Theorem prsrriota 7872

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 7867	. . 3 |- ( ( A e. R. /\ 0R <R A ) -> E! y e. P. [ <. ( y +P. 1P ) , 1P >. ] ~R = A )
2		reurex 2715	. . 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 531	. . . . 5 |- ( ( ( A e. R. /\ 0R <R A ) /\ ( y e. P. /\ [ <. ( y +P. 1P ) , 1P >. ] ~R = A ) ) -> [ <. ( y +P. 1P ) , 1P >. ] ~R = A )
5		simprl 529	. . . . . 6 |- ( ( ( A e. R. /\ 0R <R A ) /\ ( y e. P. /\ [ <. ( y +P. 1P ) , 1P >. ] ~R = A ) ) -> y e. P. )
6		srpospr 7867	. . . . . . 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 5932	. . . . . . . . . 10 |- ( x = y -> ( x +P. 1P ) = ( y +P. 1P ) )
9	8	opeq1d 3815	. . . . . . . . 9 |- ( x = y -> <. ( x +P. 1P ) , 1P >. = <. ( y +P. 1P ) , 1P >. )
10	9	eceq1d 6637	. . . . . . . 8 |- ( x = y -> [ <. ( x +P. 1P ) , 1P >. ] ~R = [ <. ( y +P. 1P ) , 1P >. ] ~R )
11	10	eqeq1d 2205	. . . . . . 7 |- ( x = y -> ( [ <. ( x +P. 1P ) , 1P >. ] ~R = A <-> [ <. ( y +P. 1P ) , 1P >. ] ~R = A ) )
12	11	riota2 5903	. . . . . 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 5932	. . . . . 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 3815	. . . . 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 6637	. . . 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 2229	. 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 2619	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 1364   e. wcel 2167   E.wrex 2476   E!wreu 2477   <.cop 3626   class class class wbr 4034   iota_crio 5879  (class class class)co 5925   [cec 6599   P.cnp 7375   1Pc1p 7376   +P. cpp 7377   ~R cer 7380   R.cnr 7381   0Rc0r 7382   <R cltr 7387

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-rmo 2483  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-po 4332  df-iso 4333  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-riota 5880  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-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-i1p 7551  df-iplp 7552  df-iltp 7554  df-enr 7810  df-nr 7811  df-ltr 7814  df-0r 7815

This theorem is referenced by:  caucvgsrlemfv  7875  caucvgsrlemgt1  7879