Theorem srpospr 8098

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 8042	. . 3 ⊢ R = ((P × P) / ~R )
2		breq2 4113	. . . 4 ⊢ ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → (0R <R [⟨𝑎, 𝑏⟩] ~R ↔ 0R <R 𝐴))
3		eqeq2 2242	. . . . 5 ⊢ ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → ([⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ↔ [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
4	3	reubidv 2729	. . . 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 8063	. . . . . . . 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 7932	. . . . . 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 7869	. . . . . . . . . 10 ⊢ 1P ∈ P
13	12	a1i 9	. . . . . . . . 9 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → 1P ∈ P)
14		addclpr 7852	. . . . . . . . 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 535	. . . . . . . 8 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → 𝑎 ∈ P)
17		simpllr 536	. . . . . . . 8 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → 𝑏 ∈ P)
18		enreceq 8051	. . . . . . . 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 1275	. . . . . . 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 7893	. . . . . . . . . . . 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 6065	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → ((𝑥 +P 1P) +P 𝑏) = ((1P +P 𝑥) +P 𝑏))
23		addassprg 7894	. . . . . . . . . . 11 ⊢ ((1P ∈ P ∧ 𝑥 ∈ P ∧ 𝑏 ∈ P) → ((1P +P 𝑥) +P 𝑏) = (1P +P (𝑥 +P 𝑏)))
24	13, 11, 17, 23	syl3anc 1274	. . . . . . . . . 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 2265	. . . . . . . . 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 2241	. . . . . . . 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 7852	. . . . . . . . . . 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 7931	. . . . . . . . . 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 1274	. . . . . . . . 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 6058	. . . . . . . . 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 7893	. . . . . . . . 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 2241	. . . . . . 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 2728	. . . . 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 6856	. 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 1398   ∈ wcel 2203  ∃!wreu 2522  ⟨cop 3692   class class class wbr 4109  (class class class)co 6050  [cec 6765  Pcnp 7606  1_Pc1p 7607   +_P cpp 7608  <_P cltp 7610   ~_R cer 7611  Rcnr 7612  0_Rc0r 7613   <_R cltr 7618

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-i1p 7782  df-iplp 7783  df-iltp 7785  df-enr 8041  df-nr 8042  df-ltr 8045  df-0r 8046

This theorem is referenced by:  prsrriota  8103  caucvgsrlemcl  8104  caucvgsrlemgt1  8110