Theorem ltapr 10069

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 10036	. 2 ⊢ dom +P = (P × P)
2		ltrelpr 10022	. 2 ⊢ <P ⊆ (P × P)
3		0npr 10016	. 2 ⊢ ¬ ∅ ∈ P
4		ltaprlem 10068	. . . . . 6 ⊢ (𝐶 ∈ P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
5	4	adantr 466	. . . . 5 ⊢ ((𝐶 ∈ P ∧ (𝐵 ∈ P ∧ 𝐴 ∈ P)) → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
6		olc 857	. . . . . . . . 9 ⊢ ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → ((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
7		ltaprlem 10068	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ P → (𝐵<P 𝐴 → (𝐶 +P 𝐵)<P (𝐶 +P 𝐴)))
8	7	adantr 466	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ P ∧ (𝐵 ∈ P ∧ 𝐴 ∈ P)) → (𝐵<P 𝐴 → (𝐶 +P 𝐵)<P (𝐶 +P 𝐴)))
9		ltsopr 10056	. . . . . . . . . . . . 13 ⊢ <P Or P
10		sotric 5196	. . . . . . . . . . . . 13 ⊢ ((<P Or P ∧ (𝐵 ∈ P ∧ 𝐴 ∈ P)) → (𝐵<P 𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴<P 𝐵)))
11	9, 10	mpan 670	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (𝐵<P 𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴<P 𝐵)))
12	11	adantl 467	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ P ∧ (𝐵 ∈ P ∧ 𝐴 ∈ P)) → (𝐵<P 𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴<P 𝐵)))
13		addclpr 10042	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ P ∧ 𝐵 ∈ P) → (𝐶 +P 𝐵) ∈ P)
14		addclpr 10042	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ P ∧ 𝐴 ∈ P) → (𝐶 +P 𝐴) ∈ P)
15	13, 14	anim12dan 605	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ P ∧ (𝐵 ∈ P ∧ 𝐴 ∈ P)) → ((𝐶 +P 𝐵) ∈ P ∧ (𝐶 +P 𝐴) ∈ P))
16		sotric 5196	. . . . . . . . . . . 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 575	. . . . . . . . . . 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 282	. . . . . . . . . 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 837	. . . . . . . 8 ⊢ ((𝐵 = 𝐴 ∨ 𝐴<P 𝐵) ↔ (¬ 𝐵 = 𝐴 → 𝐴<P 𝐵))
22	20, 21	syl6ib 241	. . . . . . 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 5663	. . . . . . . 8 ⊢ ¬ (𝐶 +P 𝐴)<P (𝐶 +P 𝐴)
25		oveq2 6801	. . . . . . . . 9 ⊢ (𝐵 = 𝐴 → (𝐶 +P 𝐵) = (𝐶 +P 𝐴))
26	25	breq2d 4798	. . . . . . . 8 ⊢ (𝐵 = 𝐴 → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐴)))
27	24, 26	mtbiri 316	. . . . . . 7 ⊢ (𝐵 = 𝐴 → ¬ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))
28	27	pm2.21d 119	. . . . . 6 ⊢ (𝐵 = 𝐴 → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → 𝐴<P 𝐵))
29	23, 28	pm2.61d2 173	. . . . 5 ⊢ ((𝐶 ∈ P ∧ (𝐵 ∈ P ∧ 𝐴 ∈ P)) → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → 𝐴<P 𝐵))
30	5, 29	impbid 202	. . . 4 ⊢ ((𝐶 ∈ P ∧ (𝐵 ∈ P ∧ 𝐴 ∈ P)) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
31	30	3impb 1107	. . 3 ⊢ ((𝐶 ∈ P ∧ 𝐵 ∈ P ∧ 𝐴 ∈ P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
32	31	3com13 1118	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
33	1, 2, 3, 32	ndmovord 6971	1 ⊢ (𝐶 ∈ P → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   ∨ wo 836   = wceq 1631   ∈ wcel 2145   class class class wbr 4786   Or wor 5169  (class class class)co 6793  Pcnp 9883   +_P cpp 9885  <_P cltp 9887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-omul 7718  df-er 7896  df-ni 9896  df-pli 9897  df-mi 9898  df-lti 9899  df-plpq 9932  df-mpq 9933  df-ltpq 9934  df-enq 9935  df-nq 9936  df-erq 9937  df-plq 9938  df-mq 9939  df-1nq 9940  df-rq 9941  df-ltnq 9942  df-np 10005  df-plp 10007  df-ltp 10009

This theorem is referenced by:  addcanpr  10070  ltsrpr  10100  gt0srpr  10101  ltsosr  10117  ltasr  10123  ltpsrpr  10132  map2psrpr  10133