Theorem ltapr 10467

Description: Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
ltapr	⊢ (𝐶 ∈ P → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))

Proof of Theorem ltapr

Step	Hyp	Ref	Expression
1		dmplp 10434	. 2 ⊢ dom +P = (P × P)
2		ltrelpr 10420	. 2 ⊢ <P ⊆ (P × P)
3		0npr 10414	. 2 ⊢ ¬ ∅ ∈ P
4		ltaprlem 10466	. . . . . 6 ⊢ (𝐶 ∈ P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
5	4	adantr 483	. . . . 5 ⊢ ((𝐶 ∈ P ∧ (𝐵 ∈ P ∧ 𝐴 ∈ P)) → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
6		olc 864	. . . . . . . . 9 ⊢ ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → ((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
7		ltaprlem 10466	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ P → (𝐵<P 𝐴 → (𝐶 +P 𝐵)<P (𝐶 +P 𝐴)))
8	7	adantr 483	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ P ∧ (𝐵 ∈ P ∧ 𝐴 ∈ P)) → (𝐵<P 𝐴 → (𝐶 +P 𝐵)<P (𝐶 +P 𝐴)))
9		ltsopr 10454	. . . . . . . . . . . . 13 ⊢ <P Or P
10		sotric 5501	. . . . . . . . . . . . 13 ⊢ ((<P Or P ∧ (𝐵 ∈ P ∧ 𝐴 ∈ P)) → (𝐵<P 𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴<P 𝐵)))
11	9, 10	mpan 688	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (𝐵<P 𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴<P 𝐵)))
12	11	adantl 484	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ P ∧ (𝐵 ∈ P ∧ 𝐴 ∈ P)) → (𝐵<P 𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴<P 𝐵)))
13		addclpr 10440	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ P ∧ 𝐵 ∈ P) → (𝐶 +P 𝐵) ∈ P)
14		addclpr 10440	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ P ∧ 𝐴 ∈ P) → (𝐶 +P 𝐴) ∈ P)
15	13, 14	anim12dan 620	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ P ∧ (𝐵 ∈ P ∧ 𝐴 ∈ P)) → ((𝐶 +P 𝐵) ∈ P ∧ (𝐶 +P 𝐴) ∈ P))
16		sotric 5501	. . . . . . . . . . . 12 ⊢ ((<P Or P ∧ ((𝐶 +P 𝐵) ∈ P ∧ (𝐶 +P 𝐴) ∈ P)) → ((𝐶 +P 𝐵)<P (𝐶 +P 𝐴) ↔ ¬ ((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))))
17	9, 15, 16	sylancr 589	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ P ∧ (𝐵 ∈ P ∧ 𝐴 ∈ P)) → ((𝐶 +P 𝐵)<P (𝐶 +P 𝐴) ↔ ¬ ((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))))
18	8, 12, 17	3imtr3d 295	. . . . . . . . . 10 ⊢ ((𝐶 ∈ P ∧ (𝐵 ∈ P ∧ 𝐴 ∈ P)) → (¬ (𝐵 = 𝐴 ∨ 𝐴<P 𝐵) → ¬ ((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))))
19	18	con4d 115	. . . . . . . . 9 ⊢ ((𝐶 ∈ P ∧ (𝐵 ∈ P ∧ 𝐴 ∈ P)) → (((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)) → (𝐵 = 𝐴 ∨ 𝐴<P 𝐵)))
20	6, 19	syl5 34	. . . . . . . 8 ⊢ ((𝐶 ∈ P ∧ (𝐵 ∈ P ∧ 𝐴 ∈ P)) → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → (𝐵 = 𝐴 ∨ 𝐴<P 𝐵)))
21		df-or 844	. . . . . . . 8 ⊢ ((𝐵 = 𝐴 ∨ 𝐴<P 𝐵) ↔ (¬ 𝐵 = 𝐴 → 𝐴<P 𝐵))
22	20, 21	syl6ib 253	. . . . . . 7 ⊢ ((𝐶 ∈ P ∧ (𝐵 ∈ P ∧ 𝐴 ∈ P)) → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → (¬ 𝐵 = 𝐴 → 𝐴<P 𝐵)))
23	22	com23 86	. . . . . 6 ⊢ ((𝐶 ∈ P ∧ (𝐵 ∈ P ∧ 𝐴 ∈ P)) → (¬ 𝐵 = 𝐴 → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → 𝐴<P 𝐵)))
24	9, 2	soirri 5986	. . . . . . . 8 ⊢ ¬ (𝐶 +P 𝐴)<P (𝐶 +P 𝐴)
25		oveq2 7164	. . . . . . . . 9 ⊢ (𝐵 = 𝐴 → (𝐶 +P 𝐵) = (𝐶 +P 𝐴))
26	25	breq2d 5078	. . . . . . . 8 ⊢ (𝐵 = 𝐴 → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐴)))
27	24, 26	mtbiri 329	. . . . . . 7 ⊢ (𝐵 = 𝐴 → ¬ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))
28	27	pm2.21d 121	. . . . . 6 ⊢ (𝐵 = 𝐴 → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → 𝐴<P 𝐵))
29	23, 28	pm2.61d2 183	. . . . 5 ⊢ ((𝐶 ∈ P ∧ (𝐵 ∈ P ∧ 𝐴 ∈ P)) → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → 𝐴<P 𝐵))
30	5, 29	impbid 214	. . . 4 ⊢ ((𝐶 ∈ P ∧ (𝐵 ∈ P ∧ 𝐴 ∈ P)) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
31	30	3impb 1111	. . 3 ⊢ ((𝐶 ∈ P ∧ 𝐵 ∈ P ∧ 𝐴 ∈ P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
32	31	3com13 1120	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
33	1, 2, 3, 32	ndmovord 7338	1 ⊢ (𝐶 ∈ P → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114   class class class wbr 5066   Or wor 5473  (class class class)co 7156  Pcnp 10281   +_P cpp 10283  <_P cltp 10285

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-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

This theorem is referenced by:  addcanpr  10468  ltsrpr  10499  gt0srpr  10500  ltsosr  10516  ltasr  10522  ltpsrpr  10531  map2psrpr  10532