Theorem prarloclemcalc 7586

Description: Some calculations for prarloc 7587. (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 7538	. . . . 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 5941	. . 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 7399	. . . . . . . . . . 11 |- 1o e. N.
9		opelxpi 4696	. . . . . . . . . . 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 7523	. . . . . . . . . . 11 |- ~Q0 e. _V
12	11	ecelqsi 6657	. . . . . . . . . 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 7509	. . . . . . . . 9 |- Q0 = ( ( om X. N. ) /. ~Q0 )
15	13, 14	eleqtrrdi 2290	. . . . . . . 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 7525	. . . . . . . 8 |- Q. C_ Q0
18	17, 1	sselid 3182	. . . . . . 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 7536	. . . . . . 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 7537	. . . . . 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 2273	. . . 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 7459	. . . . 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 7538	. . . 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 6588	. . . . . . . . . . . . . 14 |- 2o e. om
30		2on0 6493	. . . . . . . . . . . . . 14 |- 2o =/= (/)
31		elni 7392	. . . . . . . . . . . . . 14 |- ( 2o e. N. <-> ( 2o e. om /\ 2o =/= (/) ) )
32	29, 30, 31	mpbir2an 944	. . . . . . . . . . . . 13 |- 2o e. N.
33		nnppipi 7427	. . . . . . . . . . . . 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 4696	. . . . . . . . . . . 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 7444	. . . . . . . . . . . 12 |- ~Q e. _V
38	37	ecelqsi 6657	. . . . . . . . . . 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 7432	. . . . . . . . . 10 |- Q. = ( ( N. X. N. ) /. ~Q )
41	39, 40	eleqtrrdi 2290	. . . . . . . . 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 7460	. . . . . . . 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 7538	. . . . . . 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 7539	. . . . . . . . 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 7522	. . . . . . . . . . 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 5940	. . . . . . . 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 2232	. . . . . . 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 5941	. . . . . 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 2233	. . . . 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 7542	. . . . . . . . . 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 1337	. . . . . . . . 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 5940	. . . . . . 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 4696	. . . . . . . . . . . 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 6657	. . . . . . . . . . 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 2272	. . . . . . . . 9 |- [ <. 2o , 1o >. ] ~Q0 e. Q0
65		distnq0r 7547	. . . . . . . . 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 1337	. . . . . . . 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 2229	. . . . . 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 5941	. . . . 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 7549	. . . . . . . . 9 |- ( Q e. Q0 -> ( [ <. 2o , 1o >. ] ~Q0 ·Q0 Q ) = ( Q +Q0 Q ) )
71	70	oveq2d 5941	. . . . . . . 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 5941	. . . . . . 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 3182	. . . . . . 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 7535	. . . . . . . 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 7546	. . . . . . 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 1249	. . . . . 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 2232	. . . . 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 2233	. . . 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 5932	. . . . . 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 2208	. . . . 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 2240	. 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 7449	. . . . . 6 |- <Q C_ ( Q. X. Q. )
88	87	brel 4716	. . . . 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 7484	. . . . 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 1205	. . . 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 4056	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 1364   e. wcel 2167   =/= wne 2367   (/)c0 3451   <.cop 3626   class class class wbr 4034   omcom 4627   X. cxp 4662  (class class class)co 5925   1oc1o 6476   2oc2o 6477   +o coa 6480   [cec 6599   /.cqs 6600   N.cnpi 7356   ~Q ceq 7363   Q.cnq 7364   +Q cplq 7366   .Q cmq 7367   <Q cltq 7369   ~_Q0 ceq0 7370  Q₀cnq0 7371   +_Q0 cplq0 7373   ·_Q0 cmq0 7374

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-plq0 7511  df-mq0 7512

This theorem is referenced by:  prarloc  7587