Theorem archirngz 33143

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 12569	. . 3 ⊢ -1 ∈ ℤ
2		archirng.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ oGrp)
3		ogrpgrp 33017	. . . . . . . . . 10 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ Grp)
4	2, 3	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ Grp)
5		1zzd 12564	. . . . . . . . 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 2729	. . . . . . . . . 10 ⊢ (invg‘𝑊) = (invg‘𝑊)
10	7, 8, 9	mulgneg 19024	. . . . . . . . 9 ⊢ ((𝑊 ∈ Grp ∧ 1 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (-1 · 𝑋) = ((invg‘𝑊)‘(1 · 𝑋)))
11	4, 5, 6, 10	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (-1 · 𝑋) = ((invg‘𝑊)‘(1 · 𝑋)))
12	7, 8	mulg1 19013	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = 𝑋)
13	6, 12	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (1 · 𝑋) = 𝑋)
14	13	fveq2d 6862	. . . . . . . 8 ⊢ (𝜑 → ((invg‘𝑊)‘(1 · 𝑋)) = ((invg‘𝑊)‘𝑋))
15	11, 14	eqtrd 2764	. . . . . . 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 33036	. . . . . . . . 9 ⊢ ((𝑊 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ ((invg‘𝑊)‘𝑋) < 0 ))
20	19	biimpa 476	. . . . . . . 8 ⊢ (((𝑊 ∈ oGrp ∧ 𝑋 ∈ 𝐵) ∧ 0 < 𝑋) → ((invg‘𝑊)‘𝑋) < 0 )
21	2, 6, 16, 20	syl21anc 837	. . . . . . 7 ⊢ (𝜑 → ((invg‘𝑊)‘𝑋) < 0 )
22	15, 21	eqbrtrd 5129	. . . . . 6 ⊢ (𝜑 → (-1 · 𝑋) < 0 )
23	22	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (-1 · 𝑋) < 0 )
24		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝑌 = 0 )
25	23, 24	breqtrrd 5135	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (-1 · 𝑋) < 𝑌)
26		isogrp 33016	. . . . . . . . . 10 ⊢ (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
27	26	simprbi 496	. . . . . . . . 9 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
28		omndtos 33019	. . . . . . . . 9 ⊢ (𝑊 ∈ oMnd → 𝑊 ∈ Toset)
29	2, 27, 28	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ Toset)
30		tospos 18379	. . . . . . . 8 ⊢ (𝑊 ∈ Toset → 𝑊 ∈ Poset)
31	29, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Poset)
32	7, 18	grpidcl 18897	. . . . . . . 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 18279	. . . . . . 7 ⊢ ((𝑊 ∈ Poset ∧ 0 ∈ 𝐵) → 0 ≤ 0 )
36	31, 33, 35	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 0 ≤ 0 )
37	36	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 0 ≤ 0 )
38		1m1e0 12258	. . . . . . . . . 10 ⊢ (1 − 1) = 0
39	38	negeqi 11414	. . . . . . . . 9 ⊢ -(1 − 1) = -0
40		ax-1cn 11126	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
41	40, 40	negsubdii 11507	. . . . . . . . 9 ⊢ -(1 − 1) = (-1 + 1)
42		neg0 11468	. . . . . . . . 9 ⊢ -0 = 0
43	39, 41, 42	3eqtr3i 2760	. . . . . . . 8 ⊢ (-1 + 1) = 0
44	43	oveq1i 7397	. . . . . . 7 ⊢ ((-1 + 1) · 𝑋) = (0 · 𝑋)
45	7, 18, 8	mulg0 19006	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = 0 )
46	6, 45	syl 17	. . . . . . 7 ⊢ (𝜑 → (0 · 𝑋) = 0 )
47	44, 46	eqtrid 2776	. . . . . 6 ⊢ (𝜑 → ((-1 + 1) · 𝑋) = 0 )
48	47	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → ((-1 + 1) · 𝑋) = 0 )
49	37, 24, 48	3brtr4d 5139	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝑌 ≤ ((-1 + 1) · 𝑋))
50	25, 49	jca 511	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → ((-1 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((-1 + 1) · 𝑋)))
51		oveq1 7394	. . . . . 6 ⊢ (𝑛 = -1 → (𝑛 · 𝑋) = (-1 · 𝑋))
52	51	breq1d 5117	. . . . 5 ⊢ (𝑛 = -1 → ((𝑛 · 𝑋) < 𝑌 ↔ (-1 · 𝑋) < 𝑌))
53		oveq1 7394	. . . . . . 7 ⊢ (𝑛 = -1 → (𝑛 + 1) = (-1 + 1))
54	53	oveq1d 7402	. . . . . 6 ⊢ (𝑛 = -1 → ((𝑛 + 1) · 𝑋) = ((-1 + 1) · 𝑋))
55	54	breq2d 5119	. . . . 5 ⊢ (𝑛 = -1 → (𝑌 ≤ ((𝑛 + 1) · 𝑋) ↔ 𝑌 ≤ ((-1 + 1) · 𝑋)))
56	52, 55	anbi12d 632	. . . 4 ⊢ (𝑛 = -1 → (((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)) ↔ ((-1 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((-1 + 1) · 𝑋))))
