Theorem ltasr 4003

Description: Ordering property of addition.

Hypotheses

Ref	Expression
ltasr.1	⊢ A ∈ V
ltasr.2	⊢ B ∈ V

Assertion

Ref	Expression
ltasr	⊢ (C ∈ R → (A <R B ↔ (C +R A) <R (C +R B)))

Proof of Theorem ltasr

Step	Hyp	Ref	Expression
1		ltasr.2	. 2 ⊢ B ∈ V
2		dmaddsr 3988	. 2 ⊢ dom +R = (R × R)
3		ltasr.1	. 2 ⊢ A ∈ V
4		ltrelsr 3974	. 2 ⊢ <R ⊆ (R × R)
5		0nsr 3982	. 2 ⊢ ¬ ∅ ∈ R
6		df-nr 3961	. . . 4 ⊢ R = ((P × P) / ~R )
7		opreq1 3006	. . . . . 6 ⊢ ([⟨v, u⟩] ~R = C → ([⟨v, u⟩] ~R +R [⟨x, y⟩] ~R ) = (C +R [⟨x, y⟩] ~R ))
8		opreq1 3006	. . . . . 6 ⊢ ([⟨v, u⟩] ~R = C → ([⟨v, u⟩] ~R +R [⟨z, w⟩] ~R ) = (C +R [⟨z, w⟩] ~R ))
9	7, 8	breq12d 2073	. . . . 5 ⊢ ([⟨v, u⟩] ~R = C → (([⟨v, u⟩] ~R +R [⟨x, y⟩] ~R ) <R ([⟨v, u⟩] ~R +R [⟨z, w⟩] ~R ) ↔ (C +R [⟨x, y⟩] ~R ) <R (C +R [⟨z, w⟩] ~R )))
10	9	bibi2d 470	. . . 4 ⊢ ([⟨v, u⟩] ~R = C → (([⟨x, y⟩] ~R <R [⟨z, w⟩] ~R ↔ ([⟨v, u⟩] ~R +R [⟨x, y⟩] ~R ) <R ([⟨v, u⟩] ~R +R [⟨z, w⟩] ~R )) ↔ ([⟨x, y⟩] ~R <R [⟨z, w⟩] ~R ↔ (C +R [⟨x, y⟩] ~R ) <R (C +R [⟨z, w⟩] ~R ))))
11		breq1 2065	. . . . 5 ⊢ ([⟨x, y⟩] ~R = A → ([⟨x, y⟩] ~R <R [⟨z, w⟩] ~R ↔ A <R [⟨z, w⟩] ~R ))
12		opreq2 3007	. . . . . 6 ⊢ ([⟨x, y⟩] ~R = A → (C +R [⟨x, y⟩] ~R ) = (C +R A))
13	12	breq1d 2071	. . . . 5 ⊢ ([⟨x, y⟩] ~R = A → ((C +R [⟨x, y⟩] ~R ) <R (C +R [⟨z, w⟩] ~R ) ↔ (C +R A) <R (C +R [⟨z, w⟩] ~R )))
14	11, 13	bibi12d 477	. . . 4 ⊢ ([⟨x, y⟩] ~R = A → (([⟨x, y⟩] ~R <R [⟨z, w⟩] ~R ↔ (C +R [⟨x, y⟩] ~R ) <R (C +R [⟨z, w⟩] ~R )) ↔ (A <R [⟨z, w⟩] ~R ↔ (C +R A) <R (C +R [⟨z, w⟩] ~R ))))
15		breq2 2066	. . . . 5 ⊢ ([⟨z, w⟩] ~R = B → (A <R [⟨z, w⟩] ~R ↔ A <R B))
16		opreq2 3007	. . . . . 6 ⊢ ([⟨z, w⟩] ~R = B → (C +R [⟨z, w⟩] ~R ) = (C +R B))
17	16	breq2d 2072	. . . . 5 ⊢ ([⟨z, w⟩] ~R = B → ((C +R A) <R (C +R [⟨z, w⟩] ~R ) ↔ (C +R A) <R (C +R B)))
18	15, 17	bibi12d 477	. . . 4 ⊢ ([⟨z, w⟩] ~R = B → ((A <R [⟨z, w⟩] ~R ↔ (C +R A) <R (C +R [⟨z, w⟩] ~R )) ↔ (A <R B ↔ (C +R A) <R (C +R B))))
19		addclpr 3914	. . . . . . . 8 ⊢ ((v ∈ P ∧ u ∈ P) → (v +P u) ∈ P)
20	19	adantr 306	. . . . . . 7 ⊢ (((v ∈ P ∧ u ∈ P) ∧ (x ∈ P ∧ y ∈ P)) → (v +P u) ∈ P)
