Theorem prarloclemarch2 7481

Description: Like prarloclemarch 7480 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 7565. (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 7480	. . 3 |- ( ( A e. Q. /\ C e. Q. ) -> E. z e. N. A <Q ( [ <. z , 1o >. ] ~Q .Q C ) )
2	1	3adant2 1018	. 2 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> E. z e. N. A <Q ( [ <. z , 1o >. ] ~Q .Q C ) )
3		pinn 7371	. . . . . . . 8 |- ( z e. N. -> z e. om )
4		1pi 7377	. . . . . . . . . . . 12 |- 1o e. N.
5	4	elexi 2772	. . . . . . . . . . 11 |- 1o e. _V
6	5	sucid 4449	. . . . . . . . . 10 |- 1o e. suc 1o
7		df-2o 6472	. . . . . . . . . 10 |- 2o = suc 1o
8	6, 7	eleqtrri 2269	. . . . . . . . 9 |- 1o e. 2o
9		2onn 6576	. . . . . . . . . . 11 |- 2o e. om
10		nnaword2 6569	. . . . . . . . . . 11 |- ( ( 2o e. om /\ z e. om ) -> 2o C_ ( z +o 2o ) )
11	9, 10	mpan 424	. . . . . . . . . 10 |- ( z e. om -> 2o C_ ( z +o 2o ) )
12	11	sseld 3179	. . . . . . . . 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 6523	. . . . . . . . 9 |- ( 1o +o 1o ) = 2o
16		addpiord 7378	. . . . . . . . . . 11 |- ( ( 1o e. N. /\ 1o e. N. ) -> ( 1o +N 1o ) = ( 1o +o 1o ) )
17	4, 4, 16	mp2an 426	. . . . . . . . . 10 |- ( 1o +N 1o ) = ( 1o +o 1o )
18		addclpi 7389	. . . . . . . . . . 11 |- ( ( 1o e. N. /\ 1o e. N. ) -> ( 1o +N 1o ) e. N. )
19	4, 4, 18	mp2an 426	. . . . . . . . . 10 |- ( 1o +N 1o ) e. N.
20	17, 19	eqeltrri 2267	. . . . . . . . 9 |- ( 1o +o 1o ) e. N.
21	15, 20	eqeltrri 2267	. . . . . . . 8 |- 2o e. N.
22		addpiord 7378	. . . . . . . 8 |- ( ( z e. N. /\ 2o e. N. ) -> ( z +N 2o ) = ( z +o 2o ) )
23	21, 22	mpan2 425	. . . . . . 7 |- ( z e. N. -> ( z +N 2o ) = ( z +o 2o ) )
24	14, 23	eleqtrrd 2273	. . . . . 6 |- ( z e. N. -> 1o e. ( z +N 2o ) )
25		addclpi 7389	. . . . . . . 8 |- ( ( z e. N. /\ 2o e. N. ) -> ( z +N 2o ) e. N. )
26	21, 25	mpan2 425	. . . . . . 7 |- ( z e. N. -> ( z +N 2o ) e. N. )
27		ltpiord 7381	. . . . . . . 8 |- ( ( 1o e. N. /\ ( z +N 2o ) e. N. ) -> ( 1o <N ( z +N 2o ) <-> 1o e. ( z +N 2o ) ) )
28	4, 27	mpan 424	. . . . . . 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 167	. . . . 5 |- ( z e. N. -> 1o <N ( z +N 2o ) )
31	30	adantl 277	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ z e. N. ) -> 1o <N ( z +N 2o ) )
32	31	adantrr 479	. . 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 6529	. . . . . . . . . . . . . . . . 17 |- ( z e. om -> ( z +o (/) ) = z )
34		0lt1o 6495	. . . . . . . . . . . . . . . . . . . 20 |- (/) e. 1o
35		1on 6478	. . . . . . . . . . . . . . . . . . . . . 22 |- 1o e. On
36	35	onsuci 4549	. . . . . . . . . . . . . . . . . . . . 21 |- suc 1o e. On
37		ontr1 4421	. . . . . . . . . . . . . . . . . . . . 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 426	. . . . . . . . . . . . . . . . . . 19 |- (/) e. suc 1o
40	39, 7	eleqtrri 2269	. . . . . . . . . . . . . . . . . 18 |- (/) e. 2o
41		nnaordi 6563	. . . . . . . . . . . . . . . . . . 19 |- ( ( 2o e. om /\ z e. om ) -> ( (/) e. 2o -> ( z +o (/) ) e. ( z +o 2o ) ) )
42	9, 41	mpan 424	. . . . . . . . . . . . . . . . . 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 2271	. . . . . . . . . . . . . . . 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 2273	. . . . . . . . . . . . . 14 |- ( z e. N. -> z e. ( z +N 2o ) )
47		ltpiord 7381	. . . . . . . . . . . . . . 15 |- ( ( z e. N. /\ ( z +N 2o ) e. N. ) -> ( z <N ( z +N 2o ) <-> z e. ( z +N 2o ) ) )
48	26, 47	mpdan 421	. . . . . . . . . . . . . 14 |- ( z e. N. -> ( z <N ( z +N 2o ) <-> z e. ( z +N 2o ) ) )