57	56	rspcev 3588	. . 3 ⊢ ((-1 ∈ ℤ ∧ ((-1 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((-1 + 1) · 𝑋))) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))
58	1, 50, 57	sylancr 587	. 2 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))
59		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
60	59	nn0zd 12555	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℤ)
61	60	ad2antrr 726	. . . . . . 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 12640	. . . . . 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 12565	. . . . . . 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 12643	. . . . 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 12452	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℂ)
67	66	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℂ)
68		2cnd 12264	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 2 ∈ ℂ)
69	67, 68	negdi2d 11547	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -(𝑚 + 2) = (-𝑚 − 2))
70	69	oveq1d 7402	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) · 𝑋) = ((-𝑚 − 2) · 𝑋))
71	2	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ oGrp)
72		archirngz.1	. . . . . . . . . . . 12 ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp)
73	72	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (oppg‘𝑊) ∈ oGrp)
74	71, 73	jca 511	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp))
75	4	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ Grp)
76	60	peano2zd 12641	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 1) ∈ ℤ)
77	6	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑋 ∈ 𝐵)
78	7, 8	mulgcl 19023	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Grp ∧ (𝑚 + 1) ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((𝑚 + 1) · 𝑋) ∈ 𝐵)
79	75, 76, 77, 78	syl3anc 1373	. . . . . . . . . 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 12642	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 2) ∈ ℤ)
82	7, 8	mulgcl 19023	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Grp ∧ (𝑚 + 2) ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((𝑚 + 2) · 𝑋) ∈ 𝐵)
83	75, 81, 77, 82	syl3anc 1373	. . . . . . . . . 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 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 0 < 𝑋)
86		eqid 2729	. . . . . . . . . . . . 13 ⊢ (+g‘𝑊) = (+g‘𝑊)
87	7, 17, 86	ogrpaddlt 33031	. . . . . . . . . . . 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 1385	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ( 0 (+g‘𝑊)((𝑚 + 1) · 𝑋)) < (𝑋(+g‘𝑊)((𝑚 + 1) · 𝑋)))
89	7, 86, 18	grplid 18899	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Grp ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵) → ( 0 (+g‘𝑊)((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) · 𝑋))
90	75, 79, 89	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ( 0 (+g‘𝑊)((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) · 𝑋))
91		1cnd 11169	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ0 → 1 ∈ ℂ)
92	66, 91, 91	addassd 11196	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) = (𝑚 + (1 + 1)))
93		1p1e2 12306	. . . . . . . . . . . . . . . . 17 ⊢ (1 + 1) = 2
94	93	oveq2i 7398	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 + (1 + 1)) = (𝑚 + 2)
95	92, 94	eqtrdi 2780	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) = (𝑚 + 2))
96	66, 91	addcld 11193	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℂ)
97	96, 91	addcomd 11376	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) = (1 + (𝑚 + 1)))
98	95, 97	eqtr3d 2766	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 2) = (1 + (𝑚 + 1)))
99	98	oveq1d 7402	. . . . . . . . . . . . 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 12564	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 1 ∈ ℤ)
