1 ∈ ℤ)
102	7, 8, 86	mulgdir 17396	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Grp ∧ (1 ∈ ℤ ∧ (𝑚 + 1) ∈ ℤ ∧ 𝑋 ∈ 𝐵)) → ((1 + (𝑚 + 1)) · 𝑋) = ((1 · 𝑋)(+g‘𝑊)((𝑚 + 1) · 𝑋)))
103	75, 101, 76, 77, 102	syl13anc 1320	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((1 + (𝑚 + 1)) · 𝑋) = ((1 · 𝑋)(+g‘𝑊)((𝑚 + 1) · 𝑋)))
104	77, 12	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (1 · 𝑋) = 𝑋)
105	104	oveq1d 6564	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((1 · 𝑋)(+g‘𝑊)((𝑚 + 1) · 𝑋)) = (𝑋(+g‘𝑊)((𝑚 + 1) · 𝑋)))
106	100, 103, 105	3eqtrrd 2649	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑋(+g‘𝑊)((𝑚 + 1) · 𝑋)) = ((𝑚 + 2) · 𝑋))
107	88, 90, 106	3brtr3d 4614	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) · 𝑋) < ((𝑚 + 2) · 𝑋))
108	7, 17, 9	ogrpinvlt 29055	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp) ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵 ∧ ((𝑚 + 2) · 𝑋) ∈ 𝐵) → (((𝑚 + 1) · 𝑋) < ((𝑚 + 2) · 𝑋) ↔ ((invg‘𝑊)‘((𝑚 + 2) · 𝑋)) < ((invg‘𝑊)‘((𝑚 + 1) · 𝑋))))
109	108	biimpa 500	. . . . . . . . . 10 ⊢ ((((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp) ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵 ∧ ((𝑚 + 2) · 𝑋) ∈ 𝐵) ∧ ((𝑚 + 1) · 𝑋) < ((𝑚 + 2) · 𝑋)) → ((invg‘𝑊)‘((𝑚 + 2) · 𝑋)) < ((invg‘𝑊)‘((𝑚 + 1) · 𝑋)))
110	74, 79, 83, 107, 109	syl31anc 1321	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((invg‘𝑊)‘((𝑚 + 2) · 𝑋)) < ((invg‘𝑊)‘((𝑚 + 1) · 𝑋)))
111	7, 8, 9	mulgneg 17383	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Grp ∧ (𝑚 + 2) ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (-(𝑚 + 2) · 𝑋) = ((invg‘𝑊)‘((𝑚 + 2) · 𝑋)))
112	75, 81, 77, 111	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) · 𝑋) = ((invg‘𝑊)‘((𝑚 + 2) · 𝑋)))
113	7, 8, 9	mulgneg 17383	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Grp ∧ (𝑚 + 1) ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (-(𝑚 + 1) · 𝑋) = ((invg‘𝑊)‘((𝑚 + 1) · 𝑋)))
114	75, 76, 77, 113	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 1) · 𝑋) = ((invg‘𝑊)‘((𝑚 + 1) · 𝑋)))
115	110, 112, 114	3brtr4d 4615	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) · 𝑋) < (-(𝑚 + 1) · 𝑋))
116	70, 115	eqbrtrrd 4607	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-𝑚 − 2) · 𝑋) < (-(𝑚 + 1) · 𝑋))
117	116	ad2antrr 758	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌)) → ((-𝑚 − 2) · 𝑋) < (-(𝑚 + 1) · 𝑋))
118	114	ad2antrr 758	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌)) → (-(𝑚 + 1) · 𝑋) = ((invg‘𝑊)‘((𝑚 + 1) · 𝑋)))
119	31	ad4antr 764	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌)) → 𝑊 ∈ Poset)
120		archirng.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
121	7, 9	grpinvcl 17290	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((invg‘𝑊)‘𝑌) ∈ 𝐵)
122	4, 120, 121	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → ((invg‘𝑊)‘𝑌) ∈ 𝐵)
123	122	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((invg‘𝑊)‘𝑌) ∈ 𝐵)
124	123	ad2antrr 758	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌)) → ((invg‘𝑊)‘𝑌) ∈ 𝐵)
125	79	ad2antrr 758	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌)) → ((𝑚 + 1) · 𝑋) ∈ 𝐵)
126		simplrr 797	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌)) → ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))
