Theorem prarloclemarch2 7381

Description: Like prarloclemarch 7380 but the integer must be at least two, and there is also B added to the right hand side. These details follow straightforwardly but are chosen to be helpful in the proof of prarloc 7465. (Contributed by Jim Kingdon, 25-Nov-2019.)

Assertion

Ref	Expression
prarloclemarch2	|- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> E. x e. N. ( 1o <N x /\ A <Q ( B +Q ( [ <. x , 1o >. ] ~Q .Q C ) ) ) )

Distinct variable groups:   x, A   x, B   x, C

Proof of Theorem prarloclemarch2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prarloclemarch 7380	. . 3 |- ( ( A e. Q. /\ C e. Q. ) -> E. z e. N. A <Q ( [ <. z , 1o >. ] ~Q .Q C ) )
2	1	3adant2 1011	. 2 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> E. z e. N. A <Q ( [ <. z , 1o >. ] ~Q .Q C ) )
3		pinn 7271	. . . . . . . 8 |- ( z e. N. -> z e. om )
4		1pi 7277	. . . . . . . . . . . 12 |- 1o e. N.
5	4	elexi 2742	. . . . . . . . . . 11 |- 1o e. _V
6	5	sucid 4402	. . . . . . . . . 10 |- 1o e. suc 1o
7		df-2o 6396	. . . . . . . . . 10 |- 2o = suc 1o
8	6, 7	eleqtrri 2246	. . . . . . . . 9 |- 1o e. 2o
9		2onn 6500	. . . . . . . . . . 11 |- 2o e. om
10		nnaword2 6493	. . . . . . . . . . 11 |- ( ( 2o e. om /\ z e. om ) -> 2o C_ ( z +o 2o ) )
11	9, 10	mpan 422	. . . . . . . . . 10 |- ( z e. om -> 2o C_ ( z +o 2o ) )
12	11	sseld 3146	. . . . . . . . 9 |- ( z e. om -> ( 1o e. 2o -> 1o e. ( z +o 2o ) ) )
13	8, 12	mpi 15	. . . . . . . 8 |- ( z e. om -> 1o e. ( z +o 2o ) )
14	3, 13	syl 14	. . . . . . 7 |- ( z e. N. -> 1o e. ( z +o 2o ) )
15		o1p1e2 6447	. . . . . . . . 9 |- ( 1o +o 1o ) = 2o
16		addpiord 7278	. . . . . . . . . . 11 |- ( ( 1o e. N. /\ 1o e. N. ) -> ( 1o +N 1o ) = ( 1o +o 1o ) )
17	4, 4, 16	mp2an 424	. . . . . . . . . 10 |- ( 1o +N 1o ) = ( 1o +o 1o )
18		addclpi 7289	. . . . . . . . . . 11 |- ( ( 1o e. N. /\ 1o e. N. ) -> ( 1o +N 1o ) e. N. )
19	4, 4, 18	mp2an 424	. . . . . . . . . 10 |- ( 1o +N 1o ) e. N.
20	17, 19	eqeltrri 2244	. . . . . . . . 9 |- ( 1o +o 1o ) e. N.
21	15, 20	eqeltrri 2244	. . . . . . . 8 |- 2o e. N.