102	7, 8, 86	mulgdir 19038	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 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 7402	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((1 · 𝑋)(+g‘𝑊)((𝑚 + 1) · 𝑋)) = (𝑋(+g‘𝑊)((𝑚 + 1) · 𝑋)))
106	100, 103, 105	3eqtrrd 2769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑋(+g‘𝑊)((𝑚 + 1) · 𝑋)) = ((𝑚 + 2) · 𝑋))
107	88, 90, 106	3brtr3d 5138	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) · 𝑋) < ((𝑚 + 2) · 𝑋))
108	7, 17, 9	ogrpinvlt 33037	. . . . . . . . . . 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 476	. . . . . . . . . 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 1375	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((invg‘𝑊)‘((𝑚 + 2) · 𝑋)) < ((invg‘𝑊)‘((𝑚 + 1) · 𝑋)))
111	7, 8, 9	mulgneg 19024	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Grp ∧ (𝑚 + 2) ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (-(𝑚 + 2) · 𝑋) = ((invg‘𝑊)‘((𝑚 + 2) · 𝑋)))
112	75, 81, 77, 111	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) · 𝑋) = ((invg‘𝑊)‘((𝑚 + 2) · 𝑋)))
113	7, 8, 9	mulgneg 19024	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Grp ∧ (𝑚 + 1) ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (-(𝑚 + 1) · 𝑋) = ((invg‘𝑊)‘((𝑚 + 1) · 𝑋)))
114	75, 76, 77, 113	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 1) · 𝑋) = ((invg‘𝑊)‘((𝑚 + 1) · 𝑋)))
115	110, 112, 114	3brtr4d 5139	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) · 𝑋) < (-(𝑚 + 1) · 𝑋))
116	70, 115	eqbrtrrd 5131	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-𝑚 − 2) · 𝑋) < (-(𝑚 + 1) · 𝑋))
117	116	ad2antrr 726	. . . . . 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 726	. . . . . . 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 732	. . . . . . . . 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 18919	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((invg‘𝑊)‘𝑌) ∈ 𝐵)
122	4, 120, 121	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → ((invg‘𝑊)‘𝑌) ∈ 𝐵)
123	122	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((invg‘𝑊)‘𝑌) ∈ 𝐵)
124	123	ad2antrr 726	. . . . . . . . 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 726	. . . . . . . . 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 777	. . . . . . . . 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 484	. . . . . . . . 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 18280	. . . . . . . . . 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 476	. . . . . . . . 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 1380	. . . . . . . 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 6862	. . . . . . 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 18937	. . . . . . . . 9 ⊢ ((𝑊 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) = 𝑌)
133	4, 120, 132	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) = 𝑌)
134	133	ad4antr 732	. . . . . . 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 2771	. . . . . 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 5135	. . . . 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 11169	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 1 ∈ ℂ)
138	67, 68, 137	addsubassd 11553	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 2) − 1) = (𝑚 + (2 − 1)))
139		2m1e1 12307	. . . . . . . . . . . . 13 ⊢ (2 − 1) = 1
140	139	oveq2i 7398	. . . . . . . . . . . 12 ⊢ (𝑚 + (2 − 1)) = (𝑚 + 1)
141	138, 140	eqtr2di 2781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 1) = ((𝑚 + 2) − 1))
142	141	negeqd 11415	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -(𝑚 + 1) = -((𝑚 + 2) − 1))
143	67, 68	addcld 11193	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 + 2) ∈ ℂ)
