Theorem ltasr 11113

Description: Ordering property of addition. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)

Assertion

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

Proof of Theorem ltasr

Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmaddsr 11098	. 2 ⊢ dom +R = (R × R)
2		ltrelsr 11081	. 2 ⊢ <R ⊆ (R × R)
3		0nsr 11092	. 2 ⊢ ¬ ∅ ∈ R
4		df-nr 11069	. . . 4 ⊢ R = ((P × P) / ~R )
5		oveq1 7424	. . . . . 6 ⊢ ([⟨𝑣, 𝑢⟩] ~R = 𝐶 → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) = (𝐶 +R [⟨𝑥, 𝑦⟩] ~R ))
6		oveq1 7424	. . . . . 6 ⊢ ([⟨𝑣, 𝑢⟩] ~R = 𝐶 → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ))
7	5, 6	breq12d 5120	. . . . 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 )))
8	7	bibi2d 345	. . . 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 ))))
9		breq1 5110	. . . . 5 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝐴 <R [⟨𝑧, 𝑤⟩] ~R ))
10		oveq2 7425	. . . . . 6 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (𝐶 +R [⟨𝑥, 𝑦⟩] ~R ) = (𝐶 +R 𝐴))
11	10	breq1d 5117	. . . . 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 )))
12	9, 11	bibi12d 348	. . . 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 ))))
13		breq2 5111	. . . . 5 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝐴 <R 𝐵))
14		oveq2 7425	. . . . . 6 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ) = (𝐶 +R 𝐵))
15	14	breq2d 5119	. . . . 5 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((𝐶 +R 𝐴) <R (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ) ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
16	13, 15	bibi12d 348	. . . 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))))
17		addclpr 11031	. . . . . . 7 ⊢ ((𝑣 ∈ P ∧ 𝑢 ∈ P) → (𝑣 +P 𝑢) ∈ P)
18	17	3ad2ant1 1151	. . . . . 6 ⊢ (((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑣 +P 𝑢) ∈ P)
19		ltapr 11058	. . . . . . 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))))
20		ltsrpr 11090	. . . . . . 7 ⊢ ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧))
21		ltsrpr 11090	. . . . . . . 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)))
22		vex 3457	. . . . . . . . . 10 ⊢ 𝑣 ∈ V
23		vex 3457	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
24		vex 3457	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
25		addcompr 11034	. . . . . . . . . 10 ⊢ (𝑦 +P 𝑧) = (𝑧 +P 𝑦)
26		addasspr 11035	. . . . . . . . . 10 ⊢ ((𝑦 +P 𝑧) +P 𝑓) = (𝑦 +P (𝑧 +P 𝑓))
27		vex 3457	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
28	22, 23, 24, 25, 26, 27	caov4 7649	. . . . . . . . 9 ⊢ ((𝑣 +P 𝑥) +P (𝑢 +P 𝑤)) = ((𝑣 +P 𝑢) +P (𝑥 +P 𝑤))
29		addcompr 11034	. . . . . . . . . 10 ⊢ ((𝑢 +P 𝑦) +P (𝑣 +P 𝑧)) = ((𝑣 +P 𝑧) +P (𝑢 +P 𝑦))
30		vex 3457	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
31		addcompr 11034	. . . . . . . . . . 11 ⊢ (𝑥 +P 𝑤) = (𝑤 +P 𝑥)
32		addasspr 11035	. . . . . . . . . . 11 ⊢ ((𝑥 +P 𝑤) +P 𝑓) = (𝑥 +P (𝑤 +P 𝑓))
33		vex 3457	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
34	22, 30, 24, 31, 32, 33	caov42 7651	. . . . . . . . . 10 ⊢ ((𝑣 +P 𝑧) +P (𝑢 +P 𝑦)) = ((𝑣 +P 𝑢) +P (𝑦 +P 𝑧))
35	29, 34	eqtri 2785	. . . . . . . . 9 ⊢ ((𝑢 +P 𝑦) +P (𝑣 +P 𝑧)) = ((𝑣 +P 𝑢) +P (𝑦 +P 𝑧))
36	28, 35	breq12i 5116	. . . . . . . 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)))
37	21, 36	bitri 278	. . . . . . 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)))
38	19, 20, 37	3bitr4g 317	. . . . . 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 ))
39	18, 38	syl 18	. . . . 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 ))
40		addsrpr 11088	. . . . . . 7 ⊢ (((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P)) → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) = [⟨(𝑣 +P 𝑥), (𝑢 +P 𝑦)⟩] ~R )
41	40	3adant3 1150	. . . . . 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 )
42		addsrpr 11088	. . . . . . 7 ⊢ (((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = [⟨(𝑣 +P 𝑧), (𝑢 +P 𝑤)⟩] ~R )
43	42	3adant2 1149	. . . . . 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 )
44	41, 43	breq12d 5120	. . . . 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 ))
45	39, 44	bitr4d 285	. . . 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 )))
46	4, 8, 12, 16, 45	3ecoptocl 8813	. . 3 ⊢ ((𝐶 ∈ R ∧ 𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
47	46	3coml 1145	. 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
48	1, 2, 3, 47	ndmovord 7608	1 ⊢ (𝐶 ∈ R → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ⟨cop 4593   class class class wbr 5107  (class class class)co 7417  [cec 8698  Pcnp 10872   +_P cpp 10874  <_P cltp 10876   ~_R cer 10877  Rcnr 10878   +_R cplr 10882   <_R cltr 10884

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-inf2 9624

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-oadd 8463  df-omul 8464  df-er 8700  df-ec 8702  df-qs 8706  df-ni 10885  df-pli 10886  df-mi 10887  df-lti 10888  df-plpq 10921  df-mpq 10922  df-ltpq 10923  df-enq 10924  df-nq 10925  df-erq 10926  df-plq 10927  df-mq 10928  df-1nq 10929  df-rq 10930  df-ltnq 10931  df-np 10994  df-plp 10996  df-ltp 10998  df-enr 11068  df-nr 11069  df-plr 11070  df-ltr 11072

This theorem is used by:  addgt0sr  11117  sqgt0sr  11119  mappsrpr  11121  ltpsrpr  11122  map2psrpr  11123  supsrlem  11124  axpre-ltadd  11180