Theorem srpospr 7993

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 7937	. . 3 ⊢ R = ((P × P) / ~R )
2		breq2 4090	. . . 4 ⊢ ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → (0R <R [⟨𝑎, 𝑏⟩] ~R ↔ 0R <R 𝐴))
3		eqeq2 2239	. . . . 5 ⊢ ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → ([⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ↔ [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
4	3	reubidv 2716	. . . 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 234	. . 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 7958	. . . . . . . 8 ⊢ (0R <R [⟨𝑎, 𝑏⟩] ~R ↔ 𝑏<P 𝑎)
7	6	biimpi 120	. . . . . . 7 ⊢ (0R <R [⟨𝑎, 𝑏⟩] ~R → 𝑏<P 𝑎)
8	7	adantl 277	. . . . . 6 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → 𝑏<P 𝑎)
9		lteupri 7827	. . . . . 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 110	. . . . . . . . 9 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → 𝑥 ∈ P)
12		1pr 7764	. . . . . . . . . 10 ⊢ 1P ∈ P
13	12	a1i 9	. . . . . . . . 9 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → 1P ∈ P)
14		addclpr 7747	. . . . . . . . 9 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 +P 1P) ∈ P)
15	11, 13, 14	syl2anc 411	. . . . . . . 8 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → (𝑥 +P 1P) ∈ P)
16		simplll 533	. . . . . . . 8 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → 𝑎 ∈ P)
17		simpllr 534	. . . . . . . 8 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → 𝑏 ∈ P)
18		enreceq 7946	. . . . . . . 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 1272	. . . . . . 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 7788	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 +P 1P) = (1P +P 𝑥))
21	11, 13, 20	syl2anc 411	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → (𝑥 +P 1P) = (1P +P 𝑥))
22	21	oveq1d 6028	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → ((𝑥 +P 1P) +P 𝑏) = ((1P +P 𝑥) +P 𝑏))
23		addassprg 7789	. . . . . . . . . . 11 ⊢ ((1P ∈ P ∧ 𝑥 ∈ P ∧ 𝑏 ∈ P) → ((1P +P 𝑥) +P 𝑏) = (1P +P (𝑥 +P 𝑏)))
24	13, 11, 17, 23	syl3anc 1271	. . . . . . . . . 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 2262	. . . . . . . . 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 2238	. . . . . . . 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 7747	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ P ∧ 𝑏 ∈ P) → (𝑥 +P 𝑏) ∈ P)
28	11, 17, 27	syl2anc 411	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → (𝑥 +P 𝑏) ∈ P)
29		addcanprg 7826	. . . . . . . . . 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 1271	. . . . . . . . 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 6021	. . . . . . . . 9 ⊢ ((𝑥 +P 𝑏) = 𝑎 → (1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎))
32	30, 31	impbid1 142	. . . . . . . 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 188	. . . . . . 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 7788	. . . . . . . . 9 ⊢ ((𝑥 ∈ P ∧ 𝑏 ∈ P) → (𝑥 +P 𝑏) = (𝑏 +P 𝑥))
35	11, 17, 34	syl2anc 411	. . . . . . . 8 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → (𝑥 +P 𝑏) = (𝑏 +P 𝑥))
36	35	eqeq1d 2238	. . . . . . 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 215	. . . . . 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 2715	. . . . 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 147	. . . 4 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R )
40	39	ex 115	. . 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 6786	. 2 ⊢ (𝐴 ∈ R → (0R <R 𝐴 → ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
42	41	imp 124	1 ⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∃!wreu 2510  ⟨cop 3670   class class class wbr 4086  (class class class)co 6013  [cec 6695  Pcnp 7501  1_Pc1p 7502   +_P cpp 7503  <_P cltp 7505   ~_R cer 7506  Rcnr 7507  0_Rc0r 7508   <_R cltr 7513

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-2o 6578  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563  df-enq0 7634  df-nq0 7635  df-0nq0 7636  df-plq0 7637  df-mq0 7638  df-inp 7676  df-i1p 7677  df-iplp 7678  df-iltp 7680  df-enr 7936  df-nr 7937  df-ltr 7940  df-0r 7941

This theorem is referenced by:  prsrriota  7998  caucvgsrlemcl  7999  caucvgsrlemgt1  8005