Theorem nn0opth 4724

Description: An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. We can represent an ordered pair of nonnegative integers A and B by (((A + B) · (A + B)) + B). If two such ordered pairs are equal, their first elements are equal and their second elements are equal. Contrast this ordered pair representation with the standard one df-op 1815 that works for any set. (Contributed by Raph Levien, 10-Dec-02.)

Hypotheses

Ref	Expression
nn0opth.1	⊢ A ∈ ℕ0
nn0opth.2	⊢ B ∈ ℕ0
nn0opth.3	⊢ C ∈ ℕ0
nn0opth.4	⊢ D ∈ ℕ0

Assertion

Ref	Expression
nn0opth	⊢ ((((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D) ↔ (A = C ∧ B = D))

Proof of Theorem nn0opth

Step	Hyp	Ref	Expression
1		nn0opth.1	. . . . . . 7 ⊢ A ∈ ℕ0
2		nn0opth.2	. . . . . . 7 ⊢ B ∈ ℕ0
3	1, 2	nn0addge2 4561	. . . . . 6 ⊢ B ≤ (A + B)
4		nn0opth.3	. . . . . . 7 ⊢ C ∈ ℕ0
5		nn0opth.4	. . . . . . 7 ⊢ D ∈ ℕ0
6	4, 5	nn0addge2 4561	. . . . . 6 ⊢ D ≤ (C + D)
7	1, 2	nn0addcl 4552	. . . . . . . . . 10 ⊢ (A + B) ∈ ℕ0
8	4, 5	nn0addcl 4552	. . . . . . . . . 10 ⊢ (C + D) ∈ ℕ0
9	7, 2, 8, 5	nn0opthlem2 4723	. . . . . . . . 9 ⊢ ((B ≤ (A + B) ∧ D ≤ (C + D)) → ((A + B) < (C + D) → ¬ (((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D)))
10	8, 5, 7, 2	nn0opthlem2 4723	. . . . . . . . . . 11 ⊢ ((D ≤ (C + D) ∧ B ≤ (A + B)) → ((C + D) < (A + B) → ¬ (((C + D) · (C + D)) + D) = (((A + B) · (A + B)) + B)))
11	10	ancoms 334	. . . . . . . . . 10 ⊢ ((B ≤ (A + B) ∧ D ≤ (C + D)) → ((C + D) < (A + B) → ¬ (((C + D) · (C + D)) + D) = (((A + B) · (A + B)) + B)))
12		cleqcom 1103	. . . . . . . . . . 11 ⊢ ((((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D) ↔ (((C + D) · (C + D)) + D) = (((A + B) · (A + B)) + B))
13	12	negbii 162	. . . . . . . . . 10 ⊢ (¬ (((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D) ↔ ¬ (((C + D) · (C + D)) + D) = (((A + B) · (A + B)) + B))
14	11, 13	syl6ibr 186	. . . . . . . . 9 ⊢ ((B ≤ (A + B) ∧ D ≤ (C + D)) → ((C + D) < (A + B) → ¬ (((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D)))
15	9, 14	jaod 329	. . . . . . . 8 ⊢ ((B ≤ (A + B) ∧ D ≤ (C + D)) → (((A + B) < (C + D) ∨ (C + D) < (A + B)) → ¬ (((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D)))
16	1	nn0re 4544	. . . . . . . . . 10 ⊢ A ∈ ℝ
17	2	nn0re 4544	. . . . . . . . . 10 ⊢ B ∈ ℝ
18	16, 17	readdcl 4118	. . . . . . . . 9 ⊢ (A + B) ∈ ℝ
19	8	nn0re 4544	. . . . . . . . 9 ⊢ (C + D) ∈ ℝ
20	18, 19	lttri2 4295	. . . . . . . 8 ⊢ (¬ (A + B) = (C + D) ↔ ((A + B) < (C + D) ∨ (C + D) < (A + B)))
21	15, 20	syl5ib 181	. . . . . . 7 ⊢ ((B ≤ (A + B) ∧ D ≤ (C + D)) → (¬ (A + B) = (C + D) → ¬ (((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D)))
22	21	a3d 70	. . . . . 6 ⊢ ((B ≤ (A + B) ∧ D ≤ (C + D)) → ((((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D) → (A + B) = (C + D)))
23	3, 6, 22	mp2an 520	. . . . 5 ⊢ ((((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D) → (A + B) = (C + D))
24		id 9	. . . . . . . 8 ⊢ ((((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D) → (((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D))
25	23, 23	opreq12d 3014	. . . . . . . . 9 ⊢ ((((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D) → ((A + B) · (A + B)) = ((C + D) · (C + D)))
26	25	opreq1d 3012	. . . . . . . 8 ⊢ ((((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D) → (((A + B) · (A + B)) + D) = (((C + D) · (C + D)) + D))
27	24, 26	eqtr4d 1131	. . . . . . 7 ⊢ ((((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D) → (((A + B) · (A + B)) + B) = (((A + B) · (A + B)) + D))
28	1	nn0cn 4545	. . . . . . . . . 10 ⊢ A ∈ ℂ
29	2	nn0cn 4545	. . . . . . . . . 10 ⊢ B ∈ ℂ
30	28, 29	addcl 4104	. . . . . . . . 9 ⊢ (A + B) ∈ ℂ
31	30, 30	mulcl 4105	. . . . . . . 8 ⊢ ((A + B) · (A + B)) ∈ ℂ
32	5	nn0cn 4545	. . . . . . . 8 ⊢ D ∈ ℂ
33	31, 29, 32	addcan 4120	. . . . . . 7 ⊢ ((((A + B) · (A + B)) + B) = (((A + B) · (A + B)) + D) ↔ B = D)
34	27, 33	sylib 173	. . . . . 6 ⊢ ((((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D) → B = D)
35	34	opreq2d 3013	. . . . 5 ⊢ ((((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D) → (C + B) = (C + D))
36	23, 35	eqtr4d 1131	. . . 4 ⊢ ((((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D) → (A + B) = (C + B))
37	4	nn0cn 4545	. . . . 5 ⊢ C ∈ ℂ
38	28, 37, 29	addcan2 4121	. . . 4 ⊢ ((A + B) = (C + B) ↔ A = C)
39	36, 38	sylib 173	. . 3 ⊢ ((((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D) → A = C)
40	39, 34	jca 236	. 2 ⊢ ((((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D) → (A = C ∧ B = D))
41		opreq12 3008	. . . 4 ⊢ ((A = C ∧ B = D) → (A + B) = (C + D))
42	41, 41	opreq12d 3014	. . 3 ⊢ ((A = C ∧ B = D) → ((A + B) · (A + B)) = ((C + D) · (C + D)))
43		pm3.27 260	. . 3 ⊢ ((A = C ∧ B = D) → B = D)
44	42, 43	opreq12d 3014	. 2 ⊢ ((A = C ∧ B = D) → (((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D))
45	40, 44	impbi 139	1 ⊢ ((((A + B) · (A + B)) + B) = (((C + D) · (C + D)) + D) ↔ (A = C ∧ B = D))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092   class class class wbr 2054  (class class class)co 3001   + caddc 4031   · cmulc 4032   < clt 4033   ≤ cle 4092  ℕ₀cn0 4094

This theorem is referenced by:  nn0opth2 4725

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-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1st 3087  df-2nd 3088  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-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-div 4216  df-le 4277  df-n 4423  df-2 4462  df-n0 4535  df-z 4564  df-seq 4661  df-exp 4676

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