Theorem nn0opthd 10468

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 3536 that works for any set. (Contributed by Jim Kingdon, 31-Oct-2021.)

Hypotheses

Ref	Expression
nn0opthd.1	|- ( ph -> A e. NN0 )
nn0opthd.2	|- ( ph -> B e. NN0 )
nn0opthd.3	|- ( ph -> C e. NN0 )
nn0opthd.4	|- ( ph -> D e. NN0 )

Assertion

Ref	Expression
nn0opthd	|- ( ph -> ( ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) <-> ( A = C /\ B = D ) ) )

Proof of Theorem nn0opthd

Step	Hyp	Ref	Expression
1		nn0opthd.1	. . . . . . . . . . . . . . 15 |- ( ph -> A e. NN0 )
2		nn0opthd.2	. . . . . . . . . . . . . . 15 |- ( ph -> B e. NN0 )
3		nn0opthd.3	. . . . . . . . . . . . . . . 16 |- ( ph -> C e. NN0 )
4		nn0opthd.4	. . . . . . . . . . . . . . . 16 |- ( ph -> D e. NN0 )
5	3, 4	nn0addcld 9034	. . . . . . . . . . . . . . 15 |- ( ph -> ( C + D ) e. NN0 )
6	1, 2, 5, 4	nn0opthlem2d 10467	. . . . . . . . . . . . . 14 |- ( ph -> ( ( A + B ) < ( C + D ) -> ( ( ( C + D ) x. ( C + D ) ) + D ) =/= ( ( ( A + B ) x. ( A + B ) ) + B ) ) )
7	6	imp 123	. . . . . . . . . . . . 13 |- ( ( ph /\ ( A + B ) < ( C + D ) ) -> ( ( ( C + D ) x. ( C + D ) ) + D ) =/= ( ( ( A + B ) x. ( A + B ) ) + B ) )
8	7	necomd 2394	. . . . . . . . . . . 12 |- ( ( ph /\ ( A + B ) < ( C + D ) ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) =/= ( ( ( C + D ) x. ( C + D ) ) + D ) )
9	8	ex 114	. . . . . . . . . . 11 |- ( ph -> ( ( A + B ) < ( C + D ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) =/= ( ( ( C + D ) x. ( C + D ) ) + D ) ) )
10	1, 2	nn0addcld 9034	. . . . . . . . . . . 12 |- ( ph -> ( A + B ) e. NN0 )
11	3, 4, 10, 2	nn0opthlem2d 10467	. . . . . . . . . . 11 |- ( ph -> ( ( C + D ) < ( A + B ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) =/= ( ( ( C + D ) x. ( C + D ) ) + D ) ) )
12	9, 11	jaod 706	. . . . . . . . . 10 |- ( ph -> ( ( ( A + B ) < ( C + D ) \/ ( C + D ) < ( A + B ) ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) =/= ( ( ( C + D ) x. ( C + D ) ) + D ) ) )
13	10	nn0red 9031	. . . . . . . . . . 11 |- ( ph -> ( A + B ) e. RR )
14	5	nn0red 9031	. . . . . . . . . . 11 |- ( ph -> ( C + D ) e. RR )
15		reaplt 8350	. . . . . . . . . . 11 |- ( ( ( A + B ) e. RR /\ ( C + D ) e. RR ) -> ( ( A + B ) # ( C + D ) <-> ( ( A + B ) < ( C + D ) \/ ( C + D ) < ( A + B ) ) ) )
16	13, 14, 15	syl2anc 408	. . . . . . . . . 10 |- ( ph -> ( ( A + B ) # ( C + D ) <-> ( ( A + B ) < ( C + D ) \/ ( C + D ) < ( A + B ) ) ) )
17	10, 10	nn0mulcld 9035	. . . . . . . . . . . . 13 |- ( ph -> ( ( A + B ) x. ( A + B ) ) e. NN0 )
18	17, 2	nn0addcld 9034	. . . . . . . . . . . 12 |- ( ph -> ( ( ( A + B ) x. ( A + B ) ) + B ) e. NN0 )
19	18	nn0zd 9171	. . . . . . . . . . 11 |- ( ph -> ( ( ( A + B ) x. ( A + B ) ) + B ) e. ZZ )
20	5, 5	nn0mulcld 9035	. . . . . . . . . . . . 13 |- ( ph -> ( ( C + D ) x. ( C + D ) ) e. NN0 )
21	20, 4	nn0addcld 9034	. . . . . . . . . . . 12 |- ( ph -> ( ( ( C + D ) x. ( C + D ) ) + D ) e. NN0 )
22	21	nn0zd 9171	. . . . . . . . . . 11 |- ( ph -> ( ( ( C + D ) x. ( C + D ) ) + D ) e. ZZ )
23		zapne 9125	. . . . . . . . . . 11 |- ( ( ( ( ( A + B ) x. ( A + B ) ) + B ) e. ZZ /\ ( ( ( C + D ) x. ( C + D ) ) + D ) e. ZZ ) -> ( ( ( ( A + B ) x. ( A + B ) ) + B ) # ( ( ( C + D ) x. ( C + D ) ) + D ) <-> ( ( ( A + B ) x. ( A + B ) ) + B ) =/= ( ( ( C + D ) x. ( C + D ) ) + D ) ) )
24	19, 22, 23	syl2anc 408	. . . . . . . . . 10 |- ( ph -> ( ( ( ( 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	12, 16, 24	3imtr4d 202	. . . . . . . . 9 |- ( ph -> ( ( A + B ) # ( C + D ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) # ( ( ( C + D ) x. ( C + D ) ) + D ) ) )
26	25	con3d 620	. . . . . . . 8 |- ( ph -> ( -. ( ( ( A + B ) x. ( A + B ) ) + B ) # ( ( ( C + D ) x. ( C + D ) ) + D ) -> -. ( A + B ) # ( C + D ) ) )
27	18	nn0cnd 9032	. . . . . . . . 9 |- ( ph -> ( ( ( A + B ) x. ( A + B ) ) + B ) e. CC )
28	21	nn0cnd 9032	. . . . . . . . 9 |- ( ph -> ( ( ( C + D ) x. ( C + D ) ) + D ) e. CC )
