Theorem nn0opthd 10793

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 3627 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 9297	. . . . . . . . . . . . . . 15 |- ( ph -> ( C + D ) e. NN0 )
6	1, 2, 5, 4	nn0opthlem2d 10792	. . . . . . . . . . . . . 14 |- ( ph -> ( ( A + B ) < ( C + D ) -> ( ( ( C + D ) x. ( C + D ) ) + D ) =/= ( ( ( A + B ) x. ( A + B ) ) + B ) ) )
7	6	imp 124	. . . . . . . . . . . . 13 |- ( ( ph /\ ( A + B ) < ( C + D ) ) -> ( ( ( C + D ) x. ( C + D ) ) + D ) =/= ( ( ( A + B ) x. ( A + B ) ) + B ) )
8	7	necomd 2450	. . . . . . . . . . . 12 |- ( ( ph /\ ( A + B ) < ( C + D ) ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) =/= ( ( ( C + D ) x. ( C + D ) ) + D ) )
9	8	ex 115	. . . . . . . . . . 11 |- ( ph -> ( ( A + B ) < ( C + D ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) =/= ( ( ( C + D ) x. ( C + D ) ) + D ) ) )
10	1, 2	nn0addcld 9297	. . . . . . . . . . . 12 |- ( ph -> ( A + B ) e. NN0 )
11	3, 4, 10, 2	nn0opthlem2d 10792	. . . . . . . . . . 11 |- ( ph -> ( ( C + D ) < ( A + B ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) =/= ( ( ( C + D ) x. ( C + D ) ) + D ) ) )
12	9, 11	jaod 718	. . . . . . . . . 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 9294	. . . . . . . . . . 11 |- ( ph -> ( A + B ) e. RR )
14	5	nn0red 9294	. . . . . . . . . . 11 |- ( ph -> ( C + D ) e. RR )
15		reaplt 8607	. . . . . . . . . . 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 411	. . . . . . . . . 10 |- ( ph -> ( ( A + B ) # ( C + D ) <-> ( ( A + B ) < ( C + D ) \/ ( C + D ) < ( A + B ) ) ) )
17	10, 10	nn0mulcld 9298	. . . . . . . . . . . . 13 |- ( ph -> ( ( A + B ) x. ( A + B ) ) e. NN0 )
18	17, 2	nn0addcld 9297	. . . . . . . . . . . 12 |- ( ph -> ( ( ( A + B ) x. ( A + B ) ) + B ) e. NN0 )
19	18	nn0zd 9437	. . . . . . . . . . 11 |- ( ph -> ( ( ( A + B ) x. ( A + B ) ) + B ) e. ZZ )
20	5, 5	nn0mulcld 9298	. . . . . . . . . . . . 13 |- ( ph -> ( ( C + D ) x. ( C + D ) ) e. NN0 )
21	20, 4	nn0addcld 9297	. . . . . . . . . . . 12 |- ( ph -> ( ( ( C + D ) x. ( C + D ) ) + D ) e. NN0 )
22	21	nn0zd 9437	. . . . . . . . . . 11 |- ( ph -> ( ( ( C + D ) x. ( C + D ) ) + D ) e. ZZ )
23		zapne 9391	. . . . . . . . . . 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 411	. . . . . . . . . 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 203	. . . . . . . . 9 |- ( ph -> ( ( A + B ) # ( C + D ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) # ( ( ( C + D ) x. ( C + D ) ) + D ) ) )
26	25	con3d 632	. . . . . . . 8 |- ( ph -> ( -. ( ( ( A + B ) x. ( A + B ) ) + B ) # ( ( ( C + D ) x. ( C + D ) ) + D ) -> -. ( A + B ) # ( C + D ) ) )
27	18	nn0cnd 9295	. . . . . . . . 9 |- ( ph -> ( ( ( A + B ) x. ( A + B ) ) + B ) e. CC )
28	21	nn0cnd 9295	. . . . . . . . 9 |- ( ph -> ( ( ( C + D ) x. ( C + D ) ) + D ) e. CC )
29		apti 8641	. . . . . . . . 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 411	. . . . . . . 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 9295	. . . . . . . . 9 |- ( ph -> ( A + B ) e. CC )
32	5	nn0cnd 9295	. . . . . . . . 9 |- ( ph -> ( C + D ) e. CC )
33		apti 8641	. . . . . . . . 9 |- ( ( ( A + B ) e. CC /\ ( C + D ) e. CC ) -> ( ( A + B ) = ( C + D ) <-> -. ( A + B ) # ( C + D ) ) )
34	31, 32, 33	syl2anc 411	. . . . . . . 8 |- ( ph -> ( ( A + B ) = ( C + D ) <-> -. ( A + B ) # ( C + D ) ) )
35	26, 30, 34	3imtr4d 203	. . . . . . 7 |- ( ph -> ( ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) -> ( A + B ) = ( C + D ) ) )
36	35	imp 124	. . . . . 6 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( A + B ) = ( C + D ) )
37		simpr 110	. . . . . . . . 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 5936	. . . . . . . . . 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 5933	. . . . . . . . 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 2229	. . . . . . . 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 8040	. . . . . . . . . 10 |- ( ph -> ( ( A + B ) x. ( A + B ) ) e. CC )
42	2	nn0cnd 9295	. . . . . . . . . 10 |- ( ph -> B e. CC )
43	4	nn0cnd 9295	. . . . . . . . . 10 |- ( ph -> D e. CC )
44	41, 42, 43	addcand 8203	. . . . . . . . 9 |- ( ph -> ( ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( A + B ) x. ( A + B ) ) + D ) <-> B = D ) )
45	44	adantr 276	. . . . . . . 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 147	. . . . . . 7 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> B = D )
47	46	oveq2d 5934	. . . . . 6 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( C + B ) = ( C + D ) )
48	36, 47	eqtr4d 2229	. . . . 5 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( A + B ) = ( C + B ) )
49	1	nn0cnd 9295	. . . . . . 7 |- ( ph -> A e. CC )
50	3	nn0cnd 9295	. . . . . . 7 |- ( ph -> C e. CC )
51	49, 50, 42	addcan2d 8204	. . . . . 6 |- ( ph -> ( ( A + B ) = ( C + B ) <-> A = C ) )
52	51	adantr 276	. . . . 5 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( ( A + B ) = ( C + B ) <-> A = C ) )
53	48, 52	mpbid 147	. . . 4 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> A = C )
54	53, 46	jca 306	. . 3 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( A = C /\ B = D ) )
55	54	ex 115	. 2 |- ( ph -> ( ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) -> ( A = C /\ B = D ) ) )
56		oveq12 5927	. . . 4 |- ( ( A = C /\ B = D ) -> ( A + B ) = ( C + D ) )
57	56, 56	oveq12d 5936	. . 3 |- ( ( A = C /\ B = D ) -> ( ( A + B ) x. ( A + B ) ) = ( ( C + D ) x. ( C + D ) ) )
58		simpr 110	. . 3 |- ( ( A = C /\ B = D ) -> B = D )
59	57, 58	oveq12d 5936	. 2 |- ( ( A = C /\ B = D ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) )
60	55, 59	impbid1 142	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 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2164   =/= wne 2364   class class class wbr 4029  (class class class)co 5918   CCcc 7870   RRcr 7871   + caddc 7875   x. cmul 7877   < clt 8054   # cap 8600   NN0cn0 9240   ZZcz 9317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-n0 9241  df-z 9318  df-uz 9593  df-seqfrec 10519  df-exp 10610

This theorem is referenced by:  nn0opth2d  10794