22		addpiord 7278	. . . . . . . 8 |- ( ( z e. N. /\ 2o e. N. ) -> ( z +N 2o ) = ( z +o 2o ) )
23	21, 22	mpan2 423	. . . . . . 7 |- ( z e. N. -> ( z +N 2o ) = ( z +o 2o ) )
24	14, 23	eleqtrrd 2250	. . . . . 6 |- ( z e. N. -> 1o e. ( z +N 2o ) )
25		addclpi 7289	. . . . . . . 8 |- ( ( z e. N. /\ 2o e. N. ) -> ( z +N 2o ) e. N. )
26	21, 25	mpan2 423	. . . . . . 7 |- ( z e. N. -> ( z +N 2o ) e. N. )
27		ltpiord 7281	. . . . . . . 8 |- ( ( 1o e. N. /\ ( z +N 2o ) e. N. ) -> ( 1o <N ( z +N 2o ) <-> 1o e. ( z +N 2o ) ) )
28	4, 27	mpan 422	. . . . . . 7 |- ( ( z +N 2o ) e. N. -> ( 1o <N ( z +N 2o ) <-> 1o e. ( z +N 2o ) ) )
29	26, 28	syl 14	. . . . . 6 |- ( z e. N. -> ( 1o <N ( z +N 2o ) <-> 1o e. ( z +N 2o ) ) )
30	24, 29	mpbird 166	. . . . 5 |- ( z e. N. -> 1o <N ( z +N 2o ) )
31	30	adantl 275	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ z e. N. ) -> 1o <N ( z +N 2o ) )
32	31	adantrr 476	. . 3 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ ( z e. N. /\ A <Q ( [ <. z , 1o >. ] ~Q .Q C ) ) ) -> 1o <N ( z +N 2o ) )
33		nna0 6453	. . . . . . . . . . . . . . . . 17 |- ( z e. om -> ( z +o (/) ) = z )
34		0lt1o 6419	. . . . . . . . . . . . . . . . . . . 20 |- (/) e. 1o
35		1on 6402	. . . . . . . . . . . . . . . . . . . . . 22 |- 1o e. On
36	35	onsuci 4500	. . . . . . . . . . . . . . . . . . . . 21 |- suc 1o e. On
37		ontr1 4374	. . . . . . . . . . . . . . . . . . . . 21 |- ( suc 1o e. On -> ( ( (/) e. 1o /\ 1o e. suc 1o ) -> (/) e. suc 1o ) )
38	36, 37	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 |- ( ( (/) e. 1o /\ 1o e. suc 1o ) -> (/) e. suc 1o )
39	34, 6, 38	mp2an 424	. . . . . . . . . . . . . . . . . . 19 |- (/) e. suc 1o
40	39, 7	eleqtrri 2246	. . . . . . . . . . . . . . . . . 18 |- (/) e. 2o
41		nnaordi 6487	. . . . . . . . . . . . . . . . . . 19 |- ( ( 2o e. om /\ z e. om ) -> ( (/) e. 2o -> ( z +o (/) ) e. ( z +o 2o ) ) )
42	9, 41	mpan 422	. . . . . . . . . . . . . . . . . 18 |- ( z e. om -> ( (/) e. 2o -> ( z +o (/) ) e. ( z +o 2o ) ) )
43	40, 42	mpi 15	. . . . . . . . . . . . . . . . 17 |- ( z e. om -> ( z +o (/) ) e. ( z +o 2o ) )
44	33, 43	eqeltrrd 2248	. . . . . . . . . . . . . . . 16 |- ( z e. om -> z e. ( z +o 2o ) )
45	3, 44	syl 14	. . . . . . . . . . . . . . 15 |- ( z e. N. -> z e. ( z +o 2o ) )
46	45, 23	eleqtrrd 2250	. . . . . . . . . . . . . 14 |- ( z e. N. -> z e. ( z +N 2o ) )
47		ltpiord 7281	. . . . . . . . . . . . . . 15 |- ( ( z e. N. /\ ( z +N 2o ) e. N. ) -> ( z <N ( z +N 2o ) <-> z e. ( z +N 2o ) ) )
48	26, 47	mpdan 419	. . . . . . . . . . . . . 14 |- ( z e. N. -> ( z <N ( z +N 2o ) <-> z e. ( z +N 2o ) ) )
49	46, 48	mpbird 166	. . . . . . . . . . . . 13 |- ( z e. N. -> z <N ( z +N 2o ) )
50		mulidpi 7280	. . . . . . . . . . . . 13 |- ( z e. N. -> ( z .N 1o ) = z )
