Theorem map2psrpr 11122

Description: Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)

Hypothesis

Ref	Expression
map2psrpr.2	⊢ 𝐶 ∈ R

Assertion

Ref	Expression
map2psrpr	⊢ ((𝐶 +R -1R) <R 𝐴 ↔ ∃𝑥 ∈ P (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Proof of Theorem map2psrpr

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelsr 11080	. . . . 5 ⊢ <R ⊆ (R × R)
2	1	brel 5719	. . . 4 ⊢ ((𝐶 +R -1R) <R 𝐴 → ((𝐶 +R -1R) ∈ R ∧ 𝐴 ∈ R))
3	2	simprd 495	. . 3 ⊢ ((𝐶 +R -1R) <R 𝐴 → 𝐴 ∈ R)
4		map2psrpr.2	. . . . . 6 ⊢ 𝐶 ∈ R
5		ltasr 11112	. . . . . 6 ⊢ (𝐶 ∈ R → (-1R <R ((𝐶 ·R -1R) +R 𝐴) ↔ (𝐶 +R -1R) <R (𝐶 +R ((𝐶 ·R -1R) +R 𝐴))))
6	4, 5	ax-mp 5	. . . . 5 ⊢ (-1R <R ((𝐶 ·R -1R) +R 𝐴) ↔ (𝐶 +R -1R) <R (𝐶 +R ((𝐶 ·R -1R) +R 𝐴)))
7		pn0sr 11113	. . . . . . . . . 10 ⊢ (𝐶 ∈ R → (𝐶 +R (𝐶 ·R -1R)) = 0R)
8	4, 7	ax-mp 5	. . . . . . . . 9 ⊢ (𝐶 +R (𝐶 ·R -1R)) = 0R
9	8	oveq1i 7413	. . . . . . . 8 ⊢ ((𝐶 +R (𝐶 ·R -1R)) +R 𝐴) = (0R +R 𝐴)
10		addasssr 11100	. . . . . . . 8 ⊢ ((𝐶 +R (𝐶 ·R -1R)) +R 𝐴) = (𝐶 +R ((𝐶 ·R -1R) +R 𝐴))
11		addcomsr 11099	. . . . . . . 8 ⊢ (0R +R 𝐴) = (𝐴 +R 0R)
12	9, 10, 11	3eqtr3i 2766	. . . . . . 7 ⊢ (𝐶 +R ((𝐶 ·R -1R) +R 𝐴)) = (𝐴 +R 0R)
13		0idsr 11109	. . . . . . 7 ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴)
14	12, 13	eqtrid 2782	. . . . . 6 ⊢ (𝐴 ∈ R → (𝐶 +R ((𝐶 ·R -1R) +R 𝐴)) = 𝐴)
15	14	breq2d 5131	. . . . 5 ⊢ (𝐴 ∈ R → ((𝐶 +R -1R) <R (𝐶 +R ((𝐶 ·R -1R) +R 𝐴)) ↔ (𝐶 +R -1R) <R 𝐴))
16	6, 15	bitrid 283	. . . 4 ⊢ (𝐴 ∈ R → (-1R <R ((𝐶 ·R -1R) +R 𝐴) ↔ (𝐶 +R -1R) <R 𝐴))
17		m1r 11094	. . . . . . . 8 ⊢ -1R ∈ R
18		mulclsr 11096	. . . . . . . 8 ⊢ ((𝐶 ∈ R ∧ -1R ∈ R) → (𝐶 ·R -1R) ∈ R)
19	4, 17, 18	mp2an 692	. . . . . . 7 ⊢ (𝐶 ·R -1R) ∈ R
20		addclsr 11095	. . . . . . 7 ⊢ (((𝐶 ·R -1R) ∈ R ∧ 𝐴 ∈ R) → ((𝐶 ·R -1R) +R 𝐴) ∈ R)
21	19, 20	mpan 690	. . . . . 6 ⊢ (𝐴 ∈ R → ((𝐶 ·R -1R) +R 𝐴) ∈ R)
22		df-nr 11068	. . . . . . 7 ⊢ R = ((P × P) / ~R )
23		breq2 5123	. . . . . . . 8 ⊢ ([⟨𝑦, 𝑧⟩] ~R = ((𝐶 ·R -1R) +R 𝐴) → (-1R <R [⟨𝑦, 𝑧⟩] ~R ↔ -1R <R ((𝐶 ·R -1R) +R 𝐴)))
24		eqeq2 2747	. . . . . . . . 9 ⊢ ([⟨𝑦, 𝑧⟩] ~R = ((𝐶 ·R -1R) +R 𝐴) → ([⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ↔ [⟨𝑥, 1P⟩] ~R = ((𝐶 ·R -1R) +R 𝐴)))
25	24	rexbidv 3164	. . . . . . . 8 ⊢ ([⟨𝑦, 𝑧⟩] ~R = ((𝐶 ·R -1R) +R 𝐴) → (∃𝑥 ∈ P [⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ↔ ∃𝑥 ∈ P [⟨𝑥, 1P⟩] ~R = ((𝐶 ·R -1R) +R 𝐴)))
26	23, 25	imbi12d 344	. . . . . . 7 ⊢ ([⟨𝑦, 𝑧⟩] ~R = ((𝐶 ·R -1R) +R 𝐴) → ((-1R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑥 ∈ P [⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ) ↔ (-1R <R ((𝐶 ·R -1R) +R 𝐴) → ∃𝑥 ∈ P [⟨𝑥, 1P⟩] ~R = ((𝐶 ·R -1R) +R 𝐴))))