127		simpr 476	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌)) → ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌))
128	7, 34	posasymb 16775	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Poset ∧ ((invg‘𝑊)‘𝑌) ∈ 𝐵 ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵) → ((((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌)) ↔ ((invg‘𝑊)‘𝑌) = ((𝑚 + 1) · 𝑋)))
129	128	biimpa 500	. . . . . . . . 9 ⊢ (((𝑊 ∈ Poset ∧ ((invg‘𝑊)‘𝑌) ∈ 𝐵 ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵) ∧ (((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌))) → ((invg‘𝑊)‘𝑌) = ((𝑚 + 1) · 𝑋))
130	119, 124, 125, 126, 127, 129	syl32anc 1326	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌)) → ((invg‘𝑊)‘𝑌) = ((𝑚 + 1) · 𝑋))
131	130	fveq2d 6107	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌)) → ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) = ((invg‘𝑊)‘((𝑚 + 1) · 𝑋)))
132	7, 9	grpinvinv 17305	. . . . . . . . 9 ⊢ ((𝑊 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) = 𝑌)
133	4, 120, 132	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) = 𝑌)
134	133	ad4antr 764	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌)) → ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) = 𝑌)
135	118, 131, 134	3eqtr2rd 2651	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌)) → 𝑌 = (-(𝑚 + 1) · 𝑋))
136	117, 135	breqtrrd 4611	. . . . 5 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌)) → ((-𝑚 − 2) · 𝑋) < 𝑌)
137		1cnd 9935	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 1 ∈ ℂ)
138	67, 68, 137	addsubassd 10291	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 2) − 1) = (𝑚 + (2 − 1)))
139		2m1e1 11012	. . . . . . . . . . . . 13 ⊢ (2 − 1) = 1
140	139	oveq2i 6560	. . . . . . . . . . . 12 ⊢ (𝑚 + (2 − 1)) = (𝑚 + 1)
141	138, 140	syl6req 2661	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 1) = ((𝑚 + 2) − 1))
142	141	negeqd 10154	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -(𝑚 + 1) = -((𝑚 + 2) − 1))
143	67, 68	addcld 9938	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 2) ∈ ℂ)
144	143, 137	negsubdid 10286	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -((𝑚 + 2) − 1) = (-(𝑚 + 2) + 1))
145	69	oveq1d 6564	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) + 1) = ((-𝑚 − 2) + 1))
146	142, 144, 145	3eqtrrd 2649	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-𝑚 − 2) + 1) = -(𝑚 + 1))
147	146	oveq1d 6564	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((-𝑚 − 2) + 1) · 𝑋) = (-(𝑚 + 1) · 𝑋))
148	29	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ Toset)
149	148, 30	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ Poset)
150	60	znegcld 11360	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -𝑚 ∈ ℤ)
151	150, 80	zsubcld 11363	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-𝑚 − 2) ∈ ℤ)
152	151	peano2zd 11361	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-𝑚 − 2) + 1) ∈ ℤ)
153	7, 8	mulgcl 17382	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Grp ∧ ((-𝑚 − 2) + 1) ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (((-𝑚 − 2) + 1) · 𝑋) ∈ 𝐵)