51		mulcompig 7293	. . . . . . . . . . . . . . . 16 |- ( ( ( z +N 2o ) e. N. /\ 1o e. N. ) -> ( ( z +N 2o ) .N 1o ) = ( 1o .N ( z +N 2o ) ) )
52	4, 51	mpan2 423	. . . . . . . . . . . . . . 15 |- ( ( z +N 2o ) e. N. -> ( ( z +N 2o ) .N 1o ) = ( 1o .N ( z +N 2o ) ) )
53	26, 52	syl 14	. . . . . . . . . . . . . 14 |- ( z e. N. -> ( ( z +N 2o ) .N 1o ) = ( 1o .N ( z +N 2o ) ) )
54		mulidpi 7280	. . . . . . . . . . . . . . 15 |- ( ( z +N 2o ) e. N. -> ( ( z +N 2o ) .N 1o ) = ( z +N 2o ) )
55	26, 54	syl 14	. . . . . . . . . . . . . 14 |- ( z e. N. -> ( ( z +N 2o ) .N 1o ) = ( z +N 2o ) )
56	53, 55	eqtr3d 2205	. . . . . . . . . . . . 13 |- ( z e. N. -> ( 1o .N ( z +N 2o ) ) = ( z +N 2o ) )
57	49, 50, 56	3brtr4d 4021	. . . . . . . . . . . 12 |- ( z e. N. -> ( z .N 1o ) <N ( 1o .N ( z +N 2o ) ) )
58		ordpipqqs 7336	. . . . . . . . . . . . . . 15 |- ( ( ( z e. N. /\ 1o e. N. ) /\ ( ( z +N 2o ) e. N. /\ 1o e. N. ) ) -> ( [ <. z , 1o >. ] ~Q <Q [ <. ( z +N 2o ) , 1o >. ] ~Q <-> ( z .N 1o ) <N ( 1o .N ( z +N 2o ) ) ) )
59	4, 58	mpanl2 433	. . . . . . . . . . . . . 14 |- ( ( z e. N. /\ ( ( z +N 2o ) e. N. /\ 1o e. N. ) ) -> ( [ <. z , 1o >. ] ~Q <Q [ <. ( z +N 2o ) , 1o >. ] ~Q <-> ( z .N 1o ) <N ( 1o .N ( z +N 2o ) ) ) )
60	4, 59	mpanr2 436	. . . . . . . . . . . . 13 |- ( ( z e. N. /\ ( z +N 2o ) e. N. ) -> ( [ <. z , 1o >. ] ~Q <Q [ <. ( z +N 2o ) , 1o >. ] ~Q <-> ( z .N 1o ) <N ( 1o .N ( z +N 2o ) ) ) )
61	26, 60	mpdan 419	. . . . . . . . . . . 12 |- ( z e. N. -> ( [ <. z , 1o >. ] ~Q <Q [ <. ( z +N 2o ) , 1o >. ] ~Q <-> ( z .N 1o ) <N ( 1o .N ( z +N 2o ) ) ) )
62	57, 61	mpbird 166	. . . . . . . . . . 11 |- ( z e. N. -> [ <. z , 1o >. ] ~Q <Q [ <. ( z +N 2o ) , 1o >. ] ~Q )
63	62	adantl 275	. . . . . . . . . 10 |- ( ( C e. Q. /\ z e. N. ) -> [ <. z , 1o >. ] ~Q <Q [ <. ( z +N 2o ) , 1o >. ] ~Q )
64		opelxpi 4643	. . . . . . . . . . . . . . . 16 |- ( ( ( z +N 2o ) e. N. /\ 1o e. N. ) -> <. ( z +N 2o ) , 1o >. e. ( N. X. N. ) )
65	4, 64	mpan2 423	. . . . . . . . . . . . . . 15 |- ( ( z +N 2o ) e. N. -> <. ( z +N 2o ) , 1o >. e. ( N. X. N. ) )
66		enqex 7322	. . . . . . . . . . . . . . . 16 |- ~Q e. _V
67	66	ecelqsi 6567	. . . . . . . . . . . . . . 15 |- ( <. ( z +N 2o ) , 1o >. e. ( N. X. N. ) -> [ <. ( z +N 2o ) , 1o >. ] ~Q e. ( ( N. X. N. ) /. ~Q ) )
68	26, 65, 67	3syl 17	. . . . . . . . . . . . . 14 |- ( z e. N. -> [ <. ( z +N 2o ) , 1o >. ] ~Q e. ( ( N. X. N. ) /. ~Q ) )
69		df-nqqs 7310	. . . . . . . . . . . . . 14 |- Q. = ( ( N. X. N. ) /. ~Q )
70	68, 69	eleqtrrdi 2264	. . . . . . . . . . . . 13 |- ( z e. N. -> [ <. ( z +N 2o ) , 1o >. ] ~Q e. Q. )
71		opelxpi 4643	. . . . . . . . . . . . . . . . 17 |- ( ( z e. N. /\ 1o e. N. ) -> <. z , 1o >. e. ( N. X. N. ) )
72	4, 71	mpan2 423	. . . . . . . . . . . . . . . 16 |- ( z e. N. -> <. z , 1o >. e. ( N. X. N. ) )
73	66	ecelqsi 6567	. . . . . . . . . . . . . . . 16 |- ( <. z , 1o >. e. ( N. X. N. ) -> [ <. z , 1o >. ] ~Q e. ( ( N. X. N. ) /. ~Q ) )
