Theorem nn0opthd 10869

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 3642 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 9354	. . . . . . . . . . . . . . 15 |- ( ph -> ( C + D ) e. NN0 )
6	1, 2, 5, 4	nn0opthlem2d 10868	. . . . . . . . . . . . . 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 2462	. . . . . . . . . . . 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 9354	. . . . . . . . . . . 12 |- ( ph -> ( A + B ) e. NN0 )
11	3, 4, 10, 2	nn0opthlem2d 10868	. . . . . . . . . . 11 |- ( ph -> ( ( C + D ) < ( A + B ) -> ( ( ( A + B ) x. ( A + B ) ) + B ) =/= ( ( ( C + D ) x. ( C + D ) ) + D ) ) )
12	9, 11	jaod 719	. . . . . . . . . 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 9351	. . . . . . . . . . 11 |- ( ph -> ( A + B ) e. RR )
14	5	nn0red 9351	. . . . . . . . . . 11 |- ( ph -> ( C + D ) e. RR )
15		reaplt 8663	. . . . . . . . . . 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 9355	. . . . . . . . . . . . 13 |- ( ph -> ( ( A + B ) x. ( A + B ) ) e. NN0 )
18	17, 2	nn0addcld 9354	. . . . . . . . . . . 12 |- ( ph -> ( ( ( A + B ) x. ( A + B ) ) + B ) e. NN0 )
19	18	nn0zd 9495	. . . . . . . . . . 11 |- ( ph -> ( ( ( A + B ) x. ( A + B ) ) + B ) e. ZZ )
20	5, 5	nn0mulcld 9355	. . . . . . . . . . . . 13 |- ( ph -> ( ( C + D ) x. ( C + D ) ) e. NN0 )
21	20, 4	nn0addcld 9354	. . . . . . . . . . . 12 |- ( ph -> ( ( ( C + D ) x. ( C + D ) ) + D ) e. NN0 )
22	21	nn0zd 9495	. . . . . . . . . . 11 |- ( ph -> ( ( ( C + D ) x. ( C + D ) ) + D ) e. ZZ )
23		zapne 9449	. . . . . . . . . . 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 9352	. . . . . . . . 9 |- ( ph -> ( ( ( A + B ) x. ( A + B ) ) + B ) e. CC )
28	21	nn0cnd 9352	. . . . . . . . 9 |- ( ph -> ( ( ( C + D ) x. ( C + D ) ) + D ) e. CC )
29		apti 8697	. . . . . . . . 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 9352	. . . . . . . . 9 |- ( ph -> ( A + B ) e. CC )
32	5	nn0cnd 9352	. . . . . . . . 9 |- ( ph -> ( C + D ) e. CC )
33		apti 8697	. . . . . . . . 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 5964	. . . . . . . . . 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 5961	. . . . . . . . 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 2241	. . . . . . . 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 8095	. . . . . . . . . 10 |- ( ph -> ( ( A + B ) x. ( A + B ) ) e. CC )
42	2	nn0cnd 9352	. . . . . . . . . 10 |- ( ph -> B e. CC )
43	4	nn0cnd 9352	. . . . . . . . . 10 |- ( ph -> D e. CC )
44	41, 42, 43	addcand 8258	. . . . . . . . 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 5962	. . . . . 6 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( C + B ) = ( C + D ) )
48	36, 47	eqtr4d 2241	. . . . 5 |- ( ( ph /\ ( ( ( A + B ) x. ( A + B ) ) + B ) = ( ( ( C + D ) x. ( C + D ) ) + D ) ) -> ( A + B ) = ( C + B ) )
49	1	nn0cnd 9352	. . . . . . 7 |- ( ph -> A e. CC )
50	3	nn0cnd 9352	. . . . . . 7 |- ( ph -> C e. CC )
51	49, 50, 42	addcan2d 8259	. . . . . 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 5955	. . . 4 |- ( ( A = C /\ B = D ) -> ( A + B ) = ( C + D ) )
57	56, 56	oveq12d 5964	. . 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 5964	. 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 710   = wceq 1373   e. wcel 2176   =/= wne 2376   class class class wbr 4045  (class class class)co 5946   CCcc 7925   RRcr 7926   + caddc 7930   x. cmul 7932   < clt 8109   # cap 8656   NN0cn0 9297   ZZcz 9374

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-frec 6479  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-n0 9298  df-z 9375  df-uz 9651  df-seqfrec 10595  df-exp 10686

This theorem is referenced by:  nn0opth2d  10870