Theorem nn0opthd 9649

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 3407 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 8345	. . . . . . . . . . . . . . 15 |- ( ph -> ( C + D ) e. NN0 )
6	1, 2, 5, 4	nn0opthlem2d 9648	. . . . . . . . . . . . . 14 |- ( ph -> ( ( A + B ) < ( C + D ) -> ( ( ( C + D ) x. ( C + D ) ) + D ) =/= ( ( ( A + B ) x. ( A + B ) ) + B ) ) )
7	6	imp 122	. . . . . . . . . . . . 13 |- ( ( ph /\ ( A + B ) < ( C + D ) ) -> ( ( ( C + D ) x. ( C + D ) ) + D ) =/= ( ( ( A + B ) x. ( A + B ) ) + B ) )
8	7	necomd 2331	. . . . . . . . . . . 12 |- ( ( ph /\ ( A + B ) < ( C + D ) ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) =/= ( ( ( C + D ) x. ( C + D ) ) + D ) )
9	8	ex 113	. . . . . . . . . . 11 |- ( ph -> ( ( A + B ) < ( C + D ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) =/= ( ( ( C + D ) x. ( C + D ) ) + D ) ) )
10	1, 2	nn0addcld 8345	. . . . . . . . . . . 12 |- ( ph -> ( A + B ) e. NN0 )
11	3, 4, 10, 2	nn0opthlem2d 9648	. . . . . . . . . . 11 |- ( ph -> ( ( C + D ) < ( A + B ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) =/= ( ( ( C + D ) x. ( C + D ) ) + D ) ) )
12	9, 11	jaod 669	. . . . . . . . . 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 8342	. . . . . . . . . . 11 |- ( ph -> ( A + B ) e. RR )
14	5	nn0red 8342	. . . . . . . . . . 11 |- ( ph -> ( C + D ) e. RR )
15		reaplt 7688	. . . . . . . . . . 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 403	. . . . . . . . . 10 |- ( ph -> ( ( A + B ) # ( C + D ) <-> ( ( A + B ) < ( C + D ) \/ ( C + D ) < ( A + B ) ) ) )
17	10, 10	nn0mulcld 8346	. . . . . . . . . . . . 13 |- ( ph -> ( ( A + B ) x. ( A + B ) ) e. NN0 )
18	17, 2	nn0addcld 8345	. . . . . . . . . . . 12 |- ( ph -> ( ( ( A + B ) x. ( A + B ) ) + B ) e. NN0 )
19	18	nn0zd 8467	. . . . . . . . . . 11 |- ( ph -> ( ( ( A + B ) x. ( A + B ) ) + B ) e. ZZ )
20	5, 5	nn0mulcld 8346	. . . . . . . . . . . . 13 |- ( ph -> ( ( C + D ) x. ( C + D ) ) e. NN0 )
21	20, 4	nn0addcld 8345	. . . . . . . . . . . 12 |- ( ph -> ( ( ( C + D ) x. ( C + D ) ) + D ) e. NN0 )
22	21	nn0zd 8467	. . . . . . . . . . 11 |- ( ph -> ( ( ( C + D ) x. ( C + D ) ) + D ) e. ZZ )
23		zapne 8422	. . . . . . . . . . 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 403	. . . . . . . . . 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 201	. . . . . . . . 9 |- ( ph -> ( ( A + B ) # ( C + D ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) # ( ( ( C + D ) x. ( C + D ) ) + D ) ) )
26	25	con3d 593	. . . . . . . 8 |- ( ph -> ( -. ( ( ( A + B ) x. ( A + B ) ) + B ) # ( ( ( C + D ) x. ( C + D ) ) + D ) -> -. ( A + B ) # ( C + D ) ) )
27	18	nn0cnd 8343	. . . . . . . . 9 |- ( ph -> ( ( ( A + B ) x. ( A + B ) ) + B ) e. CC )
28	21	nn0cnd 8343	. . . . . . . . 9 |- ( ph -> ( ( ( C + D ) x. ( C + D ) ) + D ) e. CC )
29		apti 7722	. . . . . . . . 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 403	. . . . . . . 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 8343	. . . . . . . . 9 |- ( ph -> ( A + B ) e. CC )
32	5	nn0cnd 8343	. . . . . . . . 9 |- ( ph -> ( C + D ) e. CC )
33		apti 7722	. . . . . . . . 9 |- ( ( ( A + B ) e. CC /\ ( C + D ) e. CC ) -> ( ( A + B ) = ( C + D ) <-> -. ( A + B ) # ( C + D ) ) )
34	31, 32, 33	syl2anc 403	. . . . . . . 8 |- ( ph -> ( ( A + B ) = ( C + D ) <-> -. ( A + B ) # ( C + D ) ) )
35	26, 30, 34	3imtr4d 201	. . . . . . 7 |- ( ph -> ( ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) -> ( A + B ) = ( C + D ) ) )
36	35	imp 122	. . . . . 6 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( A + B ) = ( C + D ) )
37		simpr 108	. . . . . . . . 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 5550	. . . . . . . . . 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 5547	. . . . . . . . 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 2116	. . . . . . . 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 7139	. . . . . . . . . 10 |- ( ph -> ( ( A + B ) x. ( A + B ) ) e. CC )
42	2	nn0cnd 8343	. . . . . . . . . 10 |- ( ph -> B e. CC )
43	4	nn0cnd 8343	. . . . . . . . . 10 |- ( ph -> D e. CC )
44	41, 42, 43	addcand 7292	. . . . . . . . 9 |- ( ph -> ( ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( A + B ) x. ( A + B ) ) + D ) <-> B = D ) )
45	44	adantr 270	. . . . . . . 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 145	. . . . . . 7 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> B = D )
47	46	oveq2d 5548	. . . . . 6 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( C + B ) = ( C + D ) )
48	36, 47	eqtr4d 2116	. . . . 5 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( A + B ) = ( C + B ) )
49	1	nn0cnd 8343	. . . . . . 7 |- ( ph -> A e. CC )
50	3	nn0cnd 8343	. . . . . . 7 |- ( ph -> C e. CC )
51	49, 50, 42	addcan2d 7293	. . . . . 6 |- ( ph -> ( ( A + B ) = ( C + B ) <-> A = C ) )
52	51	adantr 270	. . . . 5 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( ( A + B ) = ( C + B ) <-> A = C ) )
53	48, 52	mpbid 145	. . . 4 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> A = C )
54	53, 46	jca 300	. . 3 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( A = C /\ B = D ) )
55	54	ex 113	. 2 |- ( ph -> ( ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) -> ( A = C /\ B = D ) ) )
56		oveq12 5541	. . . 4 |- ( ( A = C /\ B = D ) -> ( A + B ) = ( C + D ) )
57	56, 56	oveq12d 5550	. . 3 |- ( ( A = C /\ B = D ) -> ( ( A + B ) x. ( A + B ) ) = ( ( C + D ) x. ( C + D ) ) )
58		simpr 108	. . 3 |- ( ( A = C /\ B = D ) -> B = D )
59	57, 58	oveq12d 5550	. 2 |- ( ( A = C /\ B = D ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) )
60	55, 59	impbid1 140	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 102   <-> wb 103   \/ wo 661   = wceq 1284   e. wcel 1433   =/= wne 2245   class class class wbr 3785  (class class class)co 5532   CCcc 6979   RRcr 6980   + caddc 6984   x. cmul 6986   < clt 7153   # cap 7681   NN0cn0 8288   ZZcz 8351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432  df-iexp 9476

This theorem is referenced by:  nn0opth2d  9650