Theorem prarloclem5 7584

Description: A substitution of zero for y and N minus two for x. Lemma for prarloc 7587. (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 7583	. . . 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 7565	. . . . . . 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 7525	. . . . . . . . 9 |- Q. C_ Q0
8	7	sseli 3180	. . . . . . . 8 |- ( A e. Q. -> A e. Q0 )
9		nq0a0 7541	. . . . . . . 8 |- ( A e. Q0 -> ( A +Q0 0Q0 ) = A )
10	8, 9	syl 14	. . . . . . 7 |- ( A e. Q. -> ( A +Q0 0Q0 ) = A )
11	7	sseli 3180	. . . . . . . . . 10 |- ( P e. Q. -> P e. Q0 )
12		nq0m0r 7540	. . . . . . . . . 10 |- ( P e. Q0 -> (0Q0 ·Q0 P ) = 0Q0 )
13	11, 12	syl 14	. . . . . . . . 9 |- ( P e. Q. -> (0Q0 ·Q0 P ) = 0Q0 )
14		df-0nq0 7510	. . . . . . . . . 10 |- 0Q0 = [ <. (/) , 1o >. ] ~Q0
15	14	oveq1i 5935	. . . . . . . . 9 |- (0Q0 ·Q0 P ) = ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P )
16	13, 15	eqtr3di 2244	. . . . . . . 8 |- ( P e. Q. -> 0Q0 = ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) )
17	16	oveq2d 5941	. . . . . . 7 |- ( P e. Q. -> ( A +Q0 0Q0 ) = ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) )
18	10, 17	sylan9req 2250	. . . . . 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 2274	. . . 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 6588	. . . . . . . . . . . . . . 15 |- 2o e. om
23		nna0r 6545	. . . . . . . . . . . . . . 15 |- ( 2o e. om -> ( (/) +o 2o ) = 2o )
24	22, 23	ax-mp 5	. . . . . . . . . . . . . 14 |- ( (/) +o 2o ) = 2o
25	24	oveq1i 5935	. . . . . . . . . . . . 13 |- ( ( (/) +o 2o ) +o x ) = ( 2o +o x )
26	25	eqeq1i 2204	. . . . . . . . . . . 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 3815	. . . . . . . . . 10 |- ( ( 2o +o x ) = N -> <. ( ( (/) +o 2o ) +o x ) , 1o >. = <. N , 1o >. )
29	28	eceq1d 6637	. . . . . . . . 9 |- ( ( 2o +o x ) = N -> [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q = [ <. N , 1o >. ] ~Q )
30	29	oveq1d 5940	. . . . . . . 8 |- ( ( 2o +o x ) = N -> ( [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) = ( [ <. N , 1o >. ] ~Q .Q P ) )
31	30	oveq2d 5941	. . . . . . 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 2265	. . . . . 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 4631	. . . . 5 |- (/) e. om
36		opeq1 3809	. . . . . . . . . . 11 |- ( y = (/) -> <. y , 1o >. = <. (/) , 1o >. )
37	36	eceq1d 6637	. . . . . . . . . 10 |- ( y = (/) -> [ <. y , 1o >. ] ~Q0 = [ <. (/) , 1o >. ] ~Q0 )
38	37	oveq1d 5940	. . . . . . . . 9 |- ( y = (/) -> ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) = ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) )
39	38	oveq2d 5941	. . . . . . . 8 |- ( y = (/) -> ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) = ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) )
40	39	eleq1d 2265	. . . . . . 7 |- ( y = (/) -> ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L <-> ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) e. L ) )
41		oveq1 5932	. . . . . . . . . . . . 13 |- ( y = (/) -> ( y +o 2o ) = ( (/) +o 2o ) )
42	41	oveq1d 5940	. . . . . . . . . . . 12 |- ( y = (/) -> ( ( y +o 2o ) +o x ) = ( ( (/) +o 2o ) +o x ) )
43	42	opeq1d 3815	. . . . . . . . . . 11 |- ( y = (/) -> <. ( ( y +o 2o ) +o x ) , 1o >. = <. ( ( (/) +o 2o ) +o x ) , 1o >. )
44	43	eceq1d 6637	. . . . . . . . . 10 |- ( y = (/) -> [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q = [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q )
45	44	oveq1d 5940	. . . . . . . . 9 |- ( y = (/) -> ( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) = ( [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) )
46	45	oveq2d 5941	. . . . . . . 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 2265	. . . . . . 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 2868	. . . . 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 2598	. 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 2167   E.wrex 2476   (/)c0 3451   <.cop 3626   class class class wbr 4034   omcom 4627  (class class class)co 5925   1oc1o 6476   2oc2o 6477   +o coa 6480   [cec 6599   N.cnpi 7356   <N clti 7359   ~Q ceq 7363   Q.cnq 7364   +Q cplq 7366   .Q cmq 7367   ~_Q0 ceq0 7370  Q₀cnq0 7371  0_Q0c0q0 7372   +_Q0 cplq0 7373   ·_Q0 cmq0 7374   P.cnp 7375

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-mi 7390  df-lti 7391  df-enq 7431  df-nqqs 7432  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550

This theorem is referenced by:  prarloclem  7585