29		apti 8384	. . . . . . . . 9 |- ( ( ( ( ( A + B ) x. ( A + B ) ) + B ) e. CC /\ ( ( ( C + D ) x. ( C + D ) ) + D ) e. CC ) -> ( ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) <-> -. ( ( ( A + B ) x. ( A + B ) ) + B ) # ( ( ( C + D ) x. ( C + D ) ) + D ) ) )
30	27, 28, 29	syl2anc 408	. . . . . . . 8 |- ( ph -> ( ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) <-> -. ( ( ( A + B ) x. ( A + B ) ) + B ) # ( ( ( C + D ) x. ( C + D ) ) + D ) ) )
31	10	nn0cnd 9032	. . . . . . . . 9 |- ( ph -> ( A + B ) e. CC )
32	5	nn0cnd 9032	. . . . . . . . 9 |- ( ph -> ( C + D ) e. CC )
33		apti 8384	. . . . . . . . 9 |- ( ( ( A + B ) e. CC /\ ( C + D ) e. CC ) -> ( ( A + B ) = ( C + D ) <-> -. ( A + B ) # ( C + D ) ) )
34	31, 32, 33	syl2anc 408	. . . . . . . 8 |- ( ph -> ( ( A + B ) = ( C + D ) <-> -. ( A + B ) # ( C + D ) ) )
35	26, 30, 34	3imtr4d 202	. . . . . . 7 |- ( ph -> ( ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) -> ( A + B ) = ( C + D ) ) )
36	35	imp 123	. . . . . 6 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( A + B ) = ( C + D ) )
37		simpr 109	. . . . . . . . 9 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) )
38	36, 36	oveq12d 5792	. . . . . . . . . 10 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( ( A + B ) x. ( A + B ) ) = ( ( C + D ) x. ( C + D ) ) )
39	38	oveq1d 5789	. . . . . . . . 9 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( ( ( A + B ) x. ( A + B ) ) + D ) = ( ( ( C + D ) x. ( C + D ) ) + D ) )
40	37, 39	eqtr4d 2175	. . . . . . . 8 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( A + B ) x. ( A + B ) ) + D ) )
41	31, 31	mulcld 7786	. . . . . . . . . 10 |- ( ph -> ( ( A + B ) x. ( A + B ) ) e. CC )
42	2	nn0cnd 9032	. . . . . . . . . 10 |- ( ph -> B e. CC )
43	4	nn0cnd 9032	. . . . . . . . . 10 |- ( ph -> D e. CC )
44	41, 42, 43	addcand 7946	. . . . . . . . 9 |- ( ph -> ( ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( A + B ) x. ( A + B ) ) + D ) <-> B = D ) )
45	44	adantr 274	. . . . . . . 8 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( A + B ) x. ( A + B ) ) + D ) <-> B = D ) )
46	40, 45	mpbid 146	. . . . . . 7 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> B = D )
47	46	oveq2d 5790	. . . . . 6 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( C + B ) = ( C + D ) )
48	36, 47	eqtr4d 2175	. . . . 5 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( A + B ) = ( C + B ) )
49	1	nn0cnd 9032	. . . . . . 7 |- ( ph -> A e. CC )
50	3	nn0cnd 9032	. . . . . . 7 |- ( ph -> C e. CC )
51	49, 50, 42	addcan2d 7947	. . . . . 6 |- ( ph -> ( ( A + B ) = ( C + B ) <-> A = C ) )
52	51	adantr 274	. . . . 5 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( ( A + B ) = ( C + B ) <-> A = C ) )
53	48, 52	mpbid 146	. . . 4 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> A = C )
54	53, 46	jca 304	. . 3 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( A = C /\ B = D ) )
55	54	ex 114	. 2 |- ( ph -> ( ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) -> ( A = C /\ B = D ) ) )
56		oveq12 5783	. . . 4 |- ( ( A = C /\ B = D ) -> ( A + B ) = ( C + D ) )
57	56, 56	oveq12d 5792	. . 3 |- ( ( A = C /\ B = D ) -> ( ( A + B ) x. ( A + B ) ) = ( ( C + D ) x. ( C + D ) ) )
58		simpr 109	. . 3 |- ( ( A = C /\ B = D ) -> B = D )
59	57, 58	oveq12d 5792	. 2 |- ( ( A = C /\ B = D ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) )
60	55, 59	impbid1 141	1 |- ( ph -> ( ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) <-> ( A = C /\ B = D ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   = wceq 1331   e. wcel 1480   =/= wne 2308   class class class wbr 3929  (class class class)co 5774   CCcc 7618   RRcr 7619   + caddc 7623   x. cmul 7625   < clt 7800   # cap 8343   NN0cn0 8977   ZZcz 9054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-n0 8978  df-z 9055  df-uz 9327  df-seqfrec 10219  df-exp 10293

This theorem is referenced by:  nn0opth2d  10469