Theorem prarloclem5 7873

Description: A substitution of zero for y and N minus two for x. Lemma for prarloc 7876. (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 7872	. . . 4 |- ( ( N e. N. /\ 1o <N N ) -> E. x e. om ( 2o +o x ) = N )
2	1	3adant2 1047	. . 3 |- ( ( N e. N. /\ P e. Q. /\ 1o <N N ) -> E. x e. om ( 2o +o x ) = N )
3	2	3ad2ant2 1050	. 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 7854	. . . . . . 7 |- ( ( <. L , U >. e. P. /\ A e. L ) -> A e. Q. )
5	4	3ad2ant1 1049	. . . . . 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 1062	. . . . . 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 7814	. . . . . . . . 9 |- Q. C_ Q0
8	7	sseli 3244	. . . . . . . 8 |- ( A e. Q. -> A e. Q0 )
9		nq0a0 7830	. . . . . . . 8 |- ( A e. Q0 -> ( A +Q0 0Q0 ) = A )
10	8, 9	syl 14	. . . . . . 7 |- ( A e. Q. -> ( A +Q0 0Q0 ) = A )
11	7	sseli 3244	. . . . . . . . . 10 |- ( P e. Q. -> P e. Q0 )
12		nq0m0r 7829	. . . . . . . . . 10 |- ( P e. Q0 -> (0Q0 ·Q0 P ) = 0Q0 )
13	11, 12	syl 14	. . . . . . . . 9 |- ( P e. Q. -> (0Q0 ·Q0 P ) = 0Q0 )
14		df-0nq0 7799	. . . . . . . . . 10 |- 0Q0 = [ <. (/) , 1o >. ] ~Q0
15	14	oveq1i 6098	. . . . . . . . 9 |- (0Q0 ·Q0 P ) = ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P )
16	13, 15	eqtr3di 2286	. . . . . . . 8 |- ( P e. Q. -> 0Q0 = ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) )
17	16	oveq2d 6104	. . . . . . 7 |- ( P e. Q. -> ( A +Q0 0Q0 ) = ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) )
18	10, 17	sylan9req 2292	. . . . . 6 |- ( ( A e. Q. /\ P e. Q. ) -> A = ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) )
19	5, 6, 18	syl2anc 415	. . . . 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 1053	. . . . 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 2316	. . . 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 6798	. . . . . . . . . . . . . . 15 |- 2o e. om
23		nna0r 6755	. . . . . . . . . . . . . . 15 |- ( 2o e. om -> ( (/) +o 2o ) = 2o )
24	22, 23	ax-mp 5	. . . . . . . . . . . . . 14 |- ( (/) +o 2o ) = 2o
25	24	oveq1i 6098	. . . . . . . . . . . . 13 |- ( ( (/) +o 2o ) +o x ) = ( 2o +o x )
26	25	eqeq1i 2246	. . . . . . . . . . . 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 3910	. . . . . . . . . 10 |- ( ( 2o +o x ) = N -> <. ( ( (/) +o 2o ) +o x ) , 1o >. = <. N , 1o >. )
29	28	eceq1d 6847	. . . . . . . . 9 |- ( ( 2o +o x ) = N -> [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q = [ <. N , 1o >. ] ~Q )
30	29	oveq1d 6103	. . . . . . . 8 |- ( ( 2o +o x ) = N -> ( [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) = ( [ <. N , 1o >. ] ~Q .Q P ) )
31	30	oveq2d 6104	. . . . . . 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 2307	. . . . . 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 1051	. . . 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 4743	. . . . 5 |- (/) e. om
36		opeq1 3904	. . . . . . . . . . 11 |- ( y = (/) -> <. y , 1o >. = <. (/) , 1o >. )
37	36	eceq1d 6847	. . . . . . . . . 10 |- ( y = (/) -> [ <. y , 1o >. ] ~Q0 = [ <. (/) , 1o >. ] ~Q0 )
38	37	oveq1d 6103	. . . . . . . . 9 |- ( y = (/) -> ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) = ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) )
39	38	oveq2d 6104	. . . . . . . 8 |- ( y = (/) -> ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) = ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) )
40	39	eleq1d 2307	. . . . . . 7 |- ( y = (/) -> ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L <-> ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) e. L ) )
41		oveq1 6095	. . . . . . . . . . . . 13 |- ( y = (/) -> ( y +o 2o ) = ( (/) +o 2o ) )
42	41	oveq1d 6103	. . . . . . . . . . . 12 |- ( y = (/) -> ( ( y +o 2o ) +o x ) = ( ( (/) +o 2o ) +o x ) )
43	42	opeq1d 3910	. . . . . . . . . . 11 |- ( y = (/) -> <. ( ( y +o 2o ) +o x ) , 1o >. = <. ( ( (/) +o 2o ) +o x ) , 1o >. )
44	43	eceq1d 6847	. . . . . . . . . 10 |- ( y = (/) -> [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q = [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q )
45	44	oveq1d 6103	. . . . . . . . 9 |- ( y = (/) -> ( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) = ( [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) )
46	45	oveq2d 6104	. . . . . . . 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 2307	. . . . . . 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 477	. . . . . 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 2929	. . . . 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 428	. . . 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 1483	. . 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 2651	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   (/)c0 3520   <.cop 3712   class class class wbr 4131   omcom 4739  (class class class)co 6088   1oc1o 6684   2oc2o 6685   +o coa 6688   [cec 6809   N.cnpi 7645   <N clti 7648   ~Q ceq 7652   Q.cnq 7653   +Q cplq 7655   .Q cmq 7656   ~_Q0 ceq0 7659  Q₀cnq0 7660  0_Q0c0q0 7661   +_Q0 cplq0 7662   ·_Q0 cmq0 7663   P.cnp 7664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-2o 6692  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-mi 7679  df-lti 7680  df-enq 7720  df-nqqs 7721  df-enq0 7797  df-nq0 7798  df-0nq0 7799  df-plq0 7800  df-mq0 7801  df-inp 7839

This theorem is used by:  prarloclem  7874