Theorem prarloclem5 7441

Description: A substitution of zero for y and N minus two for x. Lemma for prarloc 7444. (Contributed by Jim Kingdon, 4-Nov-2019.)

Assertion

Ref	Expression
prarloclem5	|- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ ( N e. N. /\ P e. Q. /\ 1o <N N ) /\ ( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U ) -> E. x e. om E. y e. om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) )

Distinct variable groups:   x, A, y   x, L, y   x, N   x, P, y   x, U, y

Allowed substitution hint:   N( y)

Proof of Theorem prarloclem5

Step	Hyp	Ref	Expression
1		prarloclemn 7440	. . . 4 |- ( ( N e. N. /\ 1o <N N ) -> E. x e. om ( 2o +o x ) = N )
2	1	3adant2 1006	. . 3 |- ( ( N e. N. /\ P e. Q. /\ 1o <N N ) -> E. x e. om ( 2o +o x ) = N )
3	2	3ad2ant2 1009	. 2 |- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ ( N e. N. /\ P e. Q. /\ 1o <N N ) /\ ( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U ) -> E. x e. om ( 2o +o x ) = N )
4		elprnql 7422	. . . . . . 7 |- ( ( <. L , U >. e. P. /\ A e. L ) -> A e. Q. )
5	4	3ad2ant1 1008	. . . . . 6 |- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ ( N e. N. /\ P e. Q. /\ 1o <N N ) /\ ( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U ) -> A e. Q. )
6		simp22 1021	. . . . . 6 |- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ ( N e. N. /\ P e. Q. /\ 1o <N N ) /\ ( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U ) -> P e. Q. )
7		nqnq0 7382	. . . . . . . . 9 |- Q. C_ Q0
8	7	sseli 3138	. . . . . . . 8 |- ( A e. Q. -> A e. Q0 )
9		nq0a0 7398	. . . . . . . 8 |- ( A e. Q0 -> ( A +Q0 0Q0 ) = A )
10	8, 9	syl 14	. . . . . . 7 |- ( A e. Q. -> ( A +Q0 0Q0 ) = A )
11	7	sseli 3138	. . . . . . . . . 10 |- ( P e. Q. -> P e. Q0 )
12		nq0m0r 7397	. . . . . . . . . 10 |- ( P e. Q0 -> (0Q0 ·Q0 P ) = 0Q0 )
13	11, 12	syl 14	. . . . . . . . 9 |- ( P e. Q. -> (0Q0 ·Q0 P ) = 0Q0 )
14		df-0nq0 7367	. . . . . . . . . 10 |- 0Q0 = [ <. (/) , 1o >. ] ~Q0
15	14	oveq1i 5852	. . . . . . . . 9 |- (0Q0 ·Q0 P ) = ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P )
16	13, 15	eqtr3di 2214	. . . . . . . 8 |- ( P e. Q. -> 0Q0 = ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) )
17	16	oveq2d 5858	. . . . . . 7 |- ( P e. Q. -> ( A +Q0 0Q0 ) = ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) )
18	10, 17	sylan9req 2220	. . . . . 6 |- ( ( A e. Q. /\ P e. Q. ) -> A = ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) )
19	5, 6, 18	syl2anc 409	. . . . 5 |- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ ( N e. N. /\ P e. Q. /\ 1o <N N ) /\ ( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U ) -> A = ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) )
20		simp1r 1012	. . . . 5 |- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ ( N e. N. /\ P e. Q. /\ 1o <N N ) /\ ( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U ) -> A e. L )
21	19, 20	eqeltrrd 2244	. . . 4 |- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ ( N e. N. /\ P e. Q. /\ 1o <N N ) /\ ( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U ) -> ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) e. L )
22		2onn 6489	. . . . . . . . . . . . . . 15 |- 2o e. om
23		nna0r 6446	. . . . . . . . . . . . . . 15 |- ( 2o e. om -> ( (/) +o 2o ) = 2o )
24	22, 23	ax-mp 5	. . . . . . . . . . . . . 14 |- ( (/) +o 2o ) = 2o
25	24	oveq1i 5852	. . . . . . . . . . . . 13 |- ( ( (/) +o 2o ) +o x ) = ( 2o +o x )
26	25	eqeq1i 2173	. . . . . . . . . . . 12 |- ( ( ( (/) +o 2o ) +o x ) = N <-> ( 2o +o x ) = N )
27	26	biimpri 132	. . . . . . . . . . 11 |- ( ( 2o +o x ) = N -> ( ( (/) +o 2o ) +o x ) = N )
28	27	opeq1d 3764	. . . . . . . . . 10 |- ( ( 2o +o x ) = N -> <. ( ( (/) +o 2o ) +o x ) , 1o >. = <. N , 1o >. )
29	28	eceq1d 6537	. . . . . . . . 9 |- ( ( 2o +o x ) = N -> [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q = [ <. N , 1o >. ] ~Q )
30	29	oveq1d 5857	. . . . . . . 8 |- ( ( 2o +o x ) = N -> ( [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) = ( [ <. N , 1o >. ] ~Q .Q P ) )
31	30	oveq2d 5858	. . . . . . 7 |- ( ( 2o +o x ) = N -> ( A +Q ( [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) = ( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) )
32	31	eleq1d 2235	. . . . . 6 |- ( ( 2o +o x ) = N -> ( ( A +Q ( [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U <-> ( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U ) )
