Theorem prarloclemcalc 7310

Description: Some calculations for prarloc 7311. (Contributed by Jim Kingdon, 26-Oct-2019.)

Assertion

Ref	Expression
prarloclemcalc	|- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> B <Q ( A +Q P ) )

Proof of Theorem prarloclemcalc

Step	Hyp	Ref	Expression
1		simprll 526	. . . . 5 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> Q e. Q. )
2		nqnq0a 7262	. . . . 5 |- ( ( Q e. Q. /\ Q e. Q. ) -> ( Q +Q Q ) = ( Q +Q0 Q ) )
3	1, 1, 2	syl2anc 408	. . . 4 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( Q +Q Q ) = ( Q +Q0 Q ) )
4	3	oveq2d 5790	. . 3 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( A +Q0 ( Q +Q Q ) ) = ( A +Q0 ( Q +Q0 Q ) ) )
5		simpll 518	. . . . 5 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) )
6		simprrl 528	. . . . . 6 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> X e. Q. )
7		simprrr 529	. . . . . . . 8 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> M e. om )
8		1pi 7123	. . . . . . . . . . 11 |- 1o e. N.
9		opelxpi 4571	. . . . . . . . . . 11 |- ( ( M e. om /\ 1o e. N. ) -> <. M , 1o >. e. ( om X. N. ) )
10	8, 9	mpan2 421	. . . . . . . . . 10 |- ( M e. om -> <. M , 1o >. e. ( om X. N. ) )
11		enq0ex 7247	. . . . . . . . . . 11 |- ~Q0 e. _V
12	11	ecelqsi 6483	. . . . . . . . . 10 |- ( <. M , 1o >. e. ( om X. N. ) -> [ <. M , 1o >. ] ~Q0 e. ( ( om X. N. ) /. ~Q0 ) )
13	10, 12	syl 14	. . . . . . . . 9 |- ( M e. om -> [ <. M , 1o >. ] ~Q0 e. ( ( om X. N. ) /. ~Q0 ) )
14		df-nq0 7233	. . . . . . . . 9 |- Q0 = ( ( om X. N. ) /. ~Q0 )
15	13, 14	eleqtrrdi 2233	. . . . . . . 8 |- ( M e. om -> [ <. M , 1o >. ] ~Q0 e. Q0 )
16	7, 15	syl 14	. . . . . . 7 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> [ <. M , 1o >. ] ~Q0 e. Q0 )
17		nqnq0 7249	. . . . . . . 8 |- Q. C_ Q0
18	17, 1	sseldi 3095	. . . . . . 7 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> Q e. Q0 )
19		mulclnq0 7260	. . . . . . 7 |- ( ( [ <. M , 1o >. ] ~Q0 e. Q0 /\ Q e. Q0 ) -> ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) e. Q0 )
20	16, 18, 19	syl2anc 408	. . . . . 6 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) e. Q0 )
21		nqpnq0nq 7261	. . . . . 6 |- ( ( X e. Q. /\ ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) e. Q0 ) -> ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) e. Q. )
22	6, 20, 21	syl2anc 408	. . . . 5 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) e. Q. )
23	5, 22	eqeltrd 2216	. . . 4 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> A e. Q. )
24		addclnq 7183	. . . . 5 |- ( ( Q e. Q. /\ Q e. Q. ) -> ( Q +Q Q ) e. Q. )
25	1, 1, 24	syl2anc 408	. . . 4 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( Q +Q Q ) e. Q. )
26		nqnq0a 7262	. . . 4 |- ( ( A e. Q. /\ ( Q +Q Q ) e. Q. ) -> ( A +Q ( Q +Q Q ) ) = ( A +Q0 ( Q +Q Q ) ) )
27	23, 25, 26	syl2anc 408	. . 3 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( A +Q ( Q +Q Q ) ) = ( A +Q0 ( Q +Q Q ) ) )
28		simplr 519	. . . . . 6 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) )
29		2onn 6417	. . . . . . . . . . . . . 14 |- 2o e. om
30		2on0 6323	. . . . . . . . . . . . . 14 |- 2o =/= (/)
31		elni 7116	. . . . . . . . . . . . . 14 |- ( 2o e. N. <-> ( 2o e. om /\ 2o =/= (/) ) )
32	29, 30, 31	mpbir2an 926	. . . . . . . . . . . . 13 |- 2o e. N.
