Theorem srpospr 7745

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 7689	. . 3 ⊢ R = ((P × P) / ~R )
2		breq2 3993	. . . 4 ⊢ ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → (0R <R [⟨𝑎, 𝑏⟩] ~R ↔ 0R <R 𝐴))
3		eqeq2 2180	. . . . 5 ⊢ ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → ([⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ↔ [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
4	3	reubidv 2653	. . . 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 233	. . 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 7710	. . . . . . . 8 ⊢ (0R <R [⟨𝑎, 𝑏⟩] ~R ↔ 𝑏<P 𝑎)
7	6	biimpi 119	. . . . . . 7 ⊢ (0R <R [⟨𝑎, 𝑏⟩] ~R → 𝑏<P 𝑎)
8	7	adantl 275	. . . . . 6 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → 𝑏<P 𝑎)
9		lteupri 7579	. . . . . 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 109	. . . . . . . . 9 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → 𝑥 ∈ P)
12		1pr 7516	. . . . . . . . . 10 ⊢ 1P ∈ P
13	12	a1i 9	. . . . . . . . 9 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → 1P ∈ P)
14		addclpr 7499	. . . . . . . . 9 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 +P 1P) ∈ P)
15	11, 13, 14	syl2anc 409	. . . . . . . 8 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → (𝑥 +P 1P) ∈ P)
16		simplll 528	. . . . . . . 8 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → 𝑎 ∈ P)
17		simpllr 529	. . . . . . . 8 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → 𝑏 ∈ P)
18		enreceq 7698	. . . . . . . 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 1234	. . . . . . 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 7540	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 +P 1P) = (1P +P 𝑥))
21	11, 13, 20	syl2anc 409	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → (𝑥 +P 1P) = (1P +P 𝑥))
22	21	oveq1d 5868	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → ((𝑥 +P 1P) +P 𝑏) = ((1P +P 𝑥) +P 𝑏))
23		addassprg 7541	. . . . . . . . . . 11 ⊢ ((1P ∈ P ∧ 𝑥 ∈ P ∧ 𝑏 ∈ P) → ((1P +P 𝑥) +P 𝑏) = (1P +P (𝑥 +P 𝑏)))
24	13, 11, 17, 23	syl3anc 1233	. . . . . . . . . 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 2203	. . . . . . . . 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 2179	. . . . . . . 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 7499	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ P ∧ 𝑏 ∈ P) → (𝑥 +P 𝑏) ∈ P)
28	11, 17, 27	syl2anc 409	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → (𝑥 +P 𝑏) ∈ P)
29		addcanprg 7578	. . . . . . . . . 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 1233	. . . . . . . . 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 5861	. . . . . . . . 9 ⊢ ((𝑥 +P 𝑏) = 𝑎 → (1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎))
32	30, 31	impbid1 141	. . . . . . . 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 187	. . . . . . 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 7540	. . . . . . . . 9 ⊢ ((𝑥 ∈ P ∧ 𝑏 ∈ P) → (𝑥 +P 𝑏) = (𝑏 +P 𝑥))
35	11, 17, 34	syl2anc 409	. . . . . . . 8 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥 ∈ P) → (𝑥 +P 𝑏) = (𝑏 +P 𝑥))
36	35	eqeq1d 2179	. . . . . . 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 214	. . . . . 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 2652	. . . . 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 146	. . . 4 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R )
40	39	ex 114	. . 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 6600	. 2 ⊢ (𝐴 ∈ R → (0R <R 𝐴 → ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
42	41	imp 123	1 ⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141  ∃!wreu 2450  ⟨cop 3586   class class class wbr 3989  (class class class)co 5853  [cec 6511  Pcnp 7253  1_Pc1p 7254   +_P cpp 7255  <_P cltp 7257   ~_R cer 7258  Rcnr 7259  0_Rc0r 7260   <_R cltr 7265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-i1p 7429  df-iplp 7430  df-iltp 7432  df-enr 7688  df-nr 7689  df-ltr 7692  df-0r 7693

This theorem is referenced by:  prsrriota  7750  caucvgsrlemcl  7751  caucvgsrlemgt1  7757