Theorem rmxypairf1o 37476

Description: The function used to extract rational and irrational parts in df-rmx 37466 and df-rmy 37467 in fact achieves a one-to-one mapping from the quadratic irrationals to pairs of integers. (Contributed by Stefan O'Rear, 21-Sep-2014.)

Assertion

Ref	Expression
rmxypairf1o	|- ( A e. ( ZZ>= ` 2 ) -> ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) : ( NN0 X. ZZ ) -1-1-onto-> { a | E. c e. NN0 E. d e. ZZ a = ( c + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. d ) ) } )

Distinct variable group:   b, c, d, a, A

Proof of Theorem rmxypairf1o

Step	Hyp	Ref	Expression
1		ovex 6678	. . . 4 |- ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) e. _V
2		eqid 2622	. . . 4 |- ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) = ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) )
3	1, 2	fnmpti 6022	. . 3 |- ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) Fn ( NN0 X. ZZ )
4	3	a1i 11	. 2 |- ( A e. ( ZZ>= ` 2 ) -> ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) Fn ( NN0 X. ZZ ) )
5		vex 3203	. . . . . . . . . 10 |- c e. _V
6		vex 3203	. . . . . . . . . 10 |- d e. _V
7	5, 6	op1std 7178	. . . . . . . . 9 |- ( b = <. c , d >. -> ( 1st ` b ) = c )
8	5, 6	op2ndd 7179	. . . . . . . . . 10 |- ( b = <. c , d >. -> ( 2nd ` b ) = d )
9	8	oveq2d 6666	. . . . . . . . 9 |- ( b = <. c , d >. -> ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) = ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. d ) )
10	7, 9	oveq12d 6668	. . . . . . . 8 |- ( b = <. c , d >. -> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) = ( c + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. d ) ) )
11	10	eqeq2d 2632	. . . . . . 7 |- ( b = <. c , d >. -> ( a = ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) <-> a = ( c + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. d ) ) ) )
12	11	rexxp 5264	. . . . . 6 |- ( E. b e. ( NN0 X. ZZ ) a = ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) <-> E. c e. NN0 E. d e. ZZ a = ( c + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. d ) ) )
13	12	bicomi 214	. . . . 5 |- ( E. c e. NN0 E. d e. ZZ a = ( c + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. d ) ) <-> E. b e. ( NN0 X. ZZ ) a = ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) )
14	13	a1i 11	. . . 4 |- ( A e. ( ZZ>= ` 2 ) -> ( E. c e. NN0 E. d e. ZZ a = ( c + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. d ) ) <-> E. b e. ( NN0 X. ZZ ) a = ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) )
15	14	abbidv 2741	. . 3 |- ( A e. ( ZZ>= ` 2 ) -> { a | E. c e. NN0 E. d e. ZZ a = ( c + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. d ) ) } = { a | E. b e. ( NN0 X. ZZ ) a = ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) } )
16	2	rnmpt 5371	. . 3 |- ran ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) = { a | E. b e. ( NN0 X. ZZ ) a = ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) }
17	15, 16	syl6reqr 2675	. 2 |- ( A e. ( ZZ>= ` 2 ) -> ran ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) = { a | E. c e. NN0 E. d e. ZZ a = ( c + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. d ) ) } )
18		fveq2 6191	. . . . . . . 8 |- ( b = c -> ( 1st ` b ) = ( 1st ` c ) )
19		fveq2 6191	. . . . . . . . 9 |- ( b = c -> ( 2nd ` b ) = ( 2nd ` c ) )
20	19	oveq2d 6666	. . . . . . . 8 |- ( b = c -> ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) = ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` c ) ) )
21	18, 20	oveq12d 6668	. . . . . . 7 |- ( b = c -> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) = ( ( 1st ` c ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` c ) ) ) )
22		ovex 6678	. . . . . . 7 |- ( ( 1st ` c ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` c ) ) ) e. _V
23	21, 2, 22	fvmpt 6282	. . . . . 6 |- ( c e. ( NN0 X. ZZ ) -> ( ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` c ) = ( ( 1st ` c ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` c ) ) ) )
24	23	ad2antrl 764	. . . . 5 |- ( ( A e. ( ZZ>= ` 2 ) /\ ( c e. ( NN0 X. ZZ ) /\ d e. ( NN0 X. ZZ ) ) ) -> ( ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` c ) = ( ( 1st ` c ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` c ) ) ) )
25		fveq2 6191	. . . . . . . 8 |- ( b = d -> ( 1st ` b ) = ( 1st ` d ) )
26		fveq2 6191	. . . . . . . . 9 |- ( b = d -> ( 2nd ` b ) = ( 2nd ` d ) )
27	26	oveq2d 6666	. . . . . . . 8 |- ( b = d -> ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) = ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` d ) ) )
28	25, 27	oveq12d 6668	. . . . . . 7 |- ( b = d -> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) = ( ( 1st ` d ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` d ) ) ) )
29		ovex 6678	. . . . . . 7 |- ( ( 1st ` d ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` d ) ) ) e. _V
30	28, 2, 29	fvmpt 6282	. . . . . 6 |- ( d e. ( NN0 X. ZZ ) -> ( ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` d ) = ( ( 1st ` d ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` d ) ) ) )
31	30	ad2antll 765	. . . . 5 |- ( ( A e. ( ZZ>= ` 2 ) /\ ( c e. ( NN0 X. ZZ ) /\ d e. ( NN0 X. ZZ ) ) ) -> ( ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` d ) = ( ( 1st ` d ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` d ) ) ) )
32	24, 31	eqeq12d 2637	. . . 4 |- ( ( A e. ( ZZ>= ` 2 ) /\ ( c e. ( NN0 X. ZZ ) /\ d e. ( NN0 X. ZZ ) ) ) -> ( ( ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` c ) = ( ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` d ) <-> ( ( 1st ` c ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` c ) ) ) = ( ( 1st ` d ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` d ) ) ) ) )
33		rmspecsqrtnq 37470	. . . