Theorem nn0opth 6666

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 2416 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.4	. . . . . . . 8 |- D e. NN0
2	1	nn0re 6110	. . . . . . 7 |- D e. RR
3		nn0opth.3	. . . . . . 7 |- C e. NN0
4	2, 3	nn0addge2 6133	. . . . . 6 |- D <_ (C + D)
5		nn0opth.2	. . . . . . . 8 |- B e. NN0
6	5	nn0re 6110	. . . . . . 7 |- B e. RR
7		nn0opth.1	. . . . . . 7 |- A e. NN0
8	6, 7	nn0addge2 6133	. . . . . 6 |- B <_ (A + B)
9	7, 5	nn0addcl 6121	. . . . . . . . . . . 12 |- (A + B) e. NN0
10	3, 1	nn0addcl 6121	. . . . . . . . . . . 12 |- (C + D) e. NN0
11	9, 5, 10, 1	nn0opthlem2 6665	. . . . . . . . . . 11 |- ((B <_ (A + B) /\ D <_ (C + D)) -> ((A + B) < (C + D) -> (((C + D) x. (C + D)) + D) =/= (((A + B) x. (A + B)) + B)))
12	11	ancoms 436	. . . . . . . . . 10 |- ((D <_ (C + D) /\ B <_ (A + B)) -> ((A + B) < (C + D) -> (((C + D) x. (C + D)) + D) =/= (((A + B) x. (A + B)) + B)))
13		necom 1636	. . . . . . . . . 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))
14	12, 13	syl6ibr 213	. . . . . . . . 9 |- ((D <_ (C + D) /\ B <_ (A + B)) -> ((A + B) < (C + D) -> (((A + B) x. (A + B)) + B) =/= (((C + D) x. (C + D)) + D)))
15	10, 1, 9, 5	nn0opthlem2 6665	. . . . . . . . 9 |- ((D <_ (C + D) /\ B <_ (A + B)) -> ((C + D) < (A + B) -> (((A + B) x. (A + B)) + B) =/= (((C + D) x. (C + D)) + D)))
16	14, 15	jaod 424	. . . . . . . 8 |- ((D <_ (C + D) /\ B <_ (A + B)) -> (((A + B) < (C + D) \/ (C + D) < (A + B)) -> (((A + B) x. (A + B)) + B) =/= (((C + D) x. (C + D)) + D)))
17	7	nn0re 6110	. . . . . . . . . 10 |- A e. RR
18	17, 6	readdcl 5334	. . . . . . . . 9 |- (A + B) e. RR
19	10	nn0re 6110	. . . . . . . . 9 |- (C + D) e. RR
20	18, 19	lttri2 5572	. . . . . . . 8 |- ((A + B) =/= (C + D) <-> ((A + B) < (C + D) \/ (C + D) < (A + B)))
21	16, 20	syl5ib 206	. . . . . . 7 |- ((D <_ (C + D) /\ B <_ (A + B)) -> ((A + B) =/= (C + D) -> (((A + B) x. (A + B)) + B) =/= (((C + D) x. (C + D)) + D)))
22	21	necon4d 1628	. . . . . 6 |- ((D <_ (C + D) /\ B <_ (A + B)) -> ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> (A + B) = (C + D)))
23	4, 8, 22	mp2an 697	. . . . 5 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> (A + B) = (C + D))
24		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))
25	23, 23	opreq12d 3978	. . . . . . . . 9 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> ((A + B) x. (A + B)) = ((C + D) x. (C + D)))
26	25	opreq1d 3975	. . . . . . . 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))
27	24, 26	eqtr4d 1510	. . . . . . 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))
28	18	recn 5314	. . . . . . . . 9 |- (A + B) e. CC
29	28, 28	mulcl 5321	. . . . . . . 8 |- ((A + B) x. (A + B)) e. CC
30	5	nn0cn 6111	. . . . . . . 8 |- B e. CC
31	1	nn0cn 6111	. . . . . . . 8 |- D e. CC
32	29, 30, 31	addcan 5351	. . . . . . 7 |- ((((A + B) x. (A + B)) + B) = (((A + B) x. (A + B)) + D) <-> B = D)
33	27, 32	sylib 198	. . . . . 6 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> B = D)
34	33	opreq2d 3976	. . . . 5 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> (C + B) = (C + D))
35	23, 34	eqtr4d 1510	. . . 4 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> (A + B) = (C + B))
36	7	nn0cn 6111	. . . . 5 |- A e. CC
37	3	nn0cn 6111	. . . . 5 |- C e. CC
38	36, 37, 30	addcan2 5354	. . . 4 |- ((A + B) = (C + B) <-> A = C)
39	35, 38	sylib 198	. . 3 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> A = C)
40	39, 33	jca 288	. 2 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> (A = C /\ B = D))
41		opreq12 3970	. . . 4 |- ((A = C /\ B = D) -> (A + B) = (C + D))
42	41, 41	opreq12d 3978	. . 3 |- ((A = C /\ B = D) -> ((A + B) x. (A + B)) = ((C + D) x. (C + D)))
43		pm3.27 323	. . 3 |- ((A = C /\ B = D) -> B = D)
44	42, 43	opreq12d 3978	. 2 |- ((A = C /\ B = D) -> (((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D))
45	40, 44	impbi 157	1 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) <-> (A = C /\ B = D))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 956   e. wcel 958   =/= wne 1585   class class class wbr 2619  (class class class)co 3963   + caddc 5237   x. cmul 5239   <_ cle 5295  NN0cn0 5297   < clt 5486

This theorem is referenced by:  nn0opth2 6667

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-n 5925  df-2 5970  df-n0 6100  df-z 6136  df-seq1 6308  df-exp 6569

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