Theorem prarloclem5 7648

Description: A substitution of zero for y and N minus two for x. Lemma for prarloc 7651. (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 7647	. . . 4 |- ( ( N e. N. /\ 1o <N N ) -> E. x e. om ( 2o +o x ) = N )
2	1	3adant2 1019	. . 3 |- ( ( N e. N. /\ P e. Q. /\ 1o <N N ) -> E. x e. om ( 2o +o x ) = N )
3	2	3ad2ant2 1022	. 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 7629	. . . . . . 7 |- ( ( <. L , U >. e. P. /\ A e. L ) -> A e. Q. )
5	4	3ad2ant1 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 ) -> A e. Q. )
6		simp22 1034	. . . . . 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 7589	. . . . . . . . 9 |- Q. C_ Q0
8	7	sseli 3197	. . . . . . . 8 |- ( A e. Q. -> A e. Q0 )
9		nq0a0 7605	. . . . . . . 8 |- ( A e. Q0 -> ( A +Q0 0Q0 ) = A )
10	8, 9	syl 14	. . . . . . 7 |- ( A e. Q. -> ( A +Q0 0Q0 ) = A )
11	7	sseli 3197	. . . . . . . . . 10 |- ( P e. Q. -> P e. Q0 )
12		nq0m0r 7604	. . . . . . . . . 10 |- ( P e. Q0 -> (0Q0 ·Q0 P ) = 0Q0 )
13	11, 12	syl 14	. . . . . . . . 9 |- ( P e. Q. -> (0Q0 ·Q0 P ) = 0Q0 )
14		df-0nq0 7574	. . . . . . . . . 10 |- 0Q0 = [ <. (/) , 1o >. ] ~Q0
15	14	oveq1i 5977	. . . . . . . . 9 |- (0Q0 ·Q0 P ) = ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P )
16	13, 15	eqtr3di 2255	. . . . . . . 8 |- ( P e. Q. -> 0Q0 = ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) )
17	16	oveq2d 5983	. . . . . . 7 |- ( P e. Q. -> ( A +Q0 0Q0 ) = ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) )
18	10, 17	sylan9req 2261	. . . . . 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 1025	. . . . 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 2285	. . . 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 6630	. . . . . . . . . . . . . . 15 |- 2o e. om
23		nna0r 6587	. . . . . . . . . . . . . . 15 |- ( 2o e. om -> ( (/) +o 2o ) = 2o )
24	22, 23	ax-mp 5	. . . . . . . . . . . . . 14 |- ( (/) +o 2o ) = 2o
25	24	oveq1i 5977	. . . . . . . . . . . . 13 |- ( ( (/) +o 2o ) +o x ) = ( 2o +o x )
26	25	eqeq1i 2215	. . . . . . . . . . . 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 3839	. . . . . . . . . 10 |- ( ( 2o +o x ) = N -> <. ( ( (/) +o 2o ) +o x ) , 1o >. = <. N , 1o >. )
29	28	eceq1d 6679	. . . . . . . . 9 |- ( ( 2o +o x ) = N -> [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q = [ <. N , 1o >. ] ~Q )
30	29	oveq1d 5982	. . . . . . . 8 |- ( ( 2o +o x ) = N -> ( [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) = ( [ <. N , 1o >. ] ~Q .Q P ) )
31	30	oveq2d 5983	. . . . . . 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 2276	. . . . . 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 1023	. . . 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 4660	. . . . 5 |- (/) e. om
36		opeq1 3833	. . . . . . . . . . 11 |- ( y = (/) -> <. y , 1o >. = <. (/) , 1o >. )
37	36	eceq1d 6679	. . . . . . . . . 10 |- ( y = (/) -> [ <. y , 1o >. ] ~Q0 = [ <. (/) , 1o >. ] ~Q0 )
38	37	oveq1d 5982	. . . . . . . . 9 |- ( y = (/) -> ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) = ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) )
39	38	oveq2d 5983	. . . . . . . 8 |- ( y = (/) -> ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) = ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) )
40	39	eleq1d 2276	. . . . . . 7 |- ( y = (/) -> ( ( A +Q0 ( [ <. y , 1o >. ] ~Q0 ·Q0 P ) ) e. L <-> ( A +Q0 ( [ <. (/) , 1o >. ] ~Q0 ·Q0 P ) ) e. L ) )
41		oveq1 5974	. . . . . . . . . . . . 13 |- ( y = (/) -> ( y +o 2o ) = ( (/) +o 2o ) )
42	41	oveq1d 5982	. . . . . . . . . . . 12 |- ( y = (/) -> ( ( y +o 2o ) +o x ) = ( ( (/) +o 2o ) +o x ) )
43	42	opeq1d 3839	. . . . . . . . . . 11 |- ( y = (/) -> <. ( ( y +o 2o ) +o x ) , 1o >. = <. ( ( (/) +o 2o ) +o x ) , 1o >. )
44	43	eceq1d 6679	. . . . . . . . . 10 |- ( y = (/) -> [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q = [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q )
45	44	oveq1d 5982	. . . . . . . . 9 |- ( y = (/) -> ( [ <. ( ( y +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) = ( [ <. ( ( (/) +o 2o ) +o x ) , 1o >. ] ~Q .Q P ) )
46	45	oveq2d 5983	. . . . . . . 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 2276	. . . . . . 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 2884	. . . . 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 1454	. . 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 2609	. 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 981   = wceq 1373   e. wcel 2178   E.wrex 2487   (/)c0 3468   <.cop 3646   class class class wbr 4059   omcom 4656  (class class class)co 5967   1oc1o 6518   2oc2o 6519   +o coa 6522   [cec 6641   N.cnpi 7420   <N clti 7423   ~Q ceq 7427   Q.cnq 7428   +Q cplq 7430   .Q cmq 7431   ~_Q0 ceq0 7434  Q₀cnq0 7435  0_Q0c0q0 7436   +_Q0 cplq0 7437   ·_Q0 cmq0 7438   P.cnp 7439

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-eprel 4354  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-1o 6525  df-2o 6526  df-oadd 6529  df-omul 6530  df-er 6643  df-ec 6645  df-qs 6649  df-ni 7452  df-mi 7454  df-lti 7455  df-enq 7495  df-nqqs 7496  df-enq0 7572  df-nq0 7573  df-0nq0 7574  df-plq0 7575  df-mq0 7576  df-inp 7614

This theorem is referenced by:  prarloclem  7649