27		df-m1r 11074	. . . . . . . . . . 11 ⊢ -1R = [⟨1P, (1P +P 1P)⟩] ~R
28	27	breq1i 5126	. . . . . . . . . 10 ⊢ (-1R <R [⟨𝑦, 𝑧⟩] ~R ↔ [⟨1P, (1P +P 1P)⟩] ~R <R [⟨𝑦, 𝑧⟩] ~R )
29		addasspr 11034	. . . . . . . . . . . 12 ⊢ ((1P +P 1P) +P 𝑦) = (1P +P (1P +P 𝑦))
30	29	breq2i 5127	. . . . . . . . . . 11 ⊢ ((1P +P 𝑧)<P ((1P +P 1P) +P 𝑦) ↔ (1P +P 𝑧)<P (1P +P (1P +P 𝑦)))
31		ltsrpr 11089	. . . . . . . . . . 11 ⊢ ([⟨1P, (1P +P 1P)⟩] ~R <R [⟨𝑦, 𝑧⟩] ~R ↔ (1P +P 𝑧)<P ((1P +P 1P) +P 𝑦))
32		1pr 11027	. . . . . . . . . . . 12 ⊢ 1P ∈ P
33		ltapr 11057	. . . . . . . . . . . 12 ⊢ (1P ∈ P → (𝑧<P (1P +P 𝑦) ↔ (1P +P 𝑧)<P (1P +P (1P +P 𝑦))))
34	32, 33	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑧<P (1P +P 𝑦) ↔ (1P +P 𝑧)<P (1P +P (1P +P 𝑦)))
35	30, 31, 34	3bitr4i 303	. . . . . . . . . 10 ⊢ ([⟨1P, (1P +P 1P)⟩] ~R <R [⟨𝑦, 𝑧⟩] ~R ↔ 𝑧<P (1P +P 𝑦))
36	28, 35	bitri 275	. . . . . . . . 9 ⊢ (-1R <R [⟨𝑦, 𝑧⟩] ~R ↔ 𝑧<P (1P +P 𝑦))
37		ltexpri 11055	. . . . . . . . 9 ⊢ (𝑧<P (1P +P 𝑦) → ∃𝑥 ∈ P (𝑧 +P 𝑥) = (1P +P 𝑦))
38	36, 37	sylbi 217	. . . . . . . 8 ⊢ (-1R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑥 ∈ P (𝑧 +P 𝑥) = (1P +P 𝑦))
39		enreceq 11078	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ P ∧ 1P ∈ P) ∧ (𝑦 ∈ P ∧ 𝑧 ∈ P)) → ([⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ↔ (𝑥 +P 𝑧) = (1P +P 𝑦)))
40	32, 39	mpanl2 701	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ P ∧ (𝑦 ∈ P ∧ 𝑧 ∈ P)) → ([⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ↔ (𝑥 +P 𝑧) = (1P +P 𝑦)))
41		addcompr 11033	. . . . . . . . . . . 12 ⊢ (𝑧 +P 𝑥) = (𝑥 +P 𝑧)
42	41	eqeq1i 2740	. . . . . . . . . . 11 ⊢ ((𝑧 +P 𝑥) = (1P +P 𝑦) ↔ (𝑥 +P 𝑧) = (1P +P 𝑦))
43	40, 42	bitr4di 289	. . . . . . . . . 10 ⊢ ((𝑥 ∈ P ∧ (𝑦 ∈ P ∧ 𝑧 ∈ P)) → ([⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ↔ (𝑧 +P 𝑥) = (1P +P 𝑦)))
44	43	ancoms 458	. . . . . . . . 9 ⊢ (((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑥 ∈ P) → ([⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ↔ (𝑧 +P 𝑥) = (1P +P 𝑦)))
45	44	rexbidva 3162	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (∃𝑥 ∈ P [⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ↔ ∃𝑥 ∈ P (𝑧 +P 𝑥) = (1P +P 𝑦)))
46	38, 45	imbitrrid 246	. . . . . . 7 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (-1R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑥 ∈ P [⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ))
47	22, 26, 46	ecoptocl 8819	. . . . . 6 ⊢ (((𝐶 ·R -1R) +R 𝐴) ∈ R → (-1R <R ((𝐶 ·R -1R) +R 𝐴) → ∃𝑥 ∈ P [⟨𝑥, 1P⟩] ~R = ((𝐶 ·R -1R) +R 𝐴)))