21	20	3adant3 599	. . . . . 6 ⊢ (((v ∈ P ∧ u ∈ P) ∧ (x ∈ P ∧ y ∈ P) ∧ (z ∈ P ∧ w ∈ P)) → (v +P u) ∈ P)
22		oprex 3018	. . . . . . . 8 ⊢ (x +P w) ∈ V
23		oprex 3018	. . . . . . . 8 ⊢ (y +P z) ∈ V
24	22, 23	ltapr 3945	. . . . . . 7 ⊢ ((v +P u) ∈ P → ((x +P w)<P (y +P z) ↔ ((v +P u) +P (x +P w))<P ((v +P u) +P (y +P z))))
25		visset 1350	. . . . . . . 8 ⊢ x ∈ V
26		visset 1350	. . . . . . . 8 ⊢ y ∈ V
27		visset 1350	. . . . . . . 8 ⊢ z ∈ V
28		visset 1350	. . . . . . . 8 ⊢ w ∈ V
29	25, 26, 27, 28	ltsrpr 3980	. . . . . . 7 ⊢ ([⟨x, y⟩] ~R <R [⟨z, w⟩] ~R ↔ (x +P w)<P (y +P z))
30		oprex 3018	. . . . . . . . 9 ⊢ (v +P x) ∈ V
31		oprex 3018	. . . . . . . . 9 ⊢ (u +P y) ∈ V
32		oprex 3018	. . . . . . . . 9 ⊢ (v +P z) ∈ V
33		oprex 3018	. . . . . . . . 9 ⊢ (u +P w) ∈ V
34	30, 31, 32, 33	ltsrpr 3980	. . . . . . . 8 ⊢ ([⟨(v +P x), (u +P y)⟩] ~R <R [⟨(v +P z), (u +P w)⟩] ~R ↔ ((v +P x) +P (u +P w))<P ((u +P y) +P (v +P z)))
35		visset 1350	. . . . . . . . . 10 ⊢ v ∈ V
36		visset 1350	. . . . . . . . . 10 ⊢ u ∈ V
37	26, 27	addcompr 3917	. . . . . . . . . 10 ⊢ (y +P z) = (z +P y)
38		visset 1350	. . . . . . . . . . 11 ⊢ f ∈ V
39	27, 38	addasspr 3918	. . . . . . . . . 10 ⊢ ((y +P z) +P f) = (y +P (z +P f))
40	35, 25, 36, 37, 39, 28	caopr4 3078	. . . . . . . . 9 ⊢ ((v +P x) +P (u +P w)) = ((v +P u) +P (x +P w))
41	31, 32	addcompr 3917	. . . . . . . . . 10 ⊢ ((u +P y) +P (v +P z)) = ((v +P z) +P (u +P y))
42	25, 28	addcompr 3917	. . . . . . . . . . 11 ⊢ (x +P w) = (w +P x)
43	28, 38	addasspr 3918	. . . . . . . . . . 11 ⊢ ((x +P w) +P f) = (x +P (w +P f))
44	35, 27, 36, 42, 43, 26	caopr42 3080	. . . . . . . . . 10 ⊢ ((v +P z) +P (u +P y)) = ((v +P u) +P (y +P z))
45	41, 44	eqtr 1119	. . . . . . . . 9 ⊢ ((u +P y) +P (v +P z)) = ((v +P u) +P (y +P z))
46	40, 45	breq12i 2070	. . . . . . . 8 ⊢ (((v +P x) +P (u +P w))<P ((u +P y) +P (v +P z)) ↔ ((v +P u) +P (x +P w))<P ((v +P u) +P (y +P z)))
47	34, 46	bitr 151	. . . . . . 7 ⊢ ([⟨(v +P x), (u +P y)⟩] ~R <R [⟨(v +P z), (u +P w)⟩] ~R ↔ ((v +P u) +P (x +P w))<P ((v +P u) +P (y +P z)))