33		nnppipi 7151	. . . . . . . . . . . . 13 |- ( ( M e. om /\ 2o e. N. ) -> ( M +o 2o ) e. N. )
34	32, 33	mpan2 421	. . . . . . . . . . . 12 |- ( M e. om -> ( M +o 2o ) e. N. )
35		opelxpi 4571	. . . . . . . . . . . 12 |- ( ( ( M +o 2o ) e. N. /\ 1o e. N. ) -> <. ( M +o 2o ) , 1o >. e. ( N. X. N. ) )
36	34, 8, 35	sylancl 409	. . . . . . . . . . 11 |- ( M e. om -> <. ( M +o 2o ) , 1o >. e. ( N. X. N. ) )
37		enqex 7168	. . . . . . . . . . . 12 |- ~Q e. _V
38	37	ecelqsi 6483	. . . . . . . . . . 11 |- ( <. ( M +o 2o ) , 1o >. e. ( N. X. N. ) -> [ <. ( M +o 2o ) , 1o >. ] ~Q e. ( ( N. X. N. ) /. ~Q ) )
39	36, 38	syl 14	. . . . . . . . . 10 |- ( M e. om -> [ <. ( M +o 2o ) , 1o >. ] ~Q e. ( ( N. X. N. ) /. ~Q ) )
40		df-nqqs 7156	. . . . . . . . . 10 |- Q. = ( ( N. X. N. ) /. ~Q )
41	39, 40	eleqtrrdi 2233	. . . . . . . . 9 |- ( M e. om -> [ <. ( M +o 2o ) , 1o >. ] ~Q e. Q. )
42	7, 41	syl 14	. . . . . . . 8 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> [ <. ( M +o 2o ) , 1o >. ] ~Q e. Q. )
43		mulclnq 7184	. . . . . . . 8 |- ( ( [ <. ( M +o 2o ) , 1o >. ] ~Q e. Q. /\ Q e. Q. ) -> ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) e. Q. )
44	42, 1, 43	syl2anc 408	. . . . . . 7 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) e. Q. )
45		nqnq0a 7262	. . . . . . 7 |- ( ( X e. Q. /\ ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) e. Q. ) -> ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) = ( X +Q0 ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) )
46	6, 44, 45	syl2anc 408	. . . . . 6 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) = ( X +Q0 ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) )
47		nqnq0m 7263	. . . . . . . . 9 |- ( ( [ <. ( M +o 2o ) , 1o >. ] ~Q e. Q. /\ Q e. Q. ) -> ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) = ( [ <. ( M +o 2o ) , 1o >. ] ~Q ·Q0 Q ) )
48	42, 1, 47	syl2anc 408	. . . . . . . 8 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) = ( [ <. ( M +o 2o ) , 1o >. ] ~Q ·Q0 Q ) )
49		nqnq0pi 7246	. . . . . . . . . . 11 |- ( ( ( M +o 2o ) e. N. /\ 1o e. N. ) -> [ <. ( M +o 2o ) , 1o >. ] ~Q0 = [ <. ( M +o 2o ) , 1o >. ] ~Q )
50	34, 8, 49	sylancl 409	. . . . . . . . . 10 |- ( M e. om -> [ <. ( M +o 2o ) , 1o >. ] ~Q0 = [ <. ( M +o 2o ) , 1o >. ] ~Q )
51	7, 50	syl 14	. . . . . . . . 9 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> [ <. ( M +o 2o ) , 1o >. ] ~Q0 = [ <. ( M +o 2o ) , 1o >. ] ~Q )
52	51	oveq1d 5789	. . . . . . . 8 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( [ <. ( M +o 2o ) , 1o >. ] ~Q0 ·Q0 Q ) = ( [ <. ( M +o 2o ) , 1o >. ] ~Q ·Q0 Q ) )
53	48, 52	eqtr4d 2175	. . . . . . 7 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) = ( [ <. ( M +o 2o ) , 1o >. ] ~Q0 ·Q0 Q ) )
54	53	oveq2d 5790	. . . . . 6 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( X +Q0 ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) = ( X +Q0 ( [ <. ( M +o 2o ) , 1o >. ] ~Q0 ·Q0 Q ) ) )
55	28, 46, 54	3eqtrd 2176	. . . . 5 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> B = ( X +Q0 ( [ <. ( M +o 2o ) , 1o >. ] ~Q0 ·Q0 Q ) ) )
56		nnanq0 7266	. . . . . . . . . 10 |- ( ( M e. om /\ 2o e. om /\ 1o e. N. ) -> [ <. ( M +o 2o ) , 1o >. ] ~Q0 = ( [ <. M , 1o >. ] ~Q0 +Q0 [ <. 2o , 1o >. ] ~Q0 ) )
57	8, 56	mp3an3 1304	. . . . . . . . 9 |- ( ( M e. om /\ 2o e. om ) -> [ <. ( M +o 2o ) , 1o >. ] ~Q0 = ( [ <. M , 1o >. ] ~Q0 +Q0 [ <. 2o , 1o >. ] ~Q0 ) )
