Theorem map2psrpr 11004

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 10962	. . . . 5 ⊢ <R ⊆ (R × R)
2	1	brel 5684	. . . 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 10994	. . . . . 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 10995	. . . . . . . . . 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 7359	. . . . . . . 8 ⊢ ((𝐶 +R (𝐶 ·R -1R)) +R 𝐴) = (0R +R 𝐴)
10		addasssr 10982	. . . . . . . 8 ⊢ ((𝐶 +R (𝐶 ·R -1R)) +R 𝐴) = (𝐶 +R ((𝐶 ·R -1R) +R 𝐴))
11		addcomsr 10981	. . . . . . . 8 ⊢ (0R +R 𝐴) = (𝐴 +R 0R)
12	9, 10, 11	3eqtr3i 2760	. . . . . . 7 ⊢ (𝐶 +R ((𝐶 ·R -1R) +R 𝐴)) = (𝐴 +R 0R)
13		0idsr 10991	. . . . . . 7 ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴)
14	12, 13	eqtrid 2776	. . . . . 6 ⊢ (𝐴 ∈ R → (𝐶 +R ((𝐶 ·R -1R) +R 𝐴)) = 𝐴)
15	14	breq2d 5104	. . . . 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 10976	. . . . . . . 8 ⊢ -1R ∈ R
18		mulclsr 10978	. . . . . . . 8 ⊢ ((𝐶 ∈ R ∧ -1R ∈ R) → (𝐶 ·R -1R) ∈ R)
19	4, 17, 18	mp2an 692	. . . . . . 7 ⊢ (𝐶 ·R -1R) ∈ R
20		addclsr 10977	. . . . . . 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 10950	. . . . . . 7 ⊢ R = ((P × P) / ~R )
23		breq2 5096	. . . . . . . 8 ⊢ ([⟨𝑦, 𝑧⟩] ~R = ((𝐶 ·R -1R) +R 𝐴) → (-1R <R [⟨𝑦, 𝑧⟩] ~R ↔ -1R <R ((𝐶 ·R -1R) +R 𝐴)))
24		eqeq2 2741	. . . . . . . . 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 3153	. . . . . . . 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 10956	. . . . . . . . . . 11 ⊢ -1R = [⟨1P, (1P +P 1P)⟩] ~R
28	27	breq1i 5099	. . . . . . . . . 10 ⊢ (-1R <R [⟨𝑦, 𝑧⟩] ~R ↔ [⟨1P, (1P +P 1P)⟩] ~R <R [⟨𝑦, 𝑧⟩] ~R )
29		addasspr 10916	. . . . . . . . . . . 12 ⊢ ((1P +P 1P) +P 𝑦) = (1P +P (1P +P 𝑦))
30	29	breq2i 5100	. . . . . . . . . . 11 ⊢ ((1P +P 𝑧)<P ((1P +P 1P) +P 𝑦) ↔ (1P +P 𝑧)<P (1P +P (1P +P 𝑦)))
31		ltsrpr 10971	. . . . . . . . . . 11 ⊢ ([⟨1P, (1P +P 1P)⟩] ~R <R [⟨𝑦, 𝑧⟩] ~R ↔ (1P +P 𝑧)<P ((1P +P 1P) +P 𝑦))
32		1pr 10909	. . . . . . . . . . . 12 ⊢ 1P ∈ P
33		ltapr 10939	. . . . . . . . . . . 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 10937	. . . . . . . . 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 10960	. . . . . . . . . . . 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 10915	. . . . . . . . . . . 12 ⊢ (𝑧 +P 𝑥) = (𝑥 +P 𝑧)
42	41	eqeq1i 2734	. . . . . . . . . . 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 3151	. . . . . . . 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 8734	. . . . . 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 7357	. . . . . . . 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 2786	. . . . . . 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 3144	. . . . 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 11002	. . . . 5 ⊢ ((𝐶 +R -1R) <R (𝐶 +R [⟨𝑥, 1P⟩] ~R ) ↔ 𝑥 ∈ P)
57		breq2 5096	. . . . 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 3122	. 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 2109  ∃wrex 3053  ⟨cop 4583   class class class wbr 5092  (class class class)co 7349  [cec 8623  Pcnp 10753  1_Pc1p 10754   +_P cpp 10755  <_P cltp 10757   ~_R cer 10758  Rcnr 10759  0_Rc0r 10760  -1_Rcm1r 10762   +_R cplr 10763   ·_R cmr 10764   <_R cltr 10765

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-oadd 8392  df-omul 8393  df-er 8625  df-ec 8627  df-qs 8631  df-ni 10766  df-pli 10767  df-mi 10768  df-lti 10769  df-plpq 10802  df-mpq 10803  df-ltpq 10804  df-enq 10805  df-nq 10806  df-erq 10807  df-plq 10808  df-mq 10809  df-1nq 10810  df-rq 10811  df-ltnq 10812  df-np 10875  df-1p 10876  df-plp 10877  df-mp 10878  df-ltp 10879  df-enr 10949  df-nr 10950  df-plr 10951  df-mr 10952  df-ltr 10953  df-0r 10954  df-1r 10955  df-m1r 10956

This theorem is referenced by:  supsrlem  11005