144	143, 137	negsubdid 11548	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -((𝑚 + 2) − 1) = (-(𝑚 + 2) + 1))
145	69	oveq1d 7402	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 2) + 1) = ((-𝑚 − 2) + 1))
146	142, 144, 145	3eqtrrd 2769	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-𝑚 − 2) + 1) = -(𝑚 + 1))
147	146	oveq1d 7402	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((-𝑚 − 2) + 1) · 𝑋) = (-(𝑚 + 1) · 𝑋))
148	29	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ Toset)
149	148, 30	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → 𝑊 ∈ Poset)
150	60	znegcld 12640	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → -𝑚 ∈ ℤ)
151	150, 80	zsubcld 12643	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-𝑚 − 2) ∈ ℤ)
152	151	peano2zd 12641	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-𝑚 − 2) + 1) ∈ ℤ)
153	7, 8	mulgcl 19023	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Grp ∧ ((-𝑚 − 2) + 1) ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (((-𝑚 − 2) + 1) · 𝑋) ∈ 𝐵)
154	75, 152, 77, 153	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((-𝑚 − 2) + 1) · 𝑋) ∈ 𝐵)
155	7, 34	posref 18279	. . . . . . . . 9 ⊢ ((𝑊 ∈ Poset ∧ (((-𝑚 − 2) + 1) · 𝑋) ∈ 𝐵) → (((-𝑚 − 2) + 1) · 𝑋) ≤ (((-𝑚 − 2) + 1) · 𝑋))
156	149, 154, 155	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((-𝑚 − 2) + 1) · 𝑋) ≤ (((-𝑚 − 2) + 1) · 𝑋))
157	147, 156	eqbrtrrd 5131	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-(𝑚 + 1) · 𝑋) ≤ (((-𝑚 − 2) + 1) · 𝑋))
158	157	ad2antrr 726	. . . . . 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 5129	. . . . 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 7394	. . . . . . . 8 ⊢ (𝑛 = (-𝑚 − 2) → (𝑛 · 𝑋) = ((-𝑚 − 2) · 𝑋))
161	160	breq1d 5117	. . . . . . 7 ⊢ (𝑛 = (-𝑚 − 2) → ((𝑛 · 𝑋) < 𝑌 ↔ ((-𝑚 − 2) · 𝑋) < 𝑌))
162		oveq1 7394	. . . . . . . . 9 ⊢ (𝑛 = (-𝑚 − 2) → (𝑛 + 1) = ((-𝑚 − 2) + 1))
163	162	oveq1d 7402	. . . . . . . 8 ⊢ (𝑛 = (-𝑚 − 2) → ((𝑛 + 1) · 𝑋) = (((-𝑚 − 2) + 1) · 𝑋))
164	163	breq2d 5119	. . . . . . 7 ⊢ (𝑛 = (-𝑚 − 2) → (𝑌 ≤ ((𝑛 + 1) · 𝑋) ↔ 𝑌 ≤ (((-𝑚 − 2) + 1) · 𝑋)))
165	161, 164	anbi12d 632	. . . . . 6 ⊢ (𝑛 = (-𝑚 − 2) → (((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)) ↔ (((-𝑚 − 2) · 𝑋) < 𝑌 ∧ 𝑌 ≤ (((-𝑚 − 2) + 1) · 𝑋))))
166	165	rspcev 3588	. . . . 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 836	. . . 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 726	. . . . . 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 12640	. . . . 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 726	. . . . . . . . 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 726	. . . . . . . . 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 511	. . . . . . . 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 1361	. . . . . . 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 726	. . . . . . 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 726	. . . . . . 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 484	. . . . . . 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 33037	. . . . . . . 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 476	. . . . . . 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 1375	. . . . . 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 726	. . . . . . 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 2735	. . . . . 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 732	. . . . . 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 5138	. . . . 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 782	. . . . . 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 19023	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
186	75, 60, 77, 185	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (𝑚 · 𝑋) ∈ 𝐵)