48	21, 47	syl 17	. . . . 5 ⊢ (𝐴 ∈ R → (-1R <R ((𝐶 ·R -1R) +R 𝐴) → ∃𝑥 ∈ P [⟨𝑥, 1P⟩] ~R = ((𝐶 ·R -1R) +R 𝐴)))
49		oveq2 7411	. . . . . . . 8 ⊢ ([⟨𝑥, 1P⟩] ~R = ((𝐶 ·R -1R) +R 𝐴) → (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = (𝐶 +R ((𝐶 ·R -1R) +R 𝐴)))
50	49, 14	sylan9eqr 2792	. . . . . . 7 ⊢ ((𝐴 ∈ R ∧ [⟨𝑥, 1P⟩] ~R = ((𝐶 ·R -1R) +R 𝐴)) → (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴)
51	50	ex 412	. . . . . 6 ⊢ (𝐴 ∈ R → ([⟨𝑥, 1P⟩] ~R = ((𝐶 ·R -1R) +R 𝐴) → (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴))
52	51	reximdv 3155	. . . . 5 ⊢ (𝐴 ∈ R → (∃𝑥 ∈ P [⟨𝑥, 1P⟩] ~R = ((𝐶 ·R -1R) +R 𝐴) → ∃𝑥 ∈ P (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴))
53	48, 52	syld 47	. . . 4 ⊢ (𝐴 ∈ R → (-1R <R ((𝐶 ·R -1R) +R 𝐴) → ∃𝑥 ∈ P (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴))
54	16, 53	sylbird 260	. . 3 ⊢ (𝐴 ∈ R → ((𝐶 +R -1R) <R 𝐴 → ∃𝑥 ∈ P (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴))
55	3, 54	mpcom 38	. 2 ⊢ ((𝐶 +R -1R) <R 𝐴 → ∃𝑥 ∈ P (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴)
56	4	mappsrpr 11120	. . . . 5 ⊢ ((𝐶 +R -1R) <R (𝐶 +R [⟨𝑥, 1P⟩] ~R ) ↔ 𝑥 ∈ P)
57		breq2 5123	. . . . 5 ⊢ ((𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴 → ((𝐶 +R -1R) <R (𝐶 +R [⟨𝑥, 1P⟩] ~R ) ↔ (𝐶 +R -1R) <R 𝐴))
58	56, 57	bitr3id 285	. . . 4 ⊢ ((𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴 → (𝑥 ∈ P ↔ (𝐶 +R -1R) <R 𝐴))
59	58	biimpac 478	. . 3 ⊢ ((𝑥 ∈ P ∧ (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴) → (𝐶 +R -1R) <R 𝐴)
60	59	rexlimiva 3133	. 2 ⊢ (∃𝑥 ∈ P (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴 → (𝐶 +R -1R) <R 𝐴)
61	55, 60	impbii 209	1 ⊢ ((𝐶 +R -1R) <R 𝐴 ↔ ∃𝑥 ∈ P (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3060  ⟨cop 4607   class class class wbr 5119  (class class class)co 7403  [cec 8715  Pcnp 10871  1_Pc1p 10872   +_P cpp 10873  <_P cltp 10875   ~_R cer 10876  Rcnr 10877  0_Rc0r 10878  -1_Rcm1r 10880   +_R cplr 10881   ·_R cmr 10882   <_R cltr 10883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-inf2 9653

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-oadd 8482  df-omul 8483  df-er 8717  df-ec 8719  df-qs 8723  df-ni 10884  df-pli 10885  df-mi 10886  df-lti 10887  df-plpq 10920  df-mpq 10921  df-ltpq 10922  df-enq 10923  df-nq 10924  df-erq 10925  df-plq 10926  df-mq 10927  df-1nq 10928  df-rq 10929  df-ltnq 10930  df-np 10993  df-1p 10994  df-plp 10995  df-mp 10996  df-ltp 10997  df-enr 11067  df-nr 11068  df-plr 11069  df-mr 11070  df-ltr 11071  df-0r 11072  df-1r 11073  df-m1r 11074

This theorem is referenced by:  supsrlem  11123