33	32	biimprcd 159	. . . . 5 |- ( ( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U -> ( ( 2o +o x ) = N -> ( A +Q ( [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) )
34	33	3ad2ant3 1010	. . . 4 |- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ ( N e. N. /\ P e. Q. /\ 1o <N N ) /\ ( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U ) -> ( ( 2o +o x ) = N -> ( A +Q ( [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) )
35		peano1 4571	. . . . 5 |- (/) e. om
36		opeq1 3758	. . . . . . . . . . 11 |- ( y = (/) -> <. y , 1o >. = <. (/) , 1o >. )
37	36	eceq1d 6537	. . . . . . . . . 10 |- ( y = (/) -> [ <. y , 1o >. ] ~Q0 = [ <. (/) , 1o >. ] ~Q0 )
38	37	oveq1d 5857	. . . . . . . . 9 |- ( y = (/) -> ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) = ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) )
39	38	oveq2d 5858	. . . . . . . 8 |- ( y = (/) -> ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) = ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) )
40	39	eleq1d 2235	. . . . . . 7 |- ( y = (/) -> ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L <-> ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) e. L ) )
41		oveq1 5849	. . . . . . . . . . . . 13 |- ( y = (/) -> ( y +o 2o ) = ( (/) +o 2o ) )
42	41	oveq1d 5857	. . . . . . . . . . . 12 |- ( y = (/) -> ( ( y +o 2o ) +o x ) = ( ( (/) +o 2o ) +o x ) )
43	42	opeq1d 3764	. . . . . . . . . . 11 |- ( y = (/) -> <. ( ( y +o 2o ) +o x ) , 1o >. = <. ( ( (/) +o 2o ) +o x ) , 1o >. )
44	43	eceq1d 6537	. . . . . . . . . 10 |- ( y = (/) -> [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q = [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q )
45	44	oveq1d 5857	. . . . . . . . 9 |- ( y = (/) -> ( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) = ( [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) )
46	45	oveq2d 5858	. . . . . . . 8 |- ( y = (/) -> ( A +Q ( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) = ( A +Q ( [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) )
47	46	eleq1d 2235	. . . . . . 7 |- ( y = (/) -> ( ( A +Q ( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U <-> ( A +Q ( [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) )
48	40, 47	anbi12d 465	. . . . . 6 |- ( y = (/) -> ( ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) <-> ( ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) ) )
49	48	rspcev 2830	. . . . 5 |- ( ( (/) e. om /\ ( ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) ) -> E. y e. om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) )
50	35, 49	mpan 421	. . . 4 |- ( ( ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) -> E. y e. om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) )
51	21, 34, 50	syl6an 1422	. . 3 |- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ ( N e. N. /\ P e. Q. /\ 1o <N N ) /\ ( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U ) -> ( ( 2o +o x ) = N -> E. y e. om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) ) )
52	51	reximdv 2567	. 2 |- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ ( N e. N. /\ P e. Q. /\ 1o <N N ) /\ ( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U ) -> ( E. x e. om ( 2o +o x ) = N -> E. x e. om E. y e. om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) ) )
53	3, 52	mpd 13	1 |- ( ( ( <. L , U >. e. P. /\ A e. L ) /\ ( N e. N. /\ P e. Q. /\ 1o <N N ) /\ ( A +Q ( [ <. N , 1o >. ] ~Q .Q P ) ) e. U ) -> E. x e. om E. y e. om ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L /\ ( A +Q ( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) ) e. U ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   e. wcel 2136   E.wrex 2445   (/)c0 3409   <.cop 3579   class class class wbr 3982   omcom 4567  (class class class)co 5842   1oc1o 6377   2oc2o 6378   +o coa 6381   [cec 6499   N.cnpi 7213   <N clti 7216   ~Q ceq 7220   Q.cnq 7221   +Q cplq 7223   .Q cmq 7224   ~_Q0 ceq0 7227  Q₀cnq0 7228  0_Q0c0q0 7229   +_Q0 cplq0 7230   ·_Q0 cmq0 7231   P.cnp 7232

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-mi 7247  df-lti 7248  df-enq 7288  df-nqqs 7289  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407

This theorem is referenced by:  prarloclem  7442