Theorem ltapr 10464

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 10431	. 2 ⊢ dom +P = (P × P)
2		ltrelpr 10417	. 2 ⊢ <P ⊆ (P × P)
3		0npr 10411	. 2 ⊢ ¬ ∅ ∈ P
4		ltaprlem 10463	. . . . . 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 10463	. . . . . . . . . . . 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 10451	. . . . . . . . . . . . 13 ⊢ <P Or P
10		sotric 5498	. . . . . . . . . . . . 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 10437	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ P ∧ 𝐵 ∈ P) → (𝐶 +P 𝐵) ∈ P)
14		addclpr 10437	. . . . . . . . . . . . 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 5498	. . . . . . . . . . . 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 5983	. . . . . . . 8 ⊢ ¬ (𝐶 +P 𝐴)<P (𝐶 +P 𝐴)
25		oveq2 7161	. . . . . . . . 9 ⊢ (𝐵 = 𝐴 → (𝐶 +P 𝐵) = (𝐶 +P 𝐴))
26	25	breq2d 5075	. . . . . . . 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 1110	. . 3 ⊢ ((𝐶 ∈ P ∧ 𝐵 ∈ P ∧ 𝐴 ∈ P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
32	31	3com13 1119	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
33	1, 2, 3, 32	ndmovord 7335	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 1536   ∈ wcel 2113   class class class wbr 5063   Or wor 5470  (class class class)co 7153  Pcnp 10278   +_P cpp 10280  <_P cltp 10282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5327  ax-un 7458  ax-inf2 9101

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-reu 3144  df-rmo 3145  df-rab 3146  df-v 3495  df-sbc 3771  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4465  df-pw 4538  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4836  df-int 4874  df-iun 4918  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5457  df-eprel 5462  df-po 5471  df-so 5472  df-fr 5511  df-we 5513  df-xp 5558  df-rel 5559  df-cnv 5560  df-co 5561  df-dm 5562  df-rn 5563  df-res 5564  df-ima 5565  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7156  df-oprab 7157  df-mpo 7158  df-om 7578  df-1st 7686  df-2nd 7687  df-wrecs 7944  df-recs 8005  df-rdg 8043  df-1o 8099  df-oadd 8103  df-omul 8104  df-er 8286  df-ni 10291  df-pli 10292  df-mi 10293  df-lti 10294  df-plpq 10327  df-mpq 10328  df-ltpq 10329  df-enq 10330  df-nq 10331  df-erq 10332  df-plq 10333  df-mq 10334  df-1nq 10335  df-rq 10336  df-ltnq 10337  df-np 10400  df-plp 10402  df-ltp 10404

This theorem is referenced by:  addcanpr  10465  ltsrpr  10496  gt0srpr  10497  ltsosr  10513  ltasr  10519  ltpsrpr  10528  map2psrpr  10529