Theorem map2psrpr 11024

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 10982	. . . . 5 ⊢ <R ⊆ (R × R)
2	1	brel 5683	. . . 4 ⊢ ((𝐶 +R -1R) <R 𝐴 → ((𝐶 +R -1R) ∈ R ∧ 𝐴 ∈ R))
3	2	simprd 496	. . 3 ⊢ ((𝐶 +R -1R) <R 𝐴 → 𝐴 ∈ R)
4		map2psrpr.2	. . . . . 6 ⊢ 𝐶 ∈ R
5		ltasr 11014	. . . . . 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 11015	. . . . . . . . . 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 7366	. . . . . . . 8 ⊢ ((𝐶 +R (𝐶 ·R -1R)) +R 𝐴) = (0R +R 𝐴)
10		addasssr 11002	. . . . . . . 8 ⊢ ((𝐶 +R (𝐶 ·R -1R)) +R 𝐴) = (𝐶 +R ((𝐶 ·R -1R) +R 𝐴))
11		addcomsr 11001	. . . . . . . 8 ⊢ (0R +R 𝐴) = (𝐴 +R 0R)
12	9, 10, 11	3eqtr3i 2770	. . . . . . 7 ⊢ (𝐶 +R ((𝐶 ·R -1R) +R 𝐴)) = (𝐴 +R 0R)
13		0idsr 11011	. . . . . . 7 ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴)
14	12, 13	eqtrid 2786	. . . . . 6 ⊢ (𝐴 ∈ R → (𝐶 +R ((𝐶 ·R -1R) +R 𝐴)) = 𝐴)
15	14	breq2d 5084	. . . . 5 ⊢ (𝐴 ∈ R → ((𝐶 +R -1R) <R (𝐶 +R ((𝐶 ·R -1R) +R 𝐴)) ↔ (𝐶 +R -1R) <R 𝐴))
16	6, 15	bitrid 284	. . . 4 ⊢ (𝐴 ∈ R → (-1R <R ((𝐶 ·R -1R) +R 𝐴) ↔ (𝐶 +R -1R) <R 𝐴))
17		m1r 10996	. . . . . . . 8 ⊢ -1R ∈ R
18		mulclsr 10998	. . . . . . . 8 ⊢ ((𝐶 ∈ R ∧ -1R ∈ R) → (𝐶 ·R -1R) ∈ R)
19	4, 17, 18	mp2an 698	. . . . . . 7 ⊢ (𝐶 ·R -1R) ∈ R
20		addclsr 10997	. . . . . . 7 ⊢ (((𝐶 ·R -1R) ∈ R ∧ 𝐴 ∈ R) → ((𝐶 ·R -1R) +R 𝐴) ∈ R)
21	19, 20	mpan 696	. . . . . 6 ⊢ (𝐴 ∈ R → ((𝐶 ·R -1R) +R 𝐴) ∈ R)
22		df-nr 10970	. . . . . . 7 ⊢ R = ((P × P) / ~R )
23		breq2 5076	. . . . . . . 8 ⊢ ([⟨𝑦, 𝑧⟩] ~R = ((𝐶 ·R -1R) +R 𝐴) → (-1R <R [⟨𝑦, 𝑧⟩] ~R ↔ -1R <R ((𝐶 ·R -1R) +R 𝐴)))
24		eqeq2 2751	. . . . . . . . 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 3163	. . . . . . . 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 345	. . . . . . 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 10976	. . . . . . . . . . 11 ⊢ -1R = [⟨1P, (1P +P 1P)⟩] ~R
28	27	breq1i 5079	. . . . . . . . . 10 ⊢ (-1R <R [⟨𝑦, 𝑧⟩] ~R ↔ [⟨1P, (1P +P 1P)⟩] ~R <R [⟨𝑦, 𝑧⟩] ~R )
29		addasspr 10936	. . . . . . . . . . . 12 ⊢ ((1P +P 1P) +P 𝑦) = (1P +P (1P +P 𝑦))
30	29	breq2i 5080	. . . . . . . . . . 11 ⊢ ((1P +P 𝑧)<P ((1P +P 1P) +P 𝑦) ↔ (1P +P 𝑧)<P (1P +P (1P +P 𝑦)))
31		ltsrpr 10991	. . . . . . . . . . 11 ⊢ ([⟨1P, (1P +P 1P)⟩] ~R <R [⟨𝑦, 𝑧⟩] ~R ↔ (1P +P 𝑧)<P ((1P +P 1P) +P 𝑦))
32		1pr 10929	. . . . . . . . . . . 12 ⊢ 1P ∈ P
33		ltapr 10959	. . . . . . . . . . . 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 304	. . . . . . . . . 10 ⊢ ([⟨1P, (1P +P 1P)⟩] ~R <R [⟨𝑦, 𝑧⟩] ~R ↔ 𝑧<P (1P +P 𝑦))
36	28, 35	bitri 276	. . . . . . . . 9 ⊢ (-1R <R [⟨𝑦, 𝑧⟩] ~R ↔ 𝑧<P (1P +P 𝑦))
37		ltexpri 10957	. . . . . . . . 9 ⊢ (𝑧<P (1P +P 𝑦) → ∃𝑥 ∈ P (𝑧 +P 𝑥) = (1P +P 𝑦))
38	36, 37	sylbi 218	. . . . . . . 8 ⊢ (-1R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑥 ∈ P (𝑧 +P 𝑥) = (1P +P 𝑦))
39		enreceq 10980	. . . . . . . . . . . 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 707	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ P ∧ (𝑦 ∈ P ∧ 𝑧 ∈ P)) → ([⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ↔ (𝑥 +P 𝑧) = (1P +P 𝑦)))