49	46, 48	mpbird 167	. . . . . . . . . . . . 13 |- ( z e. N. -> z <N ( z +N 2o ) )
50		mulidpi 7380	. . . . . . . . . . . . 13 |- ( z e. N. -> ( z .N 1o ) = z )
51		mulcompig 7393	. . . . . . . . . . . . . . . 16 |- ( ( ( z +N 2o ) e. N. /\ 1o e. N. ) -> ( ( z +N 2o ) .N 1o ) = ( 1o .N ( z +N 2o ) ) )
52	4, 51	mpan2 425	. . . . . . . . . . . . . . 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 7380	. . . . . . . . . . . . . . 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 2228	. . . . . . . . . . . . 13 |- ( z e. N. -> ( 1o .N ( z +N 2o ) ) = ( z +N 2o ) )
57	49, 50, 56	3brtr4d 4062	. . . . . . . . . . . 12 |- ( z e. N. -> ( z .N 1o ) <N ( 1o .N ( z +N 2o ) ) )
58		ordpipqqs 7436	. . . . . . . . . . . . . . 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 435	. . . . . . . . . . . . . 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 438	. . . . . . . . . . . . 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 421	. . . . . . . . . . . 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 167	. . . . . . . . . . 11 |- ( z e. N. -> [ <. z , 1o >. ] ~Q <Q [ <. ( z +N 2o ) , 1o >. ] ~Q )
63	62	adantl 277	. . . . . . . . . 10 |- ( ( C e. Q. /\ z e. N. ) -> [ <. z , 1o >. ] ~Q <Q [ <. ( z +N 2o ) , 1o >. ] ~Q )
64		opelxpi 4692	. . . . . . . . . . . . . . . 16 |- ( ( ( z +N 2o ) e. N. /\ 1o e. N. ) -> <. ( z +N 2o ) , 1o >. e. ( N. X. N. ) )
65	4, 64	mpan2 425	. . . . . . . . . . . . . . 15 |- ( ( z +N 2o ) e. N. -> <. ( z +N 2o ) , 1o >. e. ( N. X. N. ) )
66		enqex 7422	. . . . . . . . . . . . . . . 16 |- ~Q e. _V
67	66	ecelqsi 6645	. . . . . . . . . . . . . . 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 7410	. . . . . . . . . . . . . 14 |- Q. = ( ( N. X. N. ) /. ~Q )
70	68, 69	eleqtrrdi 2287	. . . . . . . . . . . . 13 |- ( z e. N. -> [ <. ( z +N 2o ) , 1o >. ] ~Q e. Q. )
71		opelxpi 4692	. . . . . . . . . . . . . . . . 17 |- ( ( z e. N. /\ 1o e. N. ) -> <. z , 1o >. e. ( N. X. N. ) )
72	4, 71	mpan2 425	. . . . . . . . . . . . . . . 16 |- ( z e. N. -> <. z , 1o >. e. ( N. X. N. ) )
73	66	ecelqsi 6645	. . . . . . . . . . . . . . . 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 2287	. . . . . . . . . . . . . 14 |- ( z e. N. -> [ <. z , 1o >. ] ~Q e. Q. )
76		ltmnqg 7463	. . . . . . . . . . . . . 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 1282	. . . . . . . . . . . . 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 1283	. . . . . . . . . . . 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 1306	. . . . . . . . . . 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 268	. . . . . . . . . 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 147	. . . . . . . . 9 |- ( ( C e. Q. /\ z e. N. ) -> ( C .Q [ <. z , 1o >. ] ~Q ) <Q ( C .Q [ <. ( z +N 2o ) , 1o >. ] ~Q ) )
82		mulcomnqg 7445	. . . . . . . . . 10 |- ( ( C e. Q. /\ [ <. z , 1o >. ] ~Q e. Q. ) -> ( C .Q [ <. z , 1o >. ] ~Q ) = ( [ <. z , 1o >. ] ~Q .Q C ) )
83	75, 82	sylan2 286	. . . . . . . . 9 |- ( ( C e. Q. /\ z e. N. ) -> ( C .Q [ <. z , 1o >. ] ~Q ) = ( [ <. z , 1o >. ] ~Q .Q C ) )
84		mulcomnqg 7445	. . . . . . . . . 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 286	. . . . . . . . 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 4061	. . . . . . . 8 |- ( ( C e. Q. /\ z e. N. ) -> ( [ <. z , 1o >. ] ~Q .Q C ) <Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) )
87	86	3ad2antl3 1163	. . . . . . 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 479	. . . . . 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 7460	. . . . . . . . . 10 |- <Q Or Q.