58	7, 29, 57	sylancl 409	. . . . . . . 8 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> [ <. ( M +o 2o ) , 1o >. ] ~Q0 = ( [ <. M , 1o >. ] ~Q0 +Q0 [ <. 2o , 1o >. ] ~Q0 ) )
59	58	oveq1d 5789	. . . . . . 7 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( [ <. ( M +o 2o ) , 1o >. ] ~Q0 ·Q0 Q ) = ( ( [ <. M , 1o >. ] ~Q0 +Q0 [ <. 2o , 1o >. ] ~Q0 ) ·Q0 Q ) )
60		opelxpi 4571	. . . . . . . . . . . 12 |- ( ( 2o e. om /\ 1o e. N. ) -> <. 2o , 1o >. e. ( om X. N. ) )
61	29, 8, 60	mp2an 422	. . . . . . . . . . 11 |- <. 2o , 1o >. e. ( om X. N. )
62	11	ecelqsi 6483	. . . . . . . . . . 11 |- ( <. 2o , 1o >. e. ( om X. N. ) -> [ <. 2o , 1o >. ] ~Q0 e. ( ( om X. N. ) /. ~Q0 ) )
63	61, 62	ax-mp 5	. . . . . . . . . 10 |- [ <. 2o , 1o >. ] ~Q0 e. ( ( om X. N. ) /. ~Q0 )
64	63, 14	eleqtrri 2215	. . . . . . . . 9 |- [ <. 2o , 1o >. ] ~Q0 e. Q0
65		distnq0r 7271	. . . . . . . . 9 |- ( ( Q e. Q0 /\ [ <. M , 1o >. ] ~Q0 e. Q0 /\ [ <. 2o , 1o >. ] ~Q0 e. Q0 ) -> ( ( [ <. M , 1o >. ] ~Q0 +Q0 [ <. 2o , 1o >. ] ~Q0 ) ·Q0 Q ) = ( ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) +Q0 ( [ <. 2o , 1o >. ] ~Q0 ·Q0 Q ) ) )
66	64, 65	mp3an3 1304	. . . . . . . 8 |- ( ( Q e. Q0 /\ [ <. M , 1o >. ] ~Q0 e. Q0 ) -> ( ( [ <. M , 1o >. ] ~Q0 +Q0 [ <. 2o , 1o >. ] ~Q0 ) ·Q0 Q ) = ( ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) +Q0 ( [ <. 2o , 1o >. ] ~Q0 ·Q0 Q ) ) )
67	18, 16, 66	syl2anc 408	. . . . . . 7 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( ( [ <. M , 1o >. ] ~Q0 +Q0 [ <. 2o , 1o >. ] ~Q0 ) ·Q0 Q ) = ( ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) +Q0 ( [ <. 2o , 1o >. ] ~Q0 ·Q0 Q ) ) )
68	59, 67	eqtrd 2172	. . . . . 6 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( [ <. ( M +o 2o ) , 1o >. ] ~Q0 ·Q0 Q ) = ( ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) +Q0 ( [ <. 2o , 1o >. ] ~Q0 ·Q0 Q ) ) )
69	68	oveq2d 5790	. . . . 5 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( X +Q0 ( [ <. ( M +o 2o ) , 1o >. ] ~Q0 ·Q0 Q ) ) = ( X +Q0 ( ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) +Q0 ( [ <. 2o , 1o >. ] ~Q0 ·Q0 Q ) ) ) )
70		nq02m 7273	. . . . . . . . 9 |- ( Q e. Q0 -> ( [ <. 2o , 1o >. ] ~Q0 ·Q0 Q ) = ( Q +Q0 Q ) )
71	70	oveq2d 5790	. . . . . . . 8 |- ( Q e. Q0 -> ( ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) +Q0 ( [ <. 2o , 1o >. ] ~Q0 ·Q0 Q ) ) = ( ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) +Q0 ( Q +Q0 Q ) ) )
72	71	oveq2d 5790	. . . . . . 7 |- ( Q e. Q0 -> ( X +Q0 ( ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) +Q0 ( [ <. 2o , 1o >. ] ~Q0 ·Q0 Q ) ) ) = ( X +Q0 ( ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) +Q0 ( Q +Q0 Q ) ) ) )
73	18, 72	syl 14	. . . . . 6 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( X +Q0 ( ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) +Q0 ( [ <. 2o , 1o >. ] ~Q0 ·Q0 Q ) ) ) = ( X +Q0 ( ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) +Q0 ( Q +Q0 Q ) ) ) )
74	17, 6	sseldi 3095	. . . . . . 7 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> X e. Q0 )
75		addclnq0 7259	. . . . . . . 8 |- ( ( Q e. Q0 /\ Q e. Q0 ) -> ( Q +Q0 Q ) e. Q0 )
76	18, 18, 75	syl2anc 408	. . . . . . 7 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( Q +Q0 Q ) e. Q0 )