154	75, 152, 77, 153	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((-𝑚 − 2) + 1) · 𝑋) ∈ 𝐵)
155	7, 34	posref 16774	. . . . . . . . 9 ⊢ ((𝑊 ∈ Poset ∧ (((-𝑚 − 2) + 1) · 𝑋) ∈ 𝐵) → (((-𝑚 − 2) + 1) · 𝑋) ≤ (((-𝑚 − 2) + 1) · 𝑋))
156	149, 154, 155	syl2anc 691	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((-𝑚 − 2) + 1) · 𝑋) ≤ (((-𝑚 − 2) + 1) · 𝑋))
157	147, 156	eqbrtrrd 4607	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 1) · 𝑋) ≤ (((-𝑚 − 2) + 1) · 𝑋))
158	157	ad2antrr 758	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌)) → (-(𝑚 + 1) · 𝑋) ≤ (((-𝑚 − 2) + 1) · 𝑋))
159	135, 158	eqbrtrd 4605	. . . . 5 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌)) → 𝑌 ≤ (((-𝑚 − 2) + 1) · 𝑋))
160		oveq1 6556	. . . . . . . 8 ⊢ (𝑛 = (-𝑚 − 2) → (𝑛 · 𝑋) = ((-𝑚 − 2) · 𝑋))
161	160	breq1d 4593	. . . . . . 7 ⊢ (𝑛 = (-𝑚 − 2) → ((𝑛 · 𝑋) < 𝑌 ↔ ((-𝑚 − 2) · 𝑋) < 𝑌))
162		oveq1 6556	. . . . . . . . 9 ⊢ (𝑛 = (-𝑚 − 2) → (𝑛 + 1) = ((-𝑚 − 2) + 1))
163	162	oveq1d 6564	. . . . . . . 8 ⊢ (𝑛 = (-𝑚 − 2) → ((𝑛 + 1) · 𝑋) = (((-𝑚 − 2) + 1) · 𝑋))
164	163	breq2d 4595	. . . . . . 7 ⊢ (𝑛 = (-𝑚 − 2) → (𝑌 ≤ ((𝑛 + 1) · 𝑋) ↔ 𝑌 ≤ (((-𝑚 − 2) + 1) · 𝑋)))
165	161, 164	anbi12d 743	. . . . . 6 ⊢ (𝑛 = (-𝑚 − 2) → (((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)) ↔ (((-𝑚 − 2) · 𝑋) < 𝑌 ∧ 𝑌 ≤ (((-𝑚 − 2) + 1) · 𝑋))))
166	165	rspcev 3282	. . . . 5 ⊢ (((-𝑚 − 2) ∈ ℤ ∧ (((-𝑚 − 2) · 𝑋) < 𝑌 ∧ 𝑌 ≤ (((-𝑚 − 2) + 1) · 𝑋))) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))
167	65, 136, 159, 166	syl12anc 1316	. . . 4 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌)) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))
168	76	ad2antrr 758	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → (𝑚 + 1) ∈ ℤ)
169	168	znegcld 11360	. . . . 5 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → -(𝑚 + 1) ∈ ℤ)
170	2	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ (𝑚 ∈ ℕ0 ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋)) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋))) → 𝑊 ∈ oGrp)
171	72	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ (𝑚 ∈ ℕ0 ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋)) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋))) → (oppg‘𝑊) ∈ oGrp)
172	170, 171	jca 553	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ (𝑚 ∈ ℕ0 ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋)) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋))) → (𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp))
173	172	3anassrs 1282	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → (𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp))
174	123	ad2antrr 758	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((invg‘𝑊)‘𝑌) ∈ 𝐵)
175	79	ad2antrr 758	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((𝑚 + 1) · 𝑋) ∈ 𝐵)
176		simpr 476	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋))
177	7, 17, 9	ogrpinvlt 29055	. . . . . . . 8 ⊢ (((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp) ∧ ((invg‘𝑊)‘𝑌) ∈ 𝐵 ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵) → (((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋) ↔ ((invg‘𝑊)‘((𝑚 + 1) · 𝑋)) < ((invg‘𝑊)‘((invg‘𝑊)‘𝑌))))
