Theorem prarloclem5 7560

Description: A substitution of zero for y and N minus two for x. Lemma for prarloc 7563. (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 7559	. . . 4 |- ( ( N e. N. /\ 1o <N N ) -> E. x e. om ( 2o +o x ) = N )
2	1	3adant2 1018	. . 3 |- ( ( N e. N. /\ P e. Q. /\ 1o <N N ) -> E. x e. om ( 2o +o x ) = N )
3	2	3ad2ant2 1021	. 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 7541	. . . . . . 7 |- ( ( <. L , U >. e. P. /\ A e. L ) -> A e. Q. )
5	4	3ad2ant1 1020	. . . . . 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 1033	. . . . . 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 7501	. . . . . . . . 9 |- Q. C_ Q0
8	7	sseli 3175	. . . . . . . 8 |- ( A e. Q. -> A e. Q0 )
9		nq0a0 7517	. . . . . . . 8 |- ( A e. Q0 -> ( A +Q0 0Q0 ) = A )
10	8, 9	syl 14	. . . . . . 7 |- ( A e. Q. -> ( A +Q0 0Q0 ) = A )
11	7	sseli 3175	. . . . . . . . . 10 |- ( P e. Q. -> P e. Q0 )
12		nq0m0r 7516	. . . . . . . . . 10 |- ( P e. Q0 -> (0Q0 ·Q0 P ) = 0Q0 )
13	11, 12	syl 14	. . . . . . . . 9 |- ( P e. Q. -> (0Q0 ·Q0 P ) = 0Q0 )
14		df-0nq0 7486	. . . . . . . . . 10 |- 0Q0 = [ <. (/) , 1o >. ] ~Q0
15	14	oveq1i 5928	. . . . . . . . 9 |- (0Q0 ·Q0 P ) = ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P )
16	13, 15	eqtr3di 2241	. . . . . . . 8 |- ( P e. Q. -> 0Q0 = ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) )
17	16	oveq2d 5934	. . . . . . 7 |- ( P e. Q. -> ( A +Q0 0Q0 ) = ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) )
18	10, 17	sylan9req 2247	. . . . . 6 |- ( ( A e. Q. /\ P e. Q. ) -> A = ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) )
19	5, 6, 18	syl2anc 411	. . . . 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 1024	. . . . 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 2271	. . . 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 6574	. . . . . . . . . . . . . . 15 |- 2o e. om
23		nna0r 6531	. . . . . . . . . . . . . . 15 |- ( 2o e. om -> ( (/) +o 2o ) = 2o )
24	22, 23	ax-mp 5	. . . . . . . . . . . . . 14 |- ( (/) +o 2o ) = 2o
25	24	oveq1i 5928	. . . . . . . . . . . . 13 |- ( ( (/) +o 2o ) +o x ) = ( 2o +o x )
26	25	eqeq1i 2201	. . . . . . . . . . . 12 |- ( ( ( (/) +o 2o ) +o x ) = N <-> ( 2o +o x ) = N )
27	26	biimpri 133	. . . . . . . . . . 11 |- ( ( 2o +o x ) = N -> ( ( (/) +o 2o ) +o x ) = N )
28	27	opeq1d 3810	. . . . . . . . . 10 |- ( ( 2o +o x ) = N -> <. ( ( (/) +o 2o ) +o x ) , 1o >. = <. N , 1o >. )
29	28	eceq1d 6623	. . . . . . . . 9 |- ( ( 2o +o x ) = N -> [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q = [ <. N , 1o >. ] ~Q )
30	29	oveq1d 5933	. . . . . . . 8 |- ( ( 2o +o x ) = N -> ( [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) = ( [ <. N , 1o >. ] ~Q .Q P ) )
31	30	oveq2d 5934	. . . . . . 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 2262	. . . . . 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 160	. . . . 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 1022	. . . 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 4626	. . . . 5 |- (/) e. om
36		opeq1 3804	. . . . . . . . . . 11 |- ( y = (/) -> <. y , 1o >. = <. (/) , 1o >. )
37	36	eceq1d 6623	. . . . . . . . . 10 |- ( y = (/) -> [ <. y , 1o >. ] ~Q0 = [ <. (/) , 1o >. ] ~Q0 )
38	37	oveq1d 5933	. . . . . . . . 9 |- ( y = (/) -> ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) = ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) )
39	38	oveq2d 5934	. . . . . . . 8 |- ( y = (/) -> ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) = ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) )
40	39	eleq1d 2262	. . . . . . 7 |- ( y = (/) -> ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L <-> ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) e. L ) )
41		oveq1 5925	. . . . . . . . . . . . 13 |- ( y = (/) -> ( y +o 2o ) = ( (/) +o 2o ) )
42	41	oveq1d 5933	. . . . . . . . . . . 12 |- ( y = (/) -> ( ( y +o 2o ) +o x ) = ( ( (/) +o 2o ) +o x ) )
43	42	opeq1d 3810	. . . . . . . . . . 11 |- ( y = (/) -> <. ( ( y +o 2o ) +o x ) , 1o >. = <. ( ( (/) +o 2o ) +o x ) , 1o >. )
44	43	eceq1d 6623	. . . . . . . . . 10 |- ( y = (/) -> [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q = [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q )
45	44	oveq1d 5933	. . . . . . . . 9 |- ( y = (/) -> ( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) = ( [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) )
46	45	oveq2d 5934	. . . . . . . 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 2262	. . . . . . 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 473	. . . . . 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 2864	. . . . 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 424	. . . 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 1445	. . 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 2595	. 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 104   /\ w3a 980   = wceq 1364   e. wcel 2164   E.wrex 2473   (/)c0 3446   <.cop 3621   class class class wbr 4029   omcom 4622  (class class class)co 5918   1oc1o 6462   2oc2o 6463   +o coa 6466   [cec 6585   N.cnpi 7332   <N clti 7335   ~Q ceq 7339   Q.cnq 7340   +Q cplq 7342   .Q cmq 7343   ~_Q0 ceq0 7346  Q₀cnq0 7347  0_Q0c0q0 7348   +_Q0 cplq0 7349   ·_Q0 cmq0 7350   P.cnp 7351

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 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

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 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-mi 7366  df-lti 7367  df-enq 7407  df-nqqs 7408  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526

This theorem is referenced by:  prarloclem  7561