Theorem nn0opthi 6867

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 2474 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
nn0opthi	|- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) <-> (A = C /\ B = D))

Proof of Theorem nn0opthi

Step	Hyp	Ref	Expression
1		nn0opth.4	. . . . . . . 8 |- D e. NN0
2	1	nn0rei 6278	. . . . . . 7 |- D e. RR
3		nn0opth.3	. . . . . . 7 |- C e. NN0
4	2, 3	nn0addge2i 6301	. . . . . 6 |- D <_ (C + D)
5		nn0opth.2	. . . . . . . 8 |- B e. NN0
6	5	nn0rei 6278	. . . . . . 7 |- B e. RR
7		nn0opth.1	. . . . . . 7 |- A e. NN0
8	6, 7	nn0addge2i 6301	. . . . . 6 |- B <_ (A + B)
9	7, 5	nn0addcli 6289	. . . . . . . . . . . 12 |- (A + B) e. NN0
10	3, 1	nn0addcli 6289	. . . . . . . . . . . 12 |- (C + D) e. NN0
11	9, 5, 10, 1	nn0opthlem2 6866	. . . . . . . . . . 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 438	. . . . . . . . . 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 1682	. . . . . . . . . 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 211	. . . . . . . . 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 6866	. . . . . . . . 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	nn0rei 6278	. . . . . . . . . 10 |- A e. RR
18	17, 6	readdcli 5488	. . . . . . . . 9 |- (A + B) e. RR
19	10	nn0rei 6278	. . . . . . . . 9 |- (C + D) e. RR
20	18, 19	lttri2i 5726	. . . . . . . 8 |- ((A + B) =/= (C + D) <-> ((A + B) < (C + D) \/ (C + D) < (A + B)))
21	16, 20	syl5ib 204	. . . . . . 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 1672	. . . . . 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 701	. . . . 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 4036	. . . . . . . . 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 4033	. . . . . . . 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 1553	. . . . . . 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	recni 5468	. . . . . . . . 9 |- (A + B) e. CC
29	28, 28	mulcli 5475	. . . . . . . 8 |- ((A + B) x. (A + B)) e. CC
30	5	nn0cni 6279	. . . . . . . 8 |- B e. CC
31	1	nn0cni 6279	. . . . . . . 8 |- D e. CC
32	29, 30, 31	addcani 5505	. . . . . . 7 |- ((((A + B) x. (A + B)) + B) = (((A + B) x. (A + B)) + D) <-> B = D)
33	27, 32	sylib 196	. . . . . 6 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> B = D)
34	33	opreq2d 4034	. . . . 5 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> (C + B) = (C + D))
35	23, 34	eqtr4d 1553	. . . 4 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> (A + B) = (C + B))
36	7	nn0cni 6279	. . . . 5 |- A e. CC
37	3	nn0cni 6279	. . . . 5 |- C e. CC
38	36, 37, 30	addcan2i 5508	. . . 4 |- ((A + B) = (C + B) <-> A = C)
39	35, 38	sylib 196	. . 3 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> A = C)
40	39, 33	jca 286	. 2 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> (A = C /\ B = D))
41		opreq12 4028	. . . 4 |- ((A = C /\ B = D) -> (A + B) = (C + D))
42	41, 41	opreq12d 4036	. . 3 |- ((A = C /\ B = D) -> ((A + B) x. (A + B)) = ((C + D) x. (C + D)))
43		pm3.27 321	. . 3 |- ((A = C /\ B = D) -> B = D)
44	42, 43	opreq12d 4036	. 2 |- ((A = C /\ B = D) -> (((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D))
45	40, 44	impbii 155	1 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) <-> (A = C /\ B = D))

Syntax hints:   -> wi 3   <-> wb 144   \/ wo 220   /\ wa 221   = wceq 992   e. wcel 994   =/= wne 1628   class class class wbr 2692  (class class class)co 4021   + caddc 5391   x. cmul 5393   <_ cle 5449  NN0cn0 5451   < clt 5640

This theorem is referenced by:  nn0opth2i 6868

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-nel 1631  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-opr 4023  df-oprab 4024  df-1st 4140  df-2nd 4141  df-rdg 4233  df-1o 4269  df-oadd 4271  df-omul 4272  df-er 4401  df-ec 4403  df-qs 4406  df-en 4509  df-dom 4510  df-sdom 4511  df-ni 5154  df-pli 5155  df-mi 5156  df-lti 5157  df-plpq 5189  df-mpq 5190  df-enq 5191  df-nq 5192  df-plq 5193  df-mq 5194  df-rq 5195  df-ltq 5196  df-1q 5197  df-np 5240  df-1p 5241  df-plp 5242  df-mp 5243  df-ltp 5244  df-plpr 5318  df-mpr 5319  df-enr 5320  df-nr 5321  df-plr 5322  df-mr 5323  df-ltr 5324  df-0r 5325  df-1r 5326  df-m1r 5327  df-c 5394  df-0 5395  df-1 5396  df-i 5397  df-r 5398  df-plus 5399  df-mul 5400  df-lt 5401  df-sub 5510  df-neg 5512  df-pnf 5641  df-mnf 5642  df-xr 5643  df-ltxr 5644  df-le 5645  df-n 6070  df-2 6116  df-n0 6268  df-z 6304  df-seq1 6673  df-exp 6764

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