178	177	biimpa 500	. . . . . . 7 ⊢ ((((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp) ∧ ((invg‘𝑊)‘𝑌) ∈ 𝐵 ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((invg‘𝑊)‘((𝑚 + 1) · 𝑋)) < ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)))
179	173, 174, 175, 176, 178	syl31anc 1321	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((invg‘𝑊)‘((𝑚 + 1) · 𝑋)) < ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)))
180	114	ad2antrr 758	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → (-(𝑚 + 1) · 𝑋) = ((invg‘𝑊)‘((𝑚 + 1) · 𝑋)))
181	180	eqcomd 2616	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((invg‘𝑊)‘((𝑚 + 1) · 𝑋)) = (-(𝑚 + 1) · 𝑋))
182	133	ad4antr 764	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) = 𝑌)
183	179, 181, 182	3brtr3d 4614	. . . . 5 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → (-(𝑚 + 1) · 𝑋) < 𝑌)
184		simp-4l 802	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → 𝜑)
185	7, 8	mulgcl 17382	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
186	75, 60, 77, 185	syl3anc 1318	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 · 𝑋) ∈ 𝐵)
187	7, 17, 9	ogrpinvlt 29055	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp) ∧ (𝑚 · 𝑋) ∈ 𝐵 ∧ ((invg‘𝑊)‘𝑌) ∈ 𝐵) → ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ↔ ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) < ((invg‘𝑊)‘(𝑚 · 𝑋))))
188	74, 186, 123, 187	syl3anc 1318	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ↔ ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) < ((invg‘𝑊)‘(𝑚 · 𝑋))))
189	188	biimpa 500	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌)) → ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) < ((invg‘𝑊)‘(𝑚 · 𝑋)))
190	189	adantrr 749	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) → ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) < ((invg‘𝑊)‘(𝑚 · 𝑋)))
191	190	adantr 480	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) < ((invg‘𝑊)‘(𝑚 · 𝑋)))
192		negdi 10217	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝑚 + 1) = (-𝑚 + -1))
193	66, 40, 192	sylancl 693	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ0 → -(𝑚 + 1) = (-𝑚 + -1))
194	193	oveq1d 6564	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ0 → (-(𝑚 + 1) + 1) = ((-𝑚 + -1) + 1))
195	66	negcld 10258	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ0 → -𝑚 ∈ ℂ)
196	91	negcld 10258	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ0 → -1 ∈ ℂ)
197	195, 196, 91	addassd 9941	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ0 → ((-𝑚 + -1) + 1) = (-𝑚 + (-1 + 1)))
198	43	oveq2i 6560	. . . . . . . . . . . . . . 15 ⊢ (-𝑚 + (-1 + 1)) = (-𝑚 + 0)
199	198	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ0 → (-𝑚 + (-1 + 1)) = (-𝑚 + 0))
200	195	addid1d 10115	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ0 → (-𝑚 + 0) = -𝑚)
201	197, 199, 200	3eqtrd 2648	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ0 → ((-𝑚 + -1) + 1) = -𝑚)
202	194, 201	eqtrd 2644	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ0 → (-(𝑚 + 1) + 1) = -𝑚)
203	202	oveq1d 6564	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ0 → ((-(𝑚 + 1) + 1) · 𝑋) = (-𝑚 · 𝑋))
204	203	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-(𝑚 + 1) + 1) · 𝑋) = (-𝑚 · 𝑋))