41		addcompr 10935	. . . . . . . . . . . 12 ⊢ (𝑧 +P 𝑥) = (𝑥 +P 𝑧)
42	41	eqeq1i 2744	. . . . . . . . . . 11 ⊢ ((𝑧 +P 𝑥) = (1P +P 𝑦) ↔ (𝑥 +P 𝑧) = (1P +P 𝑦))
43	40, 42	bitr4di 290	. . . . . . . . . 10 ⊢ ((𝑥 ∈ P ∧ (𝑦 ∈ P ∧ 𝑧 ∈ P)) → ([⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ↔ (𝑧 +P 𝑥) = (1P +P 𝑦)))
44	43	ancoms 459	. . . . . . . . 9 ⊢ (((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑥 ∈ P) → ([⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ↔ (𝑧 +P 𝑥) = (1P +P 𝑦)))
45	44	rexbidva 3161	. . . . . . . 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 247	. . . . . . 7 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (-1R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑥 ∈ P [⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ))
47	22, 26, 46	ecoptocl 8744	. . . . . 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 7364	. . . . . . . 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 2796	. . . . . . 7 ⊢ ((𝐴 ∈ R ∧ [⟨𝑥, 1P⟩] ~R = ((𝐶 ·R -1R) +R 𝐴)) → (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴)
51	50	ex 413	. . . . . 6 ⊢ (𝐴 ∈ R → ([⟨𝑥, 1P⟩] ~R = ((𝐶 ·R -1R) +R 𝐴) → (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴))
52	51	reximdv 3154	. . . . 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 261	. . 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 11022	. . . . 5 ⊢ ((𝐶 +R -1R) <R (𝐶 +R [⟨𝑥, 1P⟩] ~R ) ↔ 𝑥 ∈ P)
57		breq2 5076	. . . . 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 286	. . . 4 ⊢ ((𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴 → (𝑥 ∈ P ↔ (𝐶 +R -1R) <R 𝐴))
59	58	biimpac 479	. . 3 ⊢ ((𝑥 ∈ P ∧ (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴) → (𝐶 +R -1R) <R 𝐴)
60	59	rexlimiva 3132	. 2 ⊢ (∃𝑥 ∈ P (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴 → (𝐶 +R -1R) <R 𝐴)
61	55, 60	impbii 210	1 ⊢ ((𝐶 +R -1R) <R 𝐴 ↔ ∃𝑥 ∈ P (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃wrex 3063  ⟨cop 4561   class class class wbr 5072  (class class class)co 7356  [cec 8631  Pcnp 10773  1_Pc1p 10774   +_P cpp 10775  <_P cltp 10777   ~_R cer 10778  Rcnr 10779  0_Rc0r 10780  -1_Rcm1r 10782   +_R cplr 10783   ·_R cmr 10784   <_R cltr 10785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-inf2 9553

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-omul 8400  df-er 8633  df-ec 8635  df-qs 8639  df-ni 10786  df-pli 10787  df-mi 10788  df-lti 10789  df-plpq 10822  df-mpq 10823  df-ltpq 10824  df-enq 10825  df-nq 10826  df-erq 10827  df-plq 10828  df-mq 10829  df-1nq 10830  df-rq 10831  df-ltnq 10832  df-np 10895  df-1p 10896  df-plp 10897  df-mp 10898  df-ltp 10899  df-enr 10969  df-nr 10970  df-plr 10971  df-mr 10972  df-ltr 10973  df-0r 10974  df-1r 10975  df-m1r 10976

This theorem is referenced by:  supsrlem  11025