187	7, 17, 9	ogrpinvlt 33037	. . . . . . . . . . 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 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ↔ ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) < ((invg‘𝑊)‘(𝑚 · 𝑋))))
189	188	biimpa 476	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌)) → ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) < ((invg‘𝑊)‘(𝑚 · 𝑋)))
190	189	adantrr 717	. . . . . . . 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 11479	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝑚 + 1) = (-𝑚 + -1))
193	66, 40, 192	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ0 → -(𝑚 + 1) = (-𝑚 + -1))
194	193	oveq1d 7402	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ0 → (-(𝑚 + 1) + 1) = ((-𝑚 + -1) + 1))
195	66	negcld 11520	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ0 → -𝑚 ∈ ℂ)
196	91	negcld 11520	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ0 → -1 ∈ ℂ)
197	195, 196, 91	addassd 11196	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ0 → ((-𝑚 + -1) + 1) = (-𝑚 + (-1 + 1)))
198	43	oveq2i 7398	. . . . . . . . . . . . . . 15 ⊢ (-𝑚 + (-1 + 1)) = (-𝑚 + 0)
199	198	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ0 → (-𝑚 + (-1 + 1)) = (-𝑚 + 0))
200	195	addridd 11374	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ0 → (-𝑚 + 0) = -𝑚)
201	197, 199, 200	3eqtrd 2768	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ0 → ((-𝑚 + -1) + 1) = -𝑚)
202	194, 201	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ0 → (-(𝑚 + 1) + 1) = -𝑚)
203	202	oveq1d 7402	. . . . . . . . . . 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 19024	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (-𝑚 · 𝑋) = ((invg‘𝑊)‘(𝑚 · 𝑋)))
206	75, 60, 77, 205	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (-𝑚 · 𝑋) = ((invg‘𝑊)‘(𝑚 · 𝑋)))
207	204, 206	eqtrd 2764	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → ((-(𝑚 + 1) + 1) · 𝑋) = ((invg‘𝑊)‘(𝑚 · 𝑋)))
208	207	ad2antrr 726	. . . . . . . 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 2735	. . . . . . 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 5138	. . . . . 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		ovexd 7422	. . . . . . 7 ⊢ (𝜑 → ((-(𝑚 + 1) + 1) · 𝑋) ∈ V)
212	34, 17	pltle 18292	. . . . . . 7 ⊢ ((𝑊 ∈ oGrp ∧ 𝑌 ∈ 𝐵 ∧ ((-(𝑚 + 1) + 1) · 𝑋) ∈ V) → (𝑌 < ((-(𝑚 + 1) + 1) · 𝑋) → 𝑌 ≤ ((-(𝑚 + 1) + 1) · 𝑋)))
213	2, 120, 211, 212	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (𝑌 < ((-(𝑚 + 1) + 1) · 𝑋) → 𝑌 ≤ ((-(𝑚 + 1) + 1) · 𝑋)))
214	184, 210, 213	sylc 65	. . . . 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))
215		oveq1 7394	. . . . . . . 8 ⊢ (𝑛 = -(𝑚 + 1) → (𝑛 · 𝑋) = (-(𝑚 + 1) · 𝑋))
216	215	breq1d 5117	. . . . . . 7 ⊢ (𝑛 = -(𝑚 + 1) → ((𝑛 · 𝑋) < 𝑌 ↔ (-(𝑚 + 1) · 𝑋) < 𝑌))
217		oveq1 7394	. . . . . . . . 9 ⊢ (𝑛 = -(𝑚 + 1) → (𝑛 + 1) = (-(𝑚 + 1) + 1))
218	217	oveq1d 7402	. . . . . . . 8 ⊢ (𝑛 = -(𝑚 + 1) → ((𝑛 + 1) · 𝑋) = ((-(𝑚 + 1) + 1) · 𝑋))
219	218	breq2d 5119	. . . . . . 7 ⊢ (𝑛 = -(𝑚 + 1) → (𝑌 ≤ ((𝑛 + 1) · 𝑋) ↔ 𝑌 ≤ ((-(𝑚 + 1) + 1) · 𝑋)))
220	216, 219	anbi12d 632	. . . . . 6 ⊢ (𝑛 = -(𝑚 + 1) → (((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)) ↔ ((-(𝑚 + 1) · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((-(𝑚 + 1) + 1) · 𝑋))))
221	220	rspcev 3588	. . . . 5 ⊢ ((-(𝑚 + 1) ∈ ℤ ∧ ((-(𝑚 + 1) · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((-(𝑚 + 1) + 1) · 𝑋))) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))
222	169, 183, 214, 221	syl12anc 836	. . . 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)))
223	7, 34, 17	tlt2 32895	. . . . . 6 ⊢ ((𝑊 ∈ Toset ∧ ((𝑚 + 1) · 𝑋) ∈ 𝐵 ∧ ((invg‘𝑊)‘𝑌) ∈ 𝐵) → (((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌) ∨ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)))
