Theorem map2psrpr 11179

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 11137	. . . . 5 ⊢ <R ⊆ (R × R)
2	1	brel 5765	. . . 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 11169	. . . . . 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 11170	. . . . . . . . . 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 7458	. . . . . . . 8 ⊢ ((𝐶 +R (𝐶 ·R -1R)) +R 𝐴) = (0R +R 𝐴)
10		addasssr 11157	. . . . . . . 8 ⊢ ((𝐶 +R (𝐶 ·R -1R)) +R 𝐴) = (𝐶 +R ((𝐶 ·R -1R) +R 𝐴))
11		addcomsr 11156	. . . . . . . 8 ⊢ (0R +R 𝐴) = (𝐴 +R 0R)
12	9, 10, 11	3eqtr3i 2776	. . . . . . 7 ⊢ (𝐶 +R ((𝐶 ·R -1R) +R 𝐴)) = (𝐴 +R 0R)
13		0idsr 11166	. . . . . . 7 ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴)
14	12, 13	eqtrid 2792	. . . . . 6 ⊢ (𝐴 ∈ R → (𝐶 +R ((𝐶 ·R -1R) +R 𝐴)) = 𝐴)
15	14	breq2d 5178	. . . . 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 11151	. . . . . . . 8 ⊢ -1R ∈ R
18		mulclsr 11153	. . . . . . . 8 ⊢ ((𝐶 ∈ R ∧ -1R ∈ R) → (𝐶 ·R -1R) ∈ R)
19	4, 17, 18	mp2an 691	. . . . . . 7 ⊢ (𝐶 ·R -1R) ∈ R
20		addclsr 11152	. . . . . . 7 ⊢ (((𝐶 ·R -1R) ∈ R ∧ 𝐴 ∈ R) → ((𝐶 ·R -1R) +R 𝐴) ∈ R)
21	19, 20	mpan 689	. . . . . 6 ⊢ (𝐴 ∈ R → ((𝐶 ·R -1R) +R 𝐴) ∈ R)
22		df-nr 11125	. . . . . . 7 ⊢ R = ((P × P) / ~R )
23		breq2 5170	. . . . . . . 8 ⊢ ([⟨𝑦, 𝑧⟩] ~R = ((𝐶 ·R -1R) +R 𝐴) → (-1R <R [⟨𝑦, 𝑧⟩] ~R ↔ -1R <R ((𝐶 ·R -1R) +R 𝐴)))
24		eqeq2 2752	. . . . . . . . 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 3185	. . . . . . . 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 11131	. . . . . . . . . . 11 ⊢ -1R = [⟨1P, (1P +P 1P)⟩] ~R
28	27	breq1i 5173	. . . . . . . . . 10 ⊢ (-1R <R [⟨𝑦, 𝑧⟩] ~R ↔ [⟨1P, (1P +P 1P)⟩] ~R <R [⟨𝑦, 𝑧⟩] ~R )
29		addasspr 11091	. . . . . . . . . . . 12 ⊢ ((1P +P 1P) +P 𝑦) = (1P +P (1P +P 𝑦))
30	29	breq2i 5174	. . . . . . . . . . 11 ⊢ ((1P +P 𝑧)<P ((1P +P 1P) +P 𝑦) ↔ (1P +P 𝑧)<P (1P +P (1P +P 𝑦)))
31		ltsrpr 11146	. . . . . . . . . . 11 ⊢ ([⟨1P, (1P +P 1P)⟩] ~R <R [⟨𝑦, 𝑧⟩] ~R ↔ (1P +P 𝑧)<P ((1P +P 1P) +P 𝑦))
32		1pr 11084	. . . . . . . . . . . 12 ⊢ 1P ∈ P
33		ltapr 11114	. . . . . . . . . . . 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 11112	. . . . . . . . 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 11135	. . . . . . . . . . . 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 700	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ P ∧ (𝑦 ∈ P ∧ 𝑧 ∈ P)) → ([⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ↔ (𝑥 +P 𝑧) = (1P +P 𝑦)))
41		addcompr 11090	. . . . . . . . . . . 12 ⊢ (𝑧 +P 𝑥) = (𝑥 +P 𝑧)
42	41	eqeq1i 2745	. . . . . . . . . . 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 3183	. . . . . . . 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 8865	. . . . . 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 7456	. . . . . . . 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 2802	. . . . . . 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 3176	. . . . 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 11177	. . . . 5 ⊢ ((𝐶 +R -1R) <R (𝐶 +R [⟨𝑥, 1P⟩] ~R ) ↔ 𝑥 ∈ P)
57		breq2 5170	. . . . 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 3153	. 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 1537   ∈ wcel 2108  ∃wrex 3076  ⟨cop 4654   class class class wbr 5166  (class class class)co 7448  [cec 8761  Pcnp 10928  1_Pc1p 10929   +_P cpp 10930  <_P cltp 10932   ~_R cer 10933  Rcnr 10934  0_Rc0r 10935  -1_Rcm1r 10937   +_R cplr 10938   ·_R cmr 10939   <_R cltr 10940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-oadd 8526  df-omul 8527  df-er 8763  df-ec 8765  df-qs 8769  df-ni 10941  df-pli 10942  df-mi 10943  df-lti 10944  df-plpq 10977  df-mpq 10978  df-ltpq 10979  df-enq 10980  df-nq 10981  df-erq 10982  df-plq 10983  df-mq 10984  df-1nq 10985  df-rq 10986  df-ltnq 10987  df-np 11050  df-1p 11051  df-plp 11052  df-mp 11053  df-ltp 11054  df-enr 11124  df-nr 11125  df-plr 11126  df-mr 11127  df-ltr 11128  df-0r 11129  df-1r 11130  df-m1r 11131

This theorem is referenced by:  supsrlem  11180