205	7, 8, 9	mulgneg 17383	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (-𝑚 · 𝑋) = ((invg‘𝑊)‘(𝑚 · 𝑋)))
206	75, 60, 77, 205	syl3anc 1318	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-𝑚 · 𝑋) = ((invg‘𝑊)‘(𝑚 · 𝑋)))
207	204, 206	eqtrd 2644	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-(𝑚 + 1) + 1) · 𝑋) = ((invg‘𝑊)‘(𝑚 · 𝑋)))
208	207	ad2antrr 758	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((-(𝑚 + 1) + 1) · 𝑋) = ((invg‘𝑊)‘(𝑚 · 𝑋)))
209	208	eqcomd 2616	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ((invg‘𝑊)‘(𝑚 · 𝑋)) = ((-(𝑚 + 1) + 1) · 𝑋))
210	191, 182, 209	3brtr3d 4614	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → 𝑌 < ((-(𝑚 + 1) + 1) · 𝑋))
211		ovex 6577	. . . . . . . 8 ⊢ ((-(𝑚 + 1) + 1) · 𝑋) ∈ V
212	211	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((-(𝑚 + 1) + 1) · 𝑋) ∈ V)
213	34, 17	pltle 16784	. . . . . . 7 ⊢ ((𝑊 ∈ oGrp ∧ 𝑌 ∈ 𝐵 ∧ ((-(𝑚 + 1) + 1) · 𝑋) ∈ V) → (𝑌 < ((-(𝑚 + 1) + 1) · 𝑋) → 𝑌 ≤ ((-(𝑚 + 1) + 1) · 𝑋)))
214	2, 120, 212, 213	syl3anc 1318	. . . . . 6 ⊢ (𝜑 → (𝑌 < ((-(𝑚 + 1) + 1) · 𝑋) → 𝑌 ≤ ((-(𝑚 + 1) + 1) · 𝑋)))
215	184, 210, 214	sylc 63	. . . . 5 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → 𝑌 ≤ ((-(𝑚 + 1) + 1) · 𝑋))
216		oveq1 6556	. . . . . . . 8 ⊢ (𝑛 = -(𝑚 + 1) → (𝑛 · 𝑋) = (-(𝑚 + 1) · 𝑋))
217	216	breq1d 4593	. . . . . . 7 ⊢ (𝑛 = -(𝑚 + 1) → ((𝑛 · 𝑋) < 𝑌 ↔ (-(𝑚 + 1) · 𝑋) < 𝑌))
218		oveq1 6556	. . . . . . . . 9 ⊢ (𝑛 = -(𝑚 + 1) → (𝑛 + 1) = (-(𝑚 + 1) + 1))
219	218	oveq1d 6564	. . . . . . . 8 ⊢ (𝑛 = -(𝑚 + 1) → ((𝑛 + 1) · 𝑋) = ((-(𝑚 + 1) + 1) · 𝑋))
220	219	breq2d 4595	. . . . . . 7 ⊢ (𝑛 = -(𝑚 + 1) → (𝑌 ≤ ((𝑛 + 1) · 𝑋) ↔ 𝑌 ≤ ((-(𝑚 + 1) + 1) · 𝑋)))
221	217, 220	anbi12d 743	. . . . . 6 ⊢ (𝑛 = -(𝑚 + 1) → (((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)) ↔ ((-(𝑚 + 1) · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((-(𝑚 + 1) + 1) · 𝑋))))
222	221	rspcev 3282	. . . . 5 ⊢ ((-(𝑚 + 1) ∈ ℤ ∧ ((-(𝑚 + 1) · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((-(𝑚 + 1) + 1) · 𝑋))) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))
223	169, 183, 215, 222	syl12anc 1316	. . . 4 ⊢ (((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) ∧ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))
224	7, 34, 17	tlt2 28995	. . . . . 6 ⊢ ((𝑊 ∈ Toset ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵 ∧ ((invg‘𝑊)‘𝑌) ∈ 𝐵) → (((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌) ∨ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)))
225	148, 79, 123, 224	syl3anc 1318	. . . . 5 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌) ∨ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)))
226	225	adantr 480	. . . 4 ⊢ ((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) → (((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌) ∨ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)))
227	167, 223, 226	mpjaodan 823	. . 3 ⊢ ((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋))) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))
228	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 < 0 ) → 𝑊 ∈ oGrp)
229		archirng.2	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ Archi)