74	72, 73	syl 14	. . . . . . . . . . . . . . 15 |- ( z e. N. -> [ <. z , 1o >. ] ~Q e. ( ( N. X. N. ) /. ~Q ) )
75	74, 69	eleqtrrdi 2264	. . . . . . . . . . . . . 14 |- ( z e. N. -> [ <. z , 1o >. ] ~Q e. Q. )
76		ltmnqg 7363	. . . . . . . . . . . . . 14 |- ( ( [ <. z , 1o >. ] ~Q e. Q. /\ [ <. ( z +N 2o ) , 1o >. ] ~Q e. Q. /\ C e. Q. ) -> ( [ <. z , 1o >. ] ~Q <Q [ <. ( z +N 2o ) , 1o >. ] ~Q <-> ( C .Q [ <. z , 1o >. ] ~Q ) <Q ( C .Q [ <. ( z +N 2o ) , 1o >. ] ~Q ) ) )
77	75, 76	syl3an1 1266	. . . . . . . . . . . . 13 |- ( ( z e. N. /\ [ <. ( z +N 2o ) , 1o >. ] ~Q e. Q. /\ C e. Q. ) -> ( [ <. z , 1o >. ] ~Q <Q [ <. ( z +N 2o ) , 1o >. ] ~Q <-> ( C .Q [ <. z , 1o >. ] ~Q ) <Q ( C .Q [ <. ( z +N 2o ) , 1o >. ] ~Q ) ) )
78	70, 77	syl3an2 1267	. . . . . . . . . . . 12 |- ( ( z e. N. /\ z e. N. /\ C e. Q. ) -> ( [ <. z , 1o >. ] ~Q <Q [ <. ( z +N 2o ) , 1o >. ] ~Q <-> ( C .Q [ <. z , 1o >. ] ~Q ) <Q ( C .Q [ <. ( z +N 2o ) , 1o >. ] ~Q ) ) )
79	78	3anidm12 1290	. . . . . . . . . . 11 |- ( ( z e. N. /\ C e. Q. ) -> ( [ <. z , 1o >. ] ~Q <Q [ <. ( z +N 2o ) , 1o >. ] ~Q <-> ( C .Q [ <. z , 1o >. ] ~Q ) <Q ( C .Q [ <. ( z +N 2o ) , 1o >. ] ~Q ) ) )
80	79	ancoms 266	. . . . . . . . . 10 |- ( ( C e. Q. /\ z e. N. ) -> ( [ <. z , 1o >. ] ~Q <Q [ <. ( z +N 2o ) , 1o >. ] ~Q <-> ( C .Q [ <. z , 1o >. ] ~Q ) <Q ( C .Q [ <. ( z +N 2o ) , 1o >. ] ~Q ) ) )
81	63, 80	mpbid 146	. . . . . . . . 9 |- ( ( C e. Q. /\ z e. N. ) -> ( C .Q [ <. z , 1o >. ] ~Q ) <Q ( C .Q [ <. ( z +N 2o ) , 1o >. ] ~Q ) )
82		mulcomnqg 7345	. . . . . . . . . 10 |- ( ( C e. Q. /\ [ <. z , 1o >. ] ~Q e. Q. ) -> ( C .Q [ <. z , 1o >. ] ~Q ) = ( [ <. z , 1o >. ] ~Q .Q C ) )
83	75, 82	sylan2 284	. . . . . . . . 9 |- ( ( C e. Q. /\ z e. N. ) -> ( C .Q [ <. z , 1o >. ] ~Q ) = ( [ <. z , 1o >. ] ~Q .Q C ) )
84		mulcomnqg 7345	. . . . . . . . . 10 |- ( ( C e. Q. /\ [ <. ( z +N 2o ) , 1o >. ] ~Q e. Q. ) -> ( C .Q [ <. ( z +N 2o ) , 1o >. ] ~Q ) = ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) )
85	70, 84	sylan2 284	. . . . . . . . 9 |- ( ( C e. Q. /\ z e. N. ) -> ( C .Q [ <. ( z +N 2o ) , 1o >. ] ~Q ) = ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) )
86	81, 83, 85	3brtr3d 4020	. . . . . . . 8 |- ( ( C e. Q. /\ z e. N. ) -> ( [ <. z , 1o >. ] ~Q .Q C ) <Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) )
87	86	3ad2antl3 1156	. . . . . . 7 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ z e. N. ) -> ( [ <. z , 1o >. ] ~Q .Q C ) <Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) )
88	87	adantrr 476	. . . . . 6 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ ( z e. N. /\ A <Q ( [ <. z , 1o >. ] ~Q .Q C ) ) ) -> ( [ <. z , 1o >. ] ~Q .Q C ) <Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) )
89		ltsonq 7360	. . . . . . . . . 10 |- <Q Or Q.