224	148, 79, 123, 223	syl3anc 1373	. . . . 5 ⊢ (((𝜑 ∧ 𝑌 < 0 ) ∧ 𝑚 ∈ ℕ0) → (((𝑚 + 1) · 𝑋) ≤ ((invg‘𝑊)‘𝑌) ∨ ((invg‘𝑊)‘𝑌) < ((𝑚 + 1) · 𝑋)))
225	224	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)))
226	167, 222, 225	mpjaodan 960	. . 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)))
227	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 < 0 ) → 𝑊 ∈ oGrp)
228		archirng.2	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ Archi)
229	228	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 < 0 ) → 𝑊 ∈ Archi)
230	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 < 0 ) → 𝑋 ∈ 𝐵)
231	122	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 < 0 ) → ((invg‘𝑊)‘𝑌) ∈ 𝐵)
232	16	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 < 0 ) → 0 < 𝑋)
233	133	breq1d 5117	. . . . . 6 ⊢ (𝜑 → (((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) < 0 ↔ 𝑌 < 0 ))
234	233	biimpar 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 < 0 ) → ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) < 0 )
235	7, 17, 9, 18	ogrpinv0lt 33036	. . . . . . 7 ⊢ ((𝑊 ∈ oGrp ∧ ((invg‘𝑊)‘𝑌) ∈ 𝐵) → ( 0 < ((invg‘𝑊)‘𝑌) ↔ ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) < 0 ))
236	2, 122, 235	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ( 0 < ((invg‘𝑊)‘𝑌) ↔ ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) < 0 ))
237	236	biimpar 477	. . . . 5 ⊢ ((𝜑 ∧ ((invg‘𝑊)‘((invg‘𝑊)‘𝑌)) < 0 ) → 0 < ((invg‘𝑊)‘𝑌))
238	234, 237	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑌 < 0 ) → 0 < ((invg‘𝑊)‘𝑌))
239	7, 18, 17, 34, 8, 227, 229, 230, 231, 232, 238	archirng 33142	. . 3 ⊢ ((𝜑 ∧ 𝑌 < 0 ) → ∃𝑚 ∈ ℕ0 ((𝑚 · 𝑋) < ((invg‘𝑊)‘𝑌) ∧ ((invg‘𝑊)‘𝑌) ≤ ((𝑚 + 1) · 𝑋)))
240	226, 239	r19.29a 3141	. 2 ⊢ ((𝜑 ∧ 𝑌 < 0 ) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))
241		nn0ssz 12552	. . 3 ⊢ ℕ0 ⊆ ℤ
242	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑌) → 𝑊 ∈ oGrp)
243	228	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑌) → 𝑊 ∈ Archi)
244	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑌) → 𝑋 ∈ 𝐵)
245	120	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑌) → 𝑌 ∈ 𝐵)
246	16	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑌) → 0 < 𝑋)
247		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑌) → 0 < 𝑌)
248	7, 18, 17, 34, 8, 242, 243, 244, 245, 246, 247	archirng 33142	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑌) → ∃𝑛 ∈ ℕ0 ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))
249		ssrexv 4016	. . 3 ⊢ (ℕ0 ⊆ ℤ → (∃𝑛 ∈ ℕ0 ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋))))
250	241, 248, 249	mpsyl 68	. 2 ⊢ ((𝜑 ∧ 0 < 𝑌) → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))
251	7, 17	tlt3 32896	. . 3 ⊢ ((𝑊 ∈ Toset ∧ 𝑌 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑌 = 0 ∨ 𝑌 < 0 ∨ 0 < 𝑌))
252	29, 120, 33, 251	syl3anc 1373	. 2 ⊢ (𝜑 → (𝑌 = 0 ∨ 𝑌 < 0 ∨ 0 < 𝑌))
253	58, 240, 250, 252	mpjao3dan 1434	1 ⊢ (𝜑 → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3447   ⊆ wss 3914   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387  ℂcc 11066  0cc0 11068  1c1 11069   + caddc 11071   − cmin 11405  -cneg 11406  2c2 12241  ℕ₀cn0 12442  ℤcz 12529  Basecbs 17179  +_gcplusg 17220  lecple 17227  0_gc0g 17402  Posetcpo 18268  ltcplt 18269  Tosetctos 18375  Grpcgrp 18865  inv_gcminusg 18866  ._gcmg 18999  opp_gcoppg 19277  oMndcomnd 33011  oGrpcogrp 33012  Archicarchi 33131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-seq 13967  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-plusg 17233  df-ple 17240  df-0g 17404  df-proset 18255  df-poset 18274  df-plt 18289  df-toset 18376  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-grp 18868  df-minusg 18869  df-mulg 19000  df-oppg 19278  df-omnd 33013  df-ogrp 33014  df-inftm 33132  df-archi 33133

This theorem is referenced by:  archiabllem2c  33149