Theorem ltapr 3945

Description: Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123.

Hypotheses

Ref	Expression
ltapr.1	⊢ A ∈ V
ltapr.2	⊢ B ∈ V

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		ltapr.2	. 2 ⊢ B ∈ V
2		dmplp 3909	. 2 ⊢ dom +P = (P × P)
3		ltapr.1	. 2 ⊢ A ∈ V
4		ltrelpr 3895	. 2 ⊢ <P ⊆ (P × P)
5		0npr 3890	. 2 ⊢ ¬ ∅ ∈ P
6	3, 1	ltaprlem 3944	. . . . . 6 ⊢ (C ∈ P → (A<P B → (C +P A)<P (C +P B)))
7	6	adantr 306	. . . . 5 ⊢ ((C ∈ P ∧ (B ∈ P ∧ A ∈ P)) → (A<P B → (C +P A)<P (C +P B)))
8	1, 3	ltaprlem 3944	. . . . . . . . . . . 12 ⊢ (C ∈ P → (B<P A → (C +P B)<P (C +P A)))
9	8	adantr 306	. . . . . . . . . . 11 ⊢ ((C ∈ P ∧ (B ∈ P ∧ A ∈ P)) → (B<P A → (C +P B)<P (C +P A)))
10		ltsopr 3930	. . . . . . . . . . . . 13 ⊢ <P Or P
11		sotric 2148	. . . . . . . . . . . . 13 ⊢ ((<P Or P ∧ (B ∈ P ∧ A ∈ P)) → (B<P A ↔ ¬ (B = A ∨ A<P B)))
12	10, 11	mpan 518	. . . . . . . . . . . 12 ⊢ ((B ∈ P ∧ A ∈ P) → (B<P A ↔ ¬ (B = A ∨ A<P B)))
13	12	adantl 305	. . . . . . . . . . 11 ⊢ ((C ∈ P ∧ (B ∈ P ∧ A ∈ P)) → (B<P A ↔ ¬ (B = A ∨ A<P B)))
14		addclpr 3914	. . . . . . . . . . . . . 14 ⊢ ((C ∈ P ∧ B ∈ P) → (C +P B) ∈ P)
15		addclpr 3914	. . . . . . . . . . . . . 14 ⊢ ((C ∈ P ∧ A ∈ P) → (C +P A) ∈ P)
16	14, 15	anim12i 268	. . . . . . . . . . . . 13 ⊢ (((C ∈ P ∧ B ∈ P) ∧ (C ∈ P ∧ A ∈ P)) → ((C +P B) ∈ P ∧ (C +P A) ∈ P))
17	16	anandis 394	. . . . . . . . . . . 12 ⊢ ((C ∈ P ∧ (B ∈ P ∧ A ∈ P)) → ((C +P B) ∈ P ∧ (C +P A) ∈ P))
18		sotric 2148	. . . . . . . . . . . . 13 ⊢ ((<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))))
19	10, 18	mpan 518	. . . . . . . . . . . 12 ⊢ (((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))))
20	17, 19	syl 12	. . . . . . . . . . 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))))
21	9, 13, 20	3imtr3d 420	. . . . . . . . . 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))))
22	21	a3d 70	. . . . . . . . 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)))
23		olc 224	. . . . . . . . 9 ⊢ ((C +P A)<P (C +P B) → ((C +P B) = (C +P A) ∨ (C +P A)<P (C +P B)))
24	22, 23	syl5 22	. . . . . . . 8 ⊢ ((C ∈ P ∧ (B ∈ P ∧ A ∈ P)) → ((C +P A)<P (C +P B) → (B = A ∨ A<P B)))
25		df-or 197	. . . . . . . 8 ⊢ ((B = A ∨ A<P B) ↔ (¬ B = A → A<P B))
26	24, 25	syl6ib 185	. . . . . . 7 ⊢ ((C ∈ P ∧ (B ∈ P ∧ A ∈ P)) → ((C +P A)<P (C +P B) → (¬ B = A → A<P B)))
27	26	com23 32	. . . . . 6 ⊢ ((C ∈ P ∧ (B ∈ P ∧ A ∈ P)) → (¬ B = A → ((C +P A)<P (C +P B) → A<P B)))
28		oprex 3018	. . . . . . . . 9 ⊢ (C +P A) ∈ V
29	28, 10, 4	soirri 2629	. . . . . . . 8 ⊢ ¬ (C +P A)<P (C +P A)
30		opreq2 3007	. . . . . . . . 9 ⊢ (B = A → (C +P B) = (C +P A))
31	30	breq2d 2072	. . . . . . . 8 ⊢ (B = A → ((C +P A)<P (C +P B) ↔ (C +P A)<P (C +P A)))
32	29, 31	mtbiri 539	. . . . . . 7 ⊢ (B = A → ¬ (C +P A)<P (C +P B))
33	32	pm2.21d 74	. . . . . 6 ⊢ (B = A → ((C +P A)<P (C +P B) → A<P B))
34	27, 33	pm2.61d2 111	. . . . 5 ⊢ ((C ∈ P ∧ (B ∈ P ∧ A ∈ P)) → ((C +P A)<P (C +P B) → A<P B))
35	7, 34	impbid 397	. . . 4 ⊢ ((C ∈ P ∧ (B ∈ P ∧ A ∈ P)) → (A<P B ↔ (C +P A)<P (C +P B)))
36	35	3impb 610	. . 3 ⊢ ((C ∈ P ∧ B ∈ P ∧ A ∈ P) → (A<P B ↔ (C +P A)<P (C +P B)))
37	36	3com13 615	. 2 ⊢ ((A ∈ P ∧ B ∈ P ∧ C ∈ P) → (A<P B ↔ (C +P A)<P (C +P B)))
38	1, 2, 3, 4, 5, 37	ndmord 3064	1 ⊢ (C ∈ P → (A<P B ↔ (C +P A)<P (C +P B)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   class class class wbr 2054   Or wor 2059  (class class class)co 3001  Pcnp 3779   +_P cpp 3781  <_P cltp 3783

This theorem is referenced by:  addcanpr 3946  ltsrpr 3980  gt0srpr 3981  ltsosr 3997  ltasr 4003  ltpsrpr 4013

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-plp 3882  df-ltp 3884

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