Theorem srpospr 7867

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 7811	. . 3 ⊢ R = ((P × P) / ~R )
2		breq2 4038	. . . 4 ⊢ ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → (0R <R [⟨𝑎, 𝑏⟩] ~R ↔ 0R <R 𝐴))
3		eqeq2 2206	. . . . 5 ⊢ ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → ([⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ↔ [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
4	3	reubidv 2681	. . . 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 7832	. . . . . . . 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 7701	. . . . . 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 7638	. . . . . . . . . 10 ⊢ 1P ∈ P
13	12	a1i 9	. . . . . . . . 9 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → 1P ∈ P)
14		addclpr 7621	. . . . . . . . 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 7820	. . . . . . . 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 1250	. . . . . . 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 7662	. . . . . . . . . . . 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 5940	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → ((𝑥 +P 1P) +P 𝑏) = ((1P +P 𝑥) +P 𝑏))
23		addassprg 7663	. . . . . . . . . . 11 ⊢ ((1P ∈ P ∧ 𝑥 ∈ P ∧ 𝑏 ∈ P) → ((1P +P 𝑥) +P 𝑏) = (1P +P (𝑥 +P 𝑏)))
24	13, 11, 17, 23	syl3anc 1249	. . . . . . . . . 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 2229	. . . . . . . . 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 2205	. . . . . . . 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 7621	. . . . . . . . . . 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 7700	. . . . . . . . . 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 1249	. . . . . . . . 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 5933	. . . . . . . . 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 7662	. . . . . . . . 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 2205	. . . . . . 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 2680	. . . . 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 6690	. 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 1364   ∈ wcel 2167  ∃!wreu 2477  ⟨cop 3626   class class class wbr 4034  (class class class)co 5925  [cec 6599  Pcnp 7375  1_Pc1p 7376   +_P cpp 7377  <_P cltp 7379   ~_R cer 7380  Rcnr 7381  0_Rc0r 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-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:  prsrriota  7872  caucvgsrlemcl  7873  caucvgsrlemgt1  7879