90		ltrelnq 7327	. . . . . . . . . 10 |- <Q C_ ( Q. X. Q. )
91	89, 90	sotri 5006	. . . . . . . . 9 |- ( ( A <Q ( [ <. z , 1o >. ] ~Q .Q C ) /\ ( [ <. z , 1o >. ] ~Q .Q C ) <Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) ) -> A <Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) )
92	91	ex 114	. . . . . . . 8 |- ( A <Q ( [ <. z , 1o >. ] ~Q .Q C ) -> ( ( [ <. z , 1o >. ] ~Q .Q C ) <Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) -> A <Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) ) )
93	92	adantl 275	. . . . . . 7 |- ( ( z e. N. /\ A <Q ( [ <. z , 1o >. ] ~Q .Q C ) ) -> ( ( [ <. z , 1o >. ] ~Q .Q C ) <Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) -> A <Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) ) )
94	93	adantl 275	. . . . . 6 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ ( z e. N. /\ A <Q ( [ <. z , 1o >. ] ~Q .Q C ) ) ) -> ( ( [ <. z , 1o >. ] ~Q .Q C ) <Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) -> A <Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) ) )
95	88, 94	mpd 13	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ ( z e. N. /\ A <Q ( [ <. z , 1o >. ] ~Q .Q C ) ) ) -> A <Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) )
96		mulclnq 7338	. . . . . . . . . 10 |- ( ( [ <. ( z +N 2o ) , 1o >. ] ~Q e. Q. /\ C e. Q. ) -> ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) e. Q. )
97	70, 96	sylan 281	. . . . . . . . 9 |- ( ( z e. N. /\ C e. Q. ) -> ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) e. Q. )
98	97	ancoms 266	. . . . . . . 8 |- ( ( C e. Q. /\ z e. N. ) -> ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) e. Q. )
99	98	3ad2antl3 1156	. . . . . . 7 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ z e. N. ) -> ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) e. Q. )
100		simpl2 996	. . . . . . 7 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ z e. N. ) -> B e. Q. )
101		ltaddnq 7369	. . . . . . 7 |- ( ( ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) e. Q. /\ B e. Q. ) -> ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) <Q ( ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) +Q B ) )
102	99, 100, 101	syl2anc 409	. . . . . 6 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ z e. N. ) -> ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) <Q ( ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) +Q B ) )
103	102	adantrr 476	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ ( z e. N. /\ A <Q ( [ <. z , 1o >. ] ~Q .Q C ) ) ) -> ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) <Q ( ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) +Q B ) )
104	89, 90	sotri 5006	. . . . 5 |- ( ( A <Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) /\ ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) <Q ( ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) +Q B ) ) -> A <Q ( ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) +Q B ) )
105	95, 103, 104	syl2anc 409	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ ( z e. N. /\ A <Q ( [ <. z , 1o >. ] ~Q .Q C ) ) ) -> A <Q ( ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) +Q B ) )
106		addcomnqg 7343	. . . . . . 7 |- ( ( ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) e. Q. /\ B e. Q. ) -> ( ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) +Q B ) = ( B +Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) ) )
107	99, 100, 106	syl2anc 409	. . . . . 6 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ z e. N. ) -> ( ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) +Q B ) = ( B +Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) ) )
