Theorem nn0opthd 10635

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 3585 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 9171	. . . . . . . . . . . . . . 15 |- ( ph -> ( C + D ) e. NN0 )
6	1, 2, 5, 4	nn0opthlem2d 10634	. . . . . . . . . . . . . 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 2422	. . . . . . . . . . . 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 9171	. . . . . . . . . . . 12 |- ( ph -> ( A + B ) e. NN0 )
11	3, 4, 10, 2	nn0opthlem2d 10634	. . . . . . . . . . 11 |- ( ph -> ( ( C + D ) < ( A + B ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) =/= ( ( ( C + D ) x. ( C + D ) ) + D ) ) )
12	9, 11	jaod 707	. . . . . . . . . 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 9168	. . . . . . . . . . 11 |- ( ph -> ( A + B ) e. RR )
14	5	nn0red 9168	. . . . . . . . . . 11 |- ( ph -> ( C + D ) e. RR )
15		reaplt 8486	. . . . . . . . . . 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 409	. . . . . . . . . 10 |- ( ph -> ( ( A + B ) # ( C + D ) <-> ( ( A + B ) < ( C + D ) \/ ( C + D ) < ( A + B ) ) ) )
17	10, 10	nn0mulcld 9172	. . . . . . . . . . . . 13 |- ( ph -> ( ( A + B ) x. ( A + B ) ) e. NN0 )
18	17, 2	nn0addcld 9171	. . . . . . . . . . . 12 |- ( ph -> ( ( ( A + B ) x. ( A + B ) ) + B ) e. NN0 )
19	18	nn0zd 9311	. . . . . . . . . . 11 |- ( ph -> ( ( ( A + B ) x. ( A + B ) ) + B ) e. ZZ )
20	5, 5	nn0mulcld 9172	. . . . . . . . . . . . 13 |- ( ph -> ( ( C + D ) x. ( C + D ) ) e. NN0 )
21	20, 4	nn0addcld 9171	. . . . . . . . . . . 12 |- ( ph -> ( ( ( C + D ) x. ( C + D ) ) + D ) e. NN0 )
22	21	nn0zd 9311	. . . . . . . . . . 11 |- ( ph -> ( ( ( C + D ) x. ( C + D ) ) + D ) e. ZZ )
23		zapne 9265	. . . . . . . . . . 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 409	. . . . . . . . . 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 621	. . . . . . . 8 |- ( ph -> ( -. ( ( ( A + B ) x. ( A + B ) ) + B ) # ( ( ( C + D ) x. ( C + D ) ) + D ) -> -. ( A + B ) # ( C + D ) ) )
27	18	nn0cnd 9169	. . . . . . . . 9 |- ( ph -> ( ( ( A + B ) x. ( A + B ) ) + B ) e. CC )
28	21	nn0cnd 9169	. . . . . . . . 9 |- ( ph -> ( ( ( C + D ) x. ( C + D ) ) + D ) e. CC )
29		apti 8520	. . . . . . . . 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 409	. . . . . . . 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 9169	. . . . . . . . 9 |- ( ph -> ( A + B ) e. CC )
32	5	nn0cnd 9169	. . . . . . . . 9 |- ( ph -> ( C + D ) e. CC )
33		apti 8520	. . . . . . . . 9 |- ( ( ( A + B ) e. CC /\ ( C + D ) e. CC ) -> ( ( A + B ) = ( C + D ) <-> -. ( A + B ) # ( C + D ) ) )
34	31, 32, 33	syl2anc 409	. . . . . . . 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 5860	. . . . . . . . . 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 5857	. . . . . . . . 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 2201	. . . . . . . 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 7919	. . . . . . . . . 10 |- ( ph -> ( ( A + B ) x. ( A + B ) ) e. CC )
42	2	nn0cnd 9169	. . . . . . . . . 10 |- ( ph -> B e. CC )
43	4	nn0cnd 9169	. . . . . . . . . 10 |- ( ph -> D e. CC )
44	41, 42, 43	addcand 8082	. . . . . . . . 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 5858	. . . . . 6 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( C + B ) = ( C + D ) )
48	36, 47	eqtr4d 2201	. . . . 5 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( A + B ) = ( C + B ) )
49	1	nn0cnd 9169	. . . . . . 7 |- ( ph -> A e. CC )
50	3	nn0cnd 9169	. . . . . . 7 |- ( ph -> C e. CC )
51	49, 50, 42	addcan2d 8083	. . . . . 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 5851	. . . 4 |- ( ( A = C /\ B = D ) -> ( A + B ) = ( C + D ) )
57	56, 56	oveq12d 5860	. . 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 5860	. 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 698   = wceq 1343   e. wcel 2136   =/= wne 2336   class class class wbr 3982  (class class class)co 5842   CCcc 7751   RRcr 7752   + caddc 7756   x. cmul 7758   < clt 7933   # cap 8479   NN0cn0 9114   ZZcz 9191

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-n0 9115  df-z 9192  df-uz 9467  df-seqfrec 10381  df-exp 10455

This theorem is referenced by:  nn0opth2d  10636