Theorem nn0opth 6604

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) x. (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 2412 that works for any set. (Contributed by Raph Levien, 10-Dec-2002.)

Hypotheses

Ref	Expression
nn0opth.1	|- A e. NN0
nn0opth.2	|- B e. NN0
nn0opth.3	|- C e. NN0
nn0opth.4	|- D e. NN0

Assertion

Ref	Expression
nn0opth	|- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) <-> (A = C /\ B = D))

Proof of Theorem nn0opth

Step	Hyp	Ref	Expression
1		nn0opth.2	. . . . . . . 8 |- B e. NN0
2	1	nn0re 6065	. . . . . . 7 |- B e. RR
3		nn0opth.1	. . . . . . 7 |- A e. NN0
4	2, 3	nn0addge2 6088	. . . . . 6 |- B <_ (A + B)
5		nn0opth.4	. . . . . . . 8 |- D e. NN0
6	5	nn0re 6065	. . . . . . 7 |- D e. RR
7		nn0opth.3	. . . . . . 7 |- C e. NN0
8	6, 7	nn0addge2 6088	. . . . . 6 |- D <_ (C + D)
9	3, 1	nn0addcl 6076	. . . . . . . . . 10 |- (A + B) e. NN0
10	7, 5	nn0addcl 6076	. . . . . . . . . 10 |- (C + D) e. NN0
11	9, 1, 10, 5	nn0opthlem2 6603	. . . . . . . . 9 |- ((B <_ (A + B) /\ D <_ (C + D)) -> ((A + B) < (C + D) -> -. (((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D)))
12	10, 5, 9, 1	nn0opthlem2 6603	. . . . . . . . . . 11 |- ((D <_ (C + D) /\ B <_ (A + B)) -> ((C + D) < (A + B) -> -. (((C + D) x. (C + D)) + D) = (((A + B) x. (A + B)) + B)))
13	12	ancoms 436	. . . . . . . . . 10 |- ((B <_ (A + B) /\ D <_ (C + D)) -> ((C + D) < (A + B) -> -. (((C + D) x. (C + D)) + D) = (((A + B) x. (A + B)) + B)))
14		eqcom 1474	. . . . . . . . . . 11 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) <-> (((C + D) x. (C + D)) + D) = (((A + B) x. (A + B)) + B))
15	14	negbii 187	. . . . . . . . . 10 |- (-. (((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) <-> -. (((C + D) x. (C + D)) + D) = (((A + B) x. (A + B)) + B))
16	13, 15	syl6ibr 213	. . . . . . . . 9 |- ((B <_ (A + B) /\ D <_ (C + D)) -> ((C + D) < (A + B) -> -. (((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D)))
17	11, 16	jaod 424	. . . . . . . 8 |- ((B <_ (A + B) /\ D <_ (C + D)) -> (((A + B) < (C + D) \/ (C + D) < (A + B)) -> -. (((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D)))
18		df-ne 1584	. . . . . . . . 9 |- ((A + B) =/= (C + D) <-> -. (A + B) = (C + D))
19	3	nn0re 6065	. . . . . . . . . . 11 |- A e. RR
20	19, 2	readdcl 5314	. . . . . . . . . 10 |- (A + B) e. RR
21	10	nn0re 6065	. . . . . . . . . 10 |- (C + D) e. RR
22	20, 21	lttri2 5553	. . . . . . . . 9 |- ((A + B) =/= (C + D) <-> ((A + B) < (C + D) \/ (C + D) < (A + B)))
23	18, 22	bitr3 175	. . . . . . . 8 |- (-. (A + B) = (C + D) <-> ((A + B) < (C + D) \/ (C + D) < (A + B)))
24	17, 23	syl5ib 206	. . . . . . 7 |- ((B <_ (A + B) /\ D <_ (C + D)) -> (-. (A + B) = (C + D) -> -. (((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D)))
25	24	a3d 75	. . . . . 6 |- ((B <_ (A + B) /\ D <_ (C + D)) -> ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> (A + B) = (C + D)))
26	4, 8, 25	mp2an 696	. . . . 5 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> (A + B) = (C + D))
27		id 59	. . . . . . . 8 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> (((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D))
28	26, 26	opreq12d 3969	. . . . . . . . 9 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> ((A + B) x. (A + B)) = ((C + D) x. (C + D)))
29	28	opreq1d 3966	. . . . . . . 8 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> (((A + B) x. (A + B)) + D) = (((C + D) x. (C + D)) + D))
30	27, 29	eqtr4d 1507	. . . . . . 7 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> (((A + B) x. (A + B)) + B) = (((A + B) x. (A + B)) + D))
31	20	recn 5294	. . . . . . . . 9 |- (A + B) e. CC
32	31, 31	mulcl 5301	. . . . . . . 8 |- ((A + B) x. (A + B)) e. CC
33	1	nn0cn 6066	. . . . . . . 8 |- B e. CC
34	5	nn0cn 6066	. . . . . . . 8 |- D e. CC
35	32, 33, 34	addcan 5331	. . . . . . 7 |- ((((A + B) x. (A + B)) + B) = (((A + B) x. (A + B)) + D) <-> B = D)
36	30, 35	sylib 198	. . . . . 6 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> B = D)
37	36	opreq2d 3967	. . . . 5 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> (C + B) = (C + D))
38	26, 37	eqtr4d 1507	. . . 4 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> (A + B) = (C + B))
39	3	nn0cn 6066	. . . . 5 |- A e. CC
40	7	nn0cn 6066	. . . . 5 |- C e. CC
41	39, 40, 33	addcan2 5334	. . . 4 |- ((A + B) = (C + B) <-> A = C)
42	38, 41	sylib 198	. . 3 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> A = C)
43	42, 36	jca 288	. 2 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> (A = C /\ B = D))
44		opreq12 3961	. . . 4 |- ((A = C /\ B = D) -> (A + B) = (C + D))
45	44, 44	opreq12d 3969	. . 3 |- ((A = C /\ B = D) -> ((A + B) x. (A + B)) = ((C + D) x. (C + D)))
46		pm3.27 323	. . 3 |- ((A = C /\ B = D) -> B = D)
47	45, 46	opreq12d 3969	. 2 |- ((A = C /\ B = D) -> (((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D))
48	43, 47	impbi 157	1 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) <-> (A = C /\ B = D))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 954   e. wcel 956   =/= wne 1582   class class class wbr 2614  (class class class)co 3954   + caddc 5217   x. cmul 5219   <_ cle 5275  NN0cn0 5277   < clt 5466

This theorem is referenced by:  nn0opth2 6605

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-en 4357  df-dom 4358  df-sdom 4359  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-lt 5227  df-sub 5336  df-neg 5338  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470  df-le 5471  df-n 5881  df-2 5925  df-n0 6055  df-z 6091  df-seq1