Theorem prarloclemcalc 7492

Description: Some calculations for prarloc 7493. (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 537	. . . . 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 7444	. . . . 5 |- ( ( Q e. Q. /\ Q e. Q. ) -> ( Q +Q Q ) = ( Q +Q0 Q ) )
3	1, 1, 2	syl2anc 411	. . . 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 5885	. . 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 527	. . . . 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 539	. . . . . 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 540	. . . . . . . 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 7305	. . . . . . . . . . 11 |- 1o e. N.
9		opelxpi 4655	. . . . . . . . . . 11 |- ( ( M e. om /\ 1o e. N. ) -> <. M , 1o >. e. ( om X. N. ) )
10	8, 9	mpan2 425	. . . . . . . . . 10 |- ( M e. om -> <. M , 1o >. e. ( om X. N. ) )
11		enq0ex 7429	. . . . . . . . . . 11 |- ~Q0 e. _V
12	11	ecelqsi 6583	. . . . . . . . . 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 7415	. . . . . . . . 9 |- Q0 = ( ( om X. N. ) /. ~Q0 )
15	13, 14	eleqtrrdi 2271	. . . . . . . 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 7431	. . . . . . . 8 |- Q. C_ Q0
18	17, 1	sselid 3153	. . . . . . 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 7442	. . . . . . 7 |- ( ( [ <. M , 1o >. ] ~Q0 e. Q0 /\ Q e. Q0 ) -> ( [ <. M , 1o >. ] ~Q0 ·Q0 Q ) e. Q0 )
20	16, 18, 19	syl2anc 411	. . . . . 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 7443	. . . . . 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 411	. . . . 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 2254	. . . 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 7365	. . . . 5 |- ( ( Q e. Q. /\ Q e. Q. ) -> ( Q +Q Q ) e. Q. )
25	1, 1, 24	syl2anc 411	. . . 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 7444	. . . 4 |- ( ( A e. Q. /\ ( Q +Q Q ) e. Q. ) -> ( A +Q ( Q +Q Q ) ) = ( A +Q0 ( Q +Q Q ) ) )
27	23, 25, 26	syl2anc 411	. . 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 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 ) ) ) -> B = ( X +Q ( [ <. ( M +o 2o ) , 1o >. ] ~Q .Q Q ) ) )
29		2onn 6516	. . . . . . . . . . . . . 14 |- 2o e. om
30		2on0 6421	. . . . . . . . . . . . . 14 |- 2o =/= (/)
31		elni 7298	. . . . . . . . . . . . . 14 |- ( 2o e. N. <-> ( 2o e. om /\ 2o =/= (/) ) )
32	29, 30, 31	mpbir2an 942	. . . . . . . . . . . . 13 |- 2o e. N.
33		nnppipi 7333	. . . . . . . . . . . . 13 |- ( ( M e. om /\ 2o e. N. ) -> ( M +o 2o ) e. N. )
34	32, 33	mpan2 425	. . . . . . . . . . . 12 |- ( M e. om -> ( M +o 2o ) e. N. )
35		opelxpi 4655	. . . . . . . . . . . 12 |- ( ( ( M +o 2o ) e. N. /\ 1o e. N. ) -> <. ( M +o 2o ) , 1o >. e. ( N. X. N. ) )
36	34, 8, 35	sylancl 413	. . . . . . . . . . 11 |- ( M e. om -> <. ( M +o 2o ) , 1o >. e. ( N. X. N. ) )
37		enqex 7350	. . . . . . . . . . . 12 |- ~Q e. _V
38	37	ecelqsi 6583	. . . . . . . . . . 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 7338	. . . . . . . . . 10 |- Q. = ( ( N. X. N. ) /. ~Q )
41	39, 40	eleqtrrdi 2271	. . . . . . . . 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 7366	. . . . . . . 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 411	. . . . . . 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 7444	. . . . . . 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 411	. . . . . 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 7445	. . . . . . . . 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 411	. . . . . . . 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 7428	. . . . . . . . . . 11 |- ( ( ( M +o 2o ) e. N. /\ 1o e. N. ) -> [ <. ( M +o 2o ) , 1o >. ] ~Q0 = [ <. ( M +o 2o ) , 1o >. ] ~Q )
50	34, 8, 49	sylancl 413	. . . . . . . . . 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 5884	. . . . . . . 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 2213	. . . . . . 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 5885	. . . . . 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 2214	. . . . 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 7448	. . . . . . . . . 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 1326	. . . . . . . . 9 |- ( ( M e. om /\ 2o e. om ) -> [ <. ( M +o 2o ) , 1o >. ] ~Q0 = ( [ <. M , 1o >. ] ~Q0 +Q0 [ <. 2o , 1o >. ] ~Q0 ) )
58	7, 29, 57	sylancl 413	. . . . . . . 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 5884	. . . . . . 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 4655	. . . . . . . . . . . 12 |- ( ( 2o e. om /\ 1o e. N. ) -> <. 2o , 1o >. e. ( om X. N. ) )
61	29, 8, 60	mp2an 426	. . . . . . . . . . 11 |- <. 2o , 1o >. e. ( om X. N. )
62	11	ecelqsi 6583	. . . . . . . . . . 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 2253	. . . . . . . . 9 |- [ <. 2o , 1o >. ] ~Q0 e. Q0
65		distnq0r 7453	. . . . . . . . 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 1326	. . . . . . . 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 411	. . . . . . 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 2210	. . . . . 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 5885	. . . . 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 7455	. . . . . . . . 9 |- ( Q e. Q0 -> ( [ <. 2o , 1o >. ] ~Q0 ·Q0 Q ) = ( Q +Q0 Q ) )
71	70	oveq2d 5885	. . . . . . . 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 5885	. . . . . . 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	sselid 3153	. . . . . . 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 7441	. . . . . . . 8 |- ( ( Q e. Q0 /\ Q e. Q0 ) -> ( Q +Q0 Q ) e. Q0 )
76	18, 18, 75	syl2anc 411	. . . . . . 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 7452	. . . . . . 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 1238	. . . . . 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 2213	. . . . 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 2214	. . . 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 5876	. . . . . 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 2189	. . . . 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 167	. . 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 2221	. 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 538	. . 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 7355	. . . . . 6 |- <Q C_ ( Q. X. Q. )
88	87	brel 4675	. . . . 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 7390	. . . . 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 1203	. . . 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 411	. . 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 147	. 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 4022	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 104   <-> wb 105   = wceq 1353   e. wcel 2148   =/= wne 2347   (/)c0 3422   <.cop 3594   class class class wbr 4000   omcom 4586   X. cxp 4621  (class class class)co 5869   1oc1o 6404   2oc2o 6405   +o coa 6408   [cec 6527   /.cqs 6528   N.cnpi 7262   ~Q ceq 7269   Q.cnq 7270   +Q cplq 7272   .Q cmq 7273   <Q cltq 7275   ~_Q0 ceq0 7276  Q₀cnq0 7277   +_Q0 cplq0 7279   ·_Q0 cmq0 7280

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-plq0 7417  df-mq0 7418

This theorem is referenced by:  prarloc  7493