Theorem srpospr 6867

Description: Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.)

Assertion

Ref	Expression
srpospr	⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem srpospr

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nr 6812	. . 3 ⊢ R = ((P × P) / ~R )
2		breq2 3768	. . . 4 ⊢ ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → (0R <R [⟨𝑎, 𝑏⟩] ~R ↔ 0R <R 𝐴))
3		eqeq2 2049	. . . . 5 ⊢ ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → ([⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ↔ [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
4	3	reubidv 2493	. . . 4 ⊢ ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → (∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ↔ ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
5	2, 4	imbi12d 223	. . 3 ⊢ ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → ((0R <R [⟨𝑎, 𝑏⟩] ~R → ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ) ↔ (0R <R 𝐴 → ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)))
6		gt0srpr 6833	. . . . . . . 8 ⊢ (0R <R [⟨𝑎, 𝑏⟩] ~R ↔ 𝑏<P 𝑎)
7	6	biimpi 113	. . . . . . 7 ⊢ (0R <R [⟨𝑎, 𝑏⟩] ~R → 𝑏<P 𝑎)
8	7	adantl 262	. . . . . 6 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → 𝑏<P 𝑎)
9		lteupri 6715	. . . . . 6 ⊢ (𝑏<P 𝑎 → ∃!𝑥 ∈ P (𝑏 +P 𝑥) = 𝑎)
10	8, 9	syl 14	. . . . 5 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → ∃!𝑥 ∈ P (𝑏 +P 𝑥) = 𝑎)
11		simpr 103	. . . . . . . . 9 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → 𝑥 ∈ P)
12		1pr 6652	. . . . . . . . . 10 ⊢ 1P ∈ P
13	12	a1i 9	. . . . . . . . 9 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → 1P ∈ P)
14		addclpr 6635	. . . . . . . . 9 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 +P 1P) ∈ P)
15	11, 13, 14	syl2anc 391	. . . . . . . 8 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → (𝑥 +P 1P) ∈ P)
16		simplll 485	. . . . . . . 8 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → 𝑎 ∈ P)
17		simpllr 486	. . . . . . . 8 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → 𝑏 ∈ P)
18		enreceq 6821	. . . . . . . 8 ⊢ ((((𝑥 +P 1P) ∈ P ∧ 1P ∈ P) ∧ (𝑎 ∈ P ∧ 𝑏 ∈ P)) → ([⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ↔ ((𝑥 +P 1P) +P 𝑏) = (1P +P 𝑎)))
19	15, 13, 16, 17, 18	syl22anc 1136	. . . . . . 7 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → ([⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ↔ ((𝑥 +P 1P) +P 𝑏) = (1P +P 𝑎)))
20		addcomprg 6676	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 +P 1P) = (1P +P 𝑥))
21	11, 13, 20	syl2anc 391	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → (𝑥 +P 1P) = (1P +P 𝑥))
22	21	oveq1d 5527	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → ((𝑥 +P 1P) +P 𝑏) = ((1P +P 𝑥) +P 𝑏))
23		addassprg 6677	. . . . . . . . . . 11 ⊢ ((1P ∈ P ∧ 𝑥 ∈ P ∧ 𝑏 ∈ P) → ((1P +P 𝑥) +P 𝑏) = (1P +P (𝑥 +P 𝑏)))
24	13, 11, 17, 23	syl3anc 1135	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → ((1P +P 𝑥) +P 𝑏) = (1P +P (𝑥 +P 𝑏)))
25	22, 24	eqtrd 2072	. . . . . . . . 9 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → ((𝑥 +P 1P) +P 𝑏) = (1P +P (𝑥 +P 𝑏)))
26	25	eqeq1d 2048	. . . . . . . 8 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → (((𝑥 +P 1P) +P 𝑏) = (1P +P 𝑎) ↔ (1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎)))
27		addclpr 6635	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ P ∧ 𝑏 ∈ P) → (𝑥 +P 𝑏) ∈ P)
28	11, 17, 27	syl2anc 391	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → (𝑥 +P 𝑏) ∈ P)
29		addcanprg 6714	. . . . . . . . . 10 ⊢ ((1P ∈ P ∧ (𝑥 +P 𝑏) ∈ P ∧ 𝑎 ∈ P) → ((1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎) → (𝑥 +P 𝑏) = 𝑎))
30	13, 28, 16, 29	syl3anc 1135	. . . . . . . . 9 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → ((1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎) → (𝑥 +P 𝑏) = 𝑎))
31		oveq2 5520	. . . . . . . . 9 ⊢ ((𝑥 +P 𝑏) = 𝑎 → (1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎))
32	30, 31	impbid1 130	. . . . . . . 8 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → ((1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎) ↔ (𝑥 +P 𝑏) = 𝑎))
33	26, 32	bitrd 177	. . . . . . 7 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → (((𝑥 +P 1P) +P 𝑏) = (1P +P 𝑎) ↔ (𝑥 +P 𝑏) = 𝑎))
34		addcomprg 6676	. . . . . . . . 9 ⊢ ((𝑥 ∈ P ∧ 𝑏 ∈ P) → (𝑥 +P 𝑏) = (𝑏 +P 𝑥))
35	11, 17, 34	syl2anc 391	. . . . . . . 8 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → (𝑥 +P 𝑏) = (𝑏 +P 𝑥))
36	35	eqeq1d 2048	. . . . . . 7 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → ((𝑥 +P 𝑏) = 𝑎 ↔ (𝑏 +P 𝑥) = 𝑎))
37	19, 33, 36	3bitrrd 204	. . . . . 6 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → ((𝑏 +P 𝑥) = 𝑎 ↔ [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ))
38	37	reubidva 2492	. . . . 5 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → (∃!𝑥 ∈ P (𝑏 +P 𝑥) = 𝑎 ↔ ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ))
39	10, 38	mpbid 135	. . . 4 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R )
40	39	ex 108	. . 3 ⊢ ((𝑎 ∈ P ∧ 𝑏 ∈ P) → (0R <R [⟨𝑎, 𝑏⟩] ~R → ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ))
41	1, 5, 40	ecoptocl 6193	. 2 ⊢ (𝐴 ∈ R → (0R <R 𝐴 → ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
42	41	imp 115	1 ⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∃!wreu 2308  ⟨cop 3378   class class class wbr 3764  (class class class)co 5512  [cec 6104  Pcnp 6389  1_Pc1p 6390   +_P cpp 6391  <_P cltp 6393   ~_R cer 6394  Rcnr 6395  0_Rc0r 6396   <_R cltr 6401

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rmo 2314  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816

This theorem is referenced by:  prsrriota  6872  caucvgsrlemcl  6873  caucvgsrlemgt1  6879