Theorem nn0opthi 7916

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 3053 that works for any set. (Contributed by Raph Levien, 10-Dec-2002. Proof shortened by Scott Fenton, 7-Sep-2010.)

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.1	. . . . . . . . . 10 |- A e. NN0
2		nn0opth.2	. . . . . . . . . 10 |- B e. NN0
3	1, 2	nn0addcli 7330	. . . . . . . . 9 |- (A + B) e. NN0
4	3	nn0rei 7319	. . . . . . . 8 |- (A + B) e. RR
5		nn0opth.3	. . . . . . . . . 10 |- C e. NN0
6		nn0opth.4	. . . . . . . . . 10 |- D e. NN0
7	5, 6	nn0addcli 7330	. . . . . . . . 9 |- (C + D) e. NN0
8	7	nn0rei 7319	. . . . . . . 8 |- (C + D) e. RR
9	4, 8	lttri2i 6747	. . . . . . 7 |- ((A + B) =/= (C + D) <-> ((A + B) < (C + D) \/ (C + D) < (A + B)))
10	1, 2, 7, 6	nn0opthlem2 7915	. . . . . . . . 9 |- ((A + B) < (C + D) -> (((C + D) x. (C + D)) + D) =/= (((A + B) x. (A + B)) + B))
11	10	necomd 2095	. . . . . . . 8 |- ((A + B) < (C + D) -> (((A + B) x. (A + B)) + B) =/= (((C + D) x. (C + D)) + D))
12	5, 6, 3, 2	nn0opthlem2 7915	. . . . . . . 8 |- ((C + D) < (A + B) -> (((A + B) x. (A + B)) + B) =/= (((C + D) x. (C + D)) + D))
13	11, 12	jaoi 368	. . . . . . 7 |- (((A + B) < (C + D) \/ (C + D) < (A + B)) -> (((A + B) x. (A + B)) + B) =/= (((C + D) x. (C + D)) + D))
14	9, 13	sylbi 216	. . . . . 6 |- ((A + B) =/= (C + D) -> (((A + B) x. (A + B)) + B) =/= (((C + D) x. (C + D)) + D))
15	14	necon4i 2069	. . . . 5 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> (A + B) = (C + D))
16		id 73	. . . . . . . 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))
17	15, 15	opreq12d 4900	. . . . . . . . 9 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> ((A + B) x. (A + B)) = ((C + D) x. (C + D)))
18	17	opreq1d 4897	. . . . . . . 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))
19	16, 18	eqtr4d 1928	. . . . . . 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))
20	3	nn0cni 7320	. . . . . . . . 9 |- (A + B) e. CC
21	20, 20	mulcli 6474	. . . . . . . 8 |- ((A + B) x. (A + B)) e. CC
22	2	nn0cni 7320	. . . . . . . 8 |- B e. CC
23	6	nn0cni 7320	. . . . . . . 8 |- D e. CC
24	21, 22, 23	addcani 6505	. . . . . . 7 |- ((((A + B) x. (A + B)) + B) = (((A + B) x. (A + B)) + D) <-> B = D)
25	19, 24	sylib 215	. . . . . 6 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> B = D)
26	25	opreq2d 4898	. . . . 5 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> (C + B) = (C + D))
27	15, 26	eqtr4d 1928	. . . 4 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> (A + B) = (C + B))
28	1	nn0cni 7320	. . . . 5 |- A e. CC
29	5	nn0cni 7320	. . . . 5 |- C e. CC
30	28, 29, 22	addcan2i 6509	. . . 4 |- ((A + B) = (C + B) <-> A = C)
31	27, 30	sylib 215	. . 3 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> A = C)
32	31, 25	jca 310	. 2 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) -> (A = C /\ B = D))
33		opreq12 4891	. . . 4 |- ((A = C /\ B = D) -> (A + B) = (C + D))
34	33, 33	opreq12d 4900	. . 3 |- ((A = C /\ B = D) -> ((A + B) x. (A + B)) = ((C + D) x. (C + D)))
35		simpr 350	. . 3 |- ((A = C /\ B = D) -> B = D)
36	34, 35	opreq12d 4900	. 2 |- ((A = C /\ B = D) -> (((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D))
37	32, 36	impbii 174	1 |- ((((A + B) x. (A + B)) + B) = (((C + D) x. (C + D)) + D) <-> (A = C /\ B = D))

Syntax hints:   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017   class class class wbr 3338  (class class class)co 4884   + caddc 6389   x. cmul 6391  NN0cn0 6450   < clt 6653

This theorem is referenced by:  nn0opth2i 7917

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812

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