90		ltrelnq 7427	. . . . . . . . . 10 |- <Q C_ ( Q. X. Q. )
91	89, 90	sotri 5062	. . . . . . . . 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 115	. . . . . . . 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 277	. . . . . . 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 277	. . . . . 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 7438	. . . . . . . . . 10 |- ( ( [ <. ( z +N 2o ) , 1o >. ] ~Q e. Q. /\ C e. Q. ) -> ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) e. Q. )
97	70, 96	sylan 283	. . . . . . . . 9 |- ( ( z e. N. /\ C e. Q. ) -> ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) e. Q. )
98	97	ancoms 268	. . . . . . . 8 |- ( ( C e. Q. /\ z e. N. ) -> ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) e. Q. )
99	98	3ad2antl3 1163	. . . . . . 7 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ z e. N. ) -> ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) e. Q. )
100		simpl2 1003	. . . . . . 7 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ z e. N. ) -> B e. Q. )
101		ltaddnq 7469	. . . . . . 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 411	. . . . . 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 479	. . . . 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 5062	. . . . 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 411	. . . 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 7443	. . . . . . 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 411	. . . . . 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 4042	. . . . 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 479	. . . 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 147	. . 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 110	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) /\ z e. N. ) -> z e. N. )
112		breq2 4034	. . . . . . . 8 |- ( x = ( z +N 2o ) -> ( 1o <N x <-> 1o <N ( z +N 2o ) ) )
113		opeq1 3805	. . . . . . . . . . . 12 |- ( x = ( z +N 2o ) -> <. x , 1o >. = <. ( z +N 2o ) , 1o >. )
114	113	eceq1d 6625	. . . . . . . . . . 11 |- ( x = ( z +N 2o ) -> [ <. x , 1o >. ] ~Q = [ <. ( z +N 2o ) , 1o >. ] ~Q )
115	114	oveq1d 5934	. . . . . . . . . 10 |- ( x = ( z +N 2o ) -> ( [ <. x , 1o >. ] ~Q .Q C ) = ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) )
116	115	oveq2d 5935	. . . . . . . . 9 |- ( x = ( z +N 2o ) -> ( B +Q ( [ <. x , 1o >. ] ~Q .Q C ) ) = ( B +Q ( [ <. ( z +N 2o ) , 1o >. ] ~Q .Q C ) ) )
117	116	breq2d 4042	. . . . . . . 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 473	. . . . . . 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 2865	. . . . . 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 115	. . . . 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 479	. . 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 433	. 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 2616	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 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   E.wrex 2473   C_ wss 3154   (/)c0 3447   <.cop 3622   class class class wbr 4030   Oncon0 4395   suc csuc 4397   omcom 4623   X. cxp 4658  (class class class)co 5919   1oc1o 6464   2oc2o 6465   +o coa 6468   [cec 6587   /.cqs 6588   N.cnpi 7334   +N cpli 7335   .N cmi 7336   <N clti 7337   ~Q ceq 7341   Q.cnq 7342   +Q cplq 7344   .Q cmq 7345   <Q cltq 7347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-1o 6471  df-2o 6472  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-pli 7367  df-mi 7368  df-lti 7369  df-plpq 7406  df-mpq 7407  df-enq 7409  df-nqqs 7410  df-plqqs 7411  df-mqqs 7412  df-1nqqs 7413  df-rq 7414  df-ltnqqs 7415

This theorem is referenced by:  prarloc  7565