77		addassnq0 7270	. . . . . . 7 |- ( ( X e. Q0 /\ ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) e. Q0 /\ ( Q +Q0 Q ) e. Q0 ) -> ( ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) +Q0 ( Q +Q0 Q ) ) = ( X +Q0 ( ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) +Q0 ( Q +Q0 Q ) ) ) )
78	74, 20, 76, 77	syl3anc 1216	. . . . . 6 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) +Q0 ( Q +Q0 Q ) ) = ( X +Q0 ( ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) +Q0 ( Q +Q0 Q ) ) ) )
79	73, 78	eqtr4d 2175	. . . . 5 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( X +Q0 ( ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) +Q0 ( [ <. 2o , 1o >. ] ~Q0 ·Q0 Q ) ) ) = ( ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) +Q0 ( Q +Q0 Q ) ) )
80	55, 69, 79	3eqtrd 2176	. . . 4 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> B = ( ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) +Q0 ( Q +Q0 Q ) ) )
81		oveq1 5781	. . . . . 6 |- ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) -> ( A +Q0 ( Q +Q0 Q ) ) = ( ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) +Q0 ( Q +Q0 Q ) ) )
82	81	eqeq2d 2151	. . . . 5 |- ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) -> ( B = ( A +Q0 ( Q +Q0 Q ) ) <-> B = ( ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) +Q0 ( Q +Q0 Q ) ) ) )
83	5, 82	syl 14	. . . 4 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( B = ( A +Q0 ( Q +Q0 Q ) ) <-> B = ( ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) +Q0 ( Q +Q0 Q ) ) ) )
84	80, 83	mpbird 166	. . 3 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> B = ( A +Q0 ( Q +Q0 Q ) ) )
85	4, 27, 84	3eqtr4rd 2183	. 2 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> B = ( A +Q ( Q +Q Q ) ) )
86		simprlr 527	. . 3 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( Q +Q Q ) <Q P )
87		ltrelnq 7173	. . . . . 6 |- <Q C_ ( Q. X. Q. )
88	87	brel 4591	. . . . 5 |- ( ( Q +Q Q ) <Q P -> ( ( Q +Q Q ) e. Q. /\ P e. Q. ) )
89	86, 88	syl 14	. . . 4 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( ( Q +Q Q ) e. Q. /\ P e. Q. ) )
90		ltanqg 7208	. . . . 5 |- ( ( ( Q +Q Q ) e. Q. /\ P e. Q. /\ A e. Q. ) -> ( ( Q +Q Q ) <Q P <-> ( A +Q ( Q +Q Q ) ) <Q ( A +Q P ) ) )
91	90	3expa 1181	. . . 4 |- ( ( ( ( Q +Q Q ) e. Q. /\ P e. Q. ) /\ A e. Q. ) -> ( ( Q +Q Q ) <Q P <-> ( A +Q ( Q +Q Q ) ) <Q ( A +Q P ) ) )
92	89, 23, 91	syl2anc 408	. . 3 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( ( Q +Q Q ) <Q P <-> ( A +Q ( Q +Q Q ) ) <Q ( A +Q P ) ) )
93	86, 92	mpbid 146	. 2 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> ( A +Q ( Q +Q Q ) ) <Q ( A +Q P ) )
94	85, 93	eqbrtrd 3950	1 |- ( ( ( A = ( X +Q0 ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) ) /\ B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) ) /\ ( ( Q e. Q. /\ ( Q +Q Q ) <Q P ) /\ ( X e. Q. /\ M e. om ) ) ) -> B <Q ( A +Q P ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   =/= wne 2308   (/)c0 3363   <.cop 3530   class class class wbr 3929   omcom 4504   X. cxp 4537  (class class class)co 5774   1oc1o 6306   2oc2o 6307   +o coa 6310   [cec 6427   /.cqs 6428   N.cnpi 7080   ~Q ceq 7087   Q.cnq 7088   +Q cplq 7090   .Q cmq 7091   <Q cltq 7093   ~_Q0 ceq0 7094  Q₀cnq0 7095   +_Q0 cplq0 7097   ·_Q0 cmq0 7098

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-plq0 7235  df-mq0 7236

This theorem is referenced by:  prarloc  7311