108	107	breq2d 4001	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ z e. N. ) -> ( A <Q ( ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) +Q B ) <-> A <Q ( B +Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) ) ) )
109	108	adantrr 476	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ ( z e. N. /\ A <Q ( [ <. z , 1o >. ] ~Q .Q C ) ) ) -> ( A <Q ( ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) +Q B ) <-> A <Q ( B +Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) ) ) )
110	105, 109	mpbid 146	. . 3 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ ( z e. N. /\ A <Q ( [ <. z , 1o >. ] ~Q .Q C ) ) ) -> A <Q ( B +Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) ) )
111		simpr 109	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ z e. N. ) -> z e. N. )
112		breq2 3993	. . . . . . . 8 |- ( x = ( z +N 2o ) -> ( 1o <N x <-> 1o <N ( z +N 2o ) ) )
113		opeq1 3765	. . . . . . . . . . . 12 |- ( x = ( z +N 2o ) -> <. x , 1o >. = <. ( z +N 2o ) , 1o >. )
114	113	eceq1d 6549	. . . . . . . . . . 11 |- ( x = ( z +N 2o ) -> [ <. x , 1o >. ] ~Q = [ <. ( z +N 2o ) , 1o >. ] ~Q )
115	114	oveq1d 5868	. . . . . . . . . 10 |- ( x = ( z +N 2o ) -> ( [ <. x , 1o >. ] ~Q .Q C ) = ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) )
116	115	oveq2d 5869	. . . . . . . . 9 |- ( x = ( z +N 2o ) -> ( B +Q ( [ <. x , 1o >. ] ~Q .Q C ) ) = ( B +Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) ) )
117	116	breq2d 4001	. . . . . . . 8 |- ( x = ( z +N 2o ) -> ( A <Q ( B +Q ( [ <. x , 1o >. ] ~Q .Q C ) ) <-> A <Q ( B +Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) ) ) )
118	112, 117	anbi12d 470	. . . . . . 7 |- ( x = ( z +N 2o ) -> ( ( 1o <N x /\ A <Q ( B +Q ( [ <. x , 1o >. ] ~Q .Q C ) ) ) <-> ( 1o <N ( z +N 2o ) /\ A <Q ( B +Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) ) ) ) )
119	118	rspcev 2834	. . . . . 6 |- ( ( ( z +N 2o ) e. N. /\ ( 1o <N ( z +N 2o ) /\ A <Q ( B +Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) ) ) ) -> E. x e. N. ( 1o <N x /\ A <Q ( B +Q ( [ <. x , 1o >. ] ~Q .Q C ) ) ) )
120	119	ex 114	. . . . 5 |- ( ( z +N 2o ) e. N. -> ( ( 1o <N ( z +N 2o ) /\ A <Q ( B +Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) ) ) -> E. x e. N. ( 1o <N x /\ A <Q ( B +Q ( [ <. x , 1o >. ] ~Q .Q C ) ) ) ) )
121	111, 26, 120	3syl 17	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ z e. N. ) -> ( ( 1o <N ( z +N 2o ) /\ A <Q ( B +Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) ) ) -> E. x e. N. ( 1o <N x /\ A <Q ( B +Q ( [ <. x , 1o >. ] ~Q .Q C ) ) ) ) )
122	121	adantrr 476	. . 3 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ ( z e. N. /\ A <Q ( [ <. z , 1o >. ] ~Q .Q C ) ) ) -> ( ( 1o <N ( z +N 2o ) /\ A <Q ( B +Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) ) ) -> E. x e. N. ( 1o <N x /\ A <Q ( B +Q ( [ <. x , 1o >. ] ~Q .Q C ) ) ) ) )
123	32, 110, 122	mp2and 431	. 2 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ ( z e. N. /\ A <Q ( [ <. z , 1o >. ] ~Q .Q C ) ) ) -> E. x e. N. ( 1o <N x /\ A <Q ( B +Q ( [ <. x , 1o >. ] ~Q .Q C ) ) ) )
124	2, 123	rexlimddv 2592	1 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> E. x e. N. ( 1o <N x /\ A <Q ( B +Q ( [ <. x , 1o >. ] ~Q .Q C ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   E.wrex 2449   C_ wss 3121   (/)c0 3414   <.cop 3586   class class class wbr 3989   Oncon0 4348   suc csuc 4350   omcom 4574   X. cxp 4609  (class class class)co 5853   1oc1o 6388   2oc2o 6389   +o coa 6392   [cec 6511   /.cqs 6512   N.cnpi 7234   +N cpli 7235   .N cmi 7236   <N clti 7237   ~Q ceq 7241   Q.cnq 7242   +Q cplq 7244   .Q cmq 7245   <Q cltq 7247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315

This theorem is referenced by:  prarloc  7465