230	229	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 < 0 ) → 𝑊 ∈ Archi)
231	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 < 0 ) → 𝑋 ∈ 𝐵)
232	122	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 < 0 ) → ((invg‘𝑊)‘𝑌) ∈ 𝐵)
233	16	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 < 0 ) → 0 < 𝑋)
234	133	breq1d 4593	. . . . . 6 ⊢ (𝜑 → (((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) < 0 ↔ 𝑌 < 0 ))
235	234	biimpar 501	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 < 0 ) → ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) < 0 )
236	7, 17, 9, 18	ogrpinv0lt 29054	. . . . . . 7 ⊢ ((𝑊 ∈ oGrp ∧ ((invg‘𝑊)‘𝑌) ∈ 𝐵) → ( 0 < ((invg‘𝑊)‘𝑌) ↔ ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) < 0 ))
237	2, 122, 236	syl2anc 691	. . . . . 6 ⊢ (𝜑 → ( 0 < ((invg‘𝑊)‘𝑌) ↔ ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) < 0 ))
238	237	biimpar 501	. . . . 5 ⊢ ((𝜑 ∧ ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) < 0 ) → 0 < ((invg‘𝑊)‘𝑌))
239	235, 238	syldan 486	. . . 4 ⊢ ((𝜑 ∧ 𝑌 < 0 ) → 0 < ((invg‘𝑊)‘𝑌))
240	7, 18, 17, 34, 8, 228, 230, 231, 232, 233, 239	archirng 29073	. . 3 ⊢ ((𝜑 ∧ 𝑌 < 0 ) → ∃𝑚 ∈ ℕ0 ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋)))
241	227, 240	r19.29a 3060	. 2 ⊢ ((𝜑 ∧ 𝑌 < 0 ) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))
242		nn0ssz 11275	. . 3 ⊢ ℕ0 ⊆ ℤ
243	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑌) → 𝑊 ∈ oGrp)
244	229	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑌) → 𝑊 ∈ Archi)
245	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑌) → 𝑋 ∈ 𝐵)
246	120	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑌) → 𝑌 ∈ 𝐵)
247	16	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑌) → 0 < 𝑋)
248		simpr 476	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑌) → 0 < 𝑌)
249	7, 18, 17, 34, 8, 243, 244, 245, 246, 247, 248	archirng 29073	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑌) → ∃𝑛 ∈ ℕ0 ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))
250		ssrexv 3630	. . 3 ⊢ (ℕ0 ⊆ ℤ → (∃𝑛 ∈ ℕ0 ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋))))
251	242, 249, 250	mpsyl 66	. 2 ⊢ ((𝜑 ∧ 0 < 𝑌) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))
252	7, 17	tlt3 28996	. . 3 ⊢ ((𝑊 ∈ Toset ∧ 𝑌 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑌 = 0 ∨ 𝑌 < 0 ∨ 0 < 𝑌))
253	29, 120, 33, 252	syl3anc 1318	. 2 ⊢ (𝜑 → (𝑌 = 0 ∨ 𝑌 < 0 ∨ 0 < 𝑌))
254	58, 241, 251, 253	mpjao3dan 1387	1 ⊢ (𝜑 → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  -cneg 10146  2c2 10947  ℕ₀cn0 11169  ℤcz 11254  Basecbs 15695  +_gcplusg 15768  lecple 15775  0_gc0g 15923  Posetcpo 16763  ltcplt 16764  Tosetctos 16856  Grpcgrp 17245  inv_gcminusg 17246  ._gcmg 17363  opp_gcoppg 17598  oMndcomnd 29028  oGrpcogrp 29029  Archicarchi 29062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-seq 12664  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-plusg 15781  df-ple 15788  df-0g 15925  df-preset 16751  df-poset 16769  df-plt 16781  df-toset 16857  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-mulg 17364  df-oppg 17599  df-omnd 29030  df-ogrp 29031  df-inftm 29063  df-archi 29064

This theorem is referenced by:  archiabllem2c  29080