48	24, 29, 47	3bitr4g 428	. . . . . 6 ⊢ ((v +P u) ∈ P → ([⟨x, y⟩] ~R <R [⟨z, w⟩] ~R ↔ [⟨(v +P x), (u +P y)⟩] ~R <R [⟨(v +P z), (u +P w)⟩] ~R ))
49	21, 48	syl 12	. . . . 5 ⊢ (((v ∈ P ∧ u ∈ P) ∧ (x ∈ P ∧ y ∈ P) ∧ (z ∈ P ∧ w ∈ P)) → ([⟨x, y⟩] ~R <R [⟨z, w⟩] ~R ↔ [⟨(v +P x), (u +P y)⟩] ~R <R [⟨(v +P z), (u +P w)⟩] ~R ))
50		addsrpr 3978	. . . . . . 7 ⊢ (((v ∈ P ∧ u ∈ P) ∧ (x ∈ P ∧ y ∈ P)) → ([⟨v, u⟩] ~R +R [⟨x, y⟩] ~R ) = [⟨(v +P x), (u +P y)⟩] ~R )
51	50	3adant3 599	. . . . . 6 ⊢ (((v ∈ P ∧ u ∈ P) ∧ (x ∈ P ∧ y ∈ P) ∧ (z ∈ P ∧ w ∈ P)) → ([⟨v, u⟩] ~R +R [⟨x, y⟩] ~R ) = [⟨(v +P x), (u +P y)⟩] ~R )
52		addsrpr 3978	. . . . . . 7 ⊢ (((v ∈ P ∧ u ∈ P) ∧ (z ∈ P ∧ w ∈ P)) → ([⟨v, u⟩] ~R +R [⟨z, w⟩] ~R ) = [⟨(v +P z), (u +P w)⟩] ~R )
53	52	3adant2 598	. . . . . 6 ⊢ (((v ∈ P ∧ u ∈ P) ∧ (x ∈ P ∧ y ∈ P) ∧ (z ∈ P ∧ w ∈ P)) → ([⟨v, u⟩] ~R +R [⟨z, w⟩] ~R ) = [⟨(v +P z), (u +P w)⟩] ~R )
54	51, 53	breq12d 2073	. . . . 5 ⊢ (((v ∈ P ∧ u ∈ P) ∧ (x ∈ P ∧ y ∈ P) ∧ (z ∈ P ∧ w ∈ P)) → (([⟨v, u⟩] ~R +R [⟨x, y⟩] ~R ) <R ([⟨v, u⟩] ~R +R [⟨z, w⟩] ~R ) ↔ [⟨(v +P x), (u +P y)⟩] ~R <R [⟨(v +P z), (u +P w)⟩] ~R ))
55	49, 54	bitr4d 409	. . . 4 ⊢ (((v ∈ P ∧ u ∈ P) ∧ (x ∈ P ∧ y ∈ P) ∧ (z ∈ P ∧ w ∈ P)) → ([⟨x, y⟩] ~R <R [⟨z, w⟩] ~R ↔ ([⟨v, u⟩] ~R +R [⟨x, y⟩] ~R ) <R ([⟨v, u⟩] ~R +R [⟨z, w⟩] ~R )))
56	6, 10, 14, 18, 55	3ecoptocl 3241	. . 3 ⊢ ((C ∈ R ∧ A ∈ R ∧ B ∈ R) → (A <R B ↔ (C +R A) <R (C +R B)))
57	56	3coml 617	. 2 ⊢ ((A ∈ R ∧ B ∈ R ∧ C ∈ R) → (A <R B ↔ (C +R A) <R (C +R B)))
58	1, 2, 3, 4, 5, 57	ndmord 3064	1 ⊢ (C ∈ R → (A <R B ↔ (C +R A) <R (C +R B)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   ∧ w3a 581   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   class class class wbr 2054  (class class class)co 3001  [cec 3198  Pcnp 3779   +_P cpp 3781  <_P cltp 3783   ~_R cer 3786  Rcnr 3787   +_R cplr 3791   <_R cltr 3793

This theorem is referenced by:  addgt0sr 4007  sqgt0sr 4009  supsrlem3 4021  axltadd 4085

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-plp 3882  df-ltp 3884  df-plpr 3958  df-enr 3960  df-nr 3961  df-plr 3962  df-ltr 3964

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