Theorem prsrriota 8119

Description: Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)

Assertion

Ref	Expression
prsrriota	|- ( ( A e. R. /\ 0R <R A ) -> [ <. ( ( iota_ x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) +P. 1P ) , 1P >. ] ~R = A )

Distinct variable group:   x, A

Proof of Theorem prsrriota

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		srpospr 8114	. . 3 |- ( ( A e. R. /\ 0R <R A ) -> E! y e. P. [ <. ( y +P. 1P ) , 1P >. ] ~R = A )
2		reurex 2765	. . 3 |- ( E! y e. P. [ <. ( y +P. 1P ) , 1P >. ] ~R = A -> E. y e. P. [ <. ( y +P. 1P ) , 1P >. ] ~R = A )
3	1, 2	syl 14	. 2 |- ( ( A e. R. /\ 0R <R A ) -> E. y e. P. [ <. ( y +P. 1P ) , 1P >. ] ~R = A )
4		simprr 533	. . . . 5 |- ( ( ( A e. R. /\ 0R <R A ) /\ ( y e. P. /\ [ <. ( y +P. 1P ) , 1P >. ] ~R = A ) ) -> [ <. ( y +P. 1P ) , 1P >. ] ~R = A )
5		simprl 531	. . . . . 6 |- ( ( ( A e. R. /\ 0R <R A ) /\ ( y e. P. /\ [ <. ( y +P. 1P ) , 1P >. ] ~R = A ) ) -> y e. P. )
6		srpospr 8114	. . . . . . 7 |- ( ( A e. R. /\ 0R <R A ) -> E! x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A )
7	6	adantr 276	. . . . . 6 |- ( ( ( A e. R. /\ 0R <R A ) /\ ( y e. P. /\ [ <. ( y +P. 1P ) , 1P >. ] ~R = A ) ) -> E! x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A )
8		oveq1 6065	. . . . . . . . . 10 |- ( x = y -> ( x +P. 1P ) = ( y +P. 1P ) )
9	8	opeq1d 3894	. . . . . . . . 9 |- ( x = y -> <. ( x +P. 1P ) , 1P >. = <. ( y +P. 1P ) , 1P >. )
10	9	eceq1d 6816	. . . . . . . 8 |- ( x = y -> [ <. ( x +P. 1P ) , 1P >. ] ~R = [ <. ( y +P. 1P ) , 1P >. ] ~R )
11	10	eqeq1d 2243	. . . . . . 7 |- ( x = y -> ( [ <. ( x +P. 1P ) , 1P >. ] ~R = A <-> [ <. ( y +P. 1P ) , 1P >. ] ~R = A ) )
12	11	riota2 6035	. . . . . 6 |- ( ( y e. P. /\ E! x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) -> ( [ <. ( y +P. 1P ) , 1P >. ] ~R = A <-> ( iota_ x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) = y ) )
13	5, 7, 12	syl2anc 411	. . . . 5 |- ( ( ( A e. R. /\ 0R <R A ) /\ ( y e. P. /\ [ <. ( y +P. 1P ) , 1P >. ] ~R = A ) ) -> ( [ <. ( y +P. 1P ) , 1P >. ] ~R = A <-> ( iota_ x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) = y ) )
14	4, 13	mpbid 147	. . . 4 |- ( ( ( A e. R. /\ 0R <R A ) /\ ( y e. P. /\ [ <. ( y +P. 1P ) , 1P >. ] ~R = A ) ) -> ( iota_ x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) = y )
15		oveq1 6065	. . . . . 6 |- ( ( iota_ x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) = y -> ( ( iota_ x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) +P. 1P ) = ( y +P. 1P ) )
16	15	opeq1d 3894	. . . . 5 |- ( ( iota_ x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) = y -> <. ( ( iota_ x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) +P. 1P ) , 1P >. = <. ( y +P. 1P ) , 1P >. )
17	16	eceq1d 6816	. . . 4 |- ( ( iota_ x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) = y -> [ <. ( ( iota_ x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) +P. 1P ) , 1P >. ] ~R = [ <. ( y +P. 1P ) , 1P >. ] ~R )
18	14, 17	syl 14	. . 3 |- ( ( ( A e. R. /\ 0R <R A ) /\ ( y e. P. /\ [ <. ( y +P. 1P ) , 1P >. ] ~R = A ) ) -> [ <. ( ( iota_ x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) +P. 1P ) , 1P >. ] ~R = [ <. ( y +P. 1P ) , 1P >. ] ~R )
19	18, 4	eqtrd 2267	. 2 |- ( ( ( A e. R. /\ 0R <R A ) /\ ( y e. P. /\ [ <. ( y +P. 1P ) , 1P >. ] ~R = A ) ) -> [ <. ( ( iota_ x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) +P. 1P ) , 1P >. ] ~R = A )
20	3, 19	rexlimddv 2667	1 |- ( ( A e. R. /\ 0R <R A ) -> [ <. ( ( iota_ x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) +P. 1P ) , 1P >. ] ~R = A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   E.wrex 2523   E!wreu 2524   <.cop 3697   class class class wbr 4114   iota_crio 6010  (class class class)co 6058   [cec 6778   P.cnp 7622   1Pc1p 7623   +P. cpp 7624   ~R cer 7627   R.cnr 7628   0Rc0r 7629   <R cltr 7634

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684  df-enq0 7755  df-nq0 7756  df-0nq0 7757  df-plq0 7758  df-mq0 7759  df-inp 7797  df-i1p 7798  df-iplp 7799  df-iltp 7801  df-enr 8057  df-nr 8058  df-ltr 8061  df-0r 8062

This theorem is referenced by:  caucvgsrlemfv  8122  caucvgsrlemgt1  8126