Theorem ltasr 10522

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 10507	. 2 ⊢ dom +R = (R × R)
2		ltrelsr 10490	. 2 ⊢ <R ⊆ (R × R)
3		0nsr 10501	. 2 ⊢ ¬ ∅ ∈ R
4		df-nr 10478	. . . 4 ⊢ R = ((P × P) / ~R )
5		oveq1 7163	. . . . . 6 ⊢ ([⟨𝑣, 𝑢⟩] ~R = 𝐶 → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) = (𝐶 +R [⟨𝑥, 𝑦⟩] ~R ))
6		oveq1 7163	. . . . . 6 ⊢ ([⟨𝑣, 𝑢⟩] ~R = 𝐶 → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ))
7	5, 6	breq12d 5079	. . . . 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 5069	. . . . 5 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝐴 <R [⟨𝑧, 𝑤⟩] ~R ))
10		oveq2 7164	. . . . . 6 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (𝐶 +R [⟨𝑥, 𝑦⟩] ~R ) = (𝐶 +R 𝐴))
11	10	breq1d 5076	. . . . 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 5070	. . . . 5 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝐴 <R 𝐵))
14		oveq2 7164	. . . . . 6 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ) = (𝐶 +R 𝐵))
15	14	breq2d 5078	. . . . 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 10440	. . . . . . 7 ⊢ ((𝑣 ∈ P ∧ 𝑢 ∈ P) → (𝑣 +P 𝑢) ∈ P)
18	17	3ad2ant1 1129	. . . . . 6 ⊢ (((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑣 +P 𝑢) ∈ P)
19		ltapr 10467	. . . . . . 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 10499	. . . . . . 7 ⊢ ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧))
21		ltsrpr 10499	. . . . . . . 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 3497	. . . . . . . . . 10 ⊢ 𝑣 ∈ V
23		vex 3497	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
24		vex 3497	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
25		addcompr 10443	. . . . . . . . . 10 ⊢ (𝑦 +P 𝑧) = (𝑧 +P 𝑦)
26		addasspr 10444	. . . . . . . . . 10 ⊢ ((𝑦 +P 𝑧) +P 𝑓) = (𝑦 +P (𝑧 +P 𝑓))
27		vex 3497	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
28	22, 23, 24, 25, 26, 27	caov4 7379	. . . . . . . . 9 ⊢ ((𝑣 +P 𝑥) +P (𝑢 +P 𝑤)) = ((𝑣 +P 𝑢) +P (𝑥 +P 𝑤))
29		addcompr 10443	. . . . . . . . . 10 ⊢ ((𝑢 +P 𝑦) +P (𝑣 +P 𝑧)) = ((𝑣 +P 𝑧) +P (𝑢 +P 𝑦))
30		vex 3497	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
31		addcompr 10443	. . . . . . . . . . 11 ⊢ (𝑥 +P 𝑤) = (𝑤 +P 𝑥)
32		addasspr 10444	. . . . . . . . . . 11 ⊢ ((𝑥 +P 𝑤) +P 𝑓) = (𝑥 +P (𝑤 +P 𝑓))
33		vex 3497	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
34	22, 30, 24, 31, 32, 33	caov42 7381	. . . . . . . . . 10 ⊢ ((𝑣 +P 𝑧) +P (𝑢 +P 𝑦)) = ((𝑣 +P 𝑢) +P (𝑦 +P 𝑧))
35	29, 34	eqtri 2844	. . . . . . . . 9 ⊢ ((𝑢 +P 𝑦) +P (𝑣 +P 𝑧)) = ((𝑣 +P 𝑢) +P (𝑦 +P 𝑧))
36	28, 35	breq12i 5075	. . . . . . . 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 277	. . . . . . 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 316	. . . . . 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 17	. . . . 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 10497	. . . . . . 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 1128	. . . . . 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 10497	. . . . . . 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 1127	. . . . . 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 5079	. . . . 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 284	. . . 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 8389	. . 3 ⊢ ((𝐶 ∈ R ∧ 𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
47	46	3coml 1123	. 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
48	1, 2, 3, 47	ndmovord 7338	1 ⊢ (𝐶 ∈ R → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ⟨cop 4573   class class class wbr 5066  (class class class)co 7156  [cec 8287  Pcnp 10281   +_P cpp 10283  <_P cltp 10285   ~_R cer 10286  Rcnr 10287   +_R cplr 10291   <_R cltr 10293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-omul 8107  df-er 8289  df-ec 8291  df-qs 8295  df-ni 10294  df-pli 10295  df-mi 10296  df-lti 10297  df-plpq 10330  df-mpq 10331  df-ltpq 10332  df-enq 10333  df-nq 10334  df-erq 10335  df-plq 10336  df-mq 10337  df-1nq 10338  df-rq 10339  df-ltnq 10340  df-np 10403  df-plp 10405  df-ltp 10407  df-enr 10477  df-nr 10478  df-plr 10479  df-ltr 10481

This theorem is referenced by:  addgt0sr  10526  sqgt0sr  10528  mappsrpr  10530  ltpsrpr  10531  map2psrpr  10532  supsrlem  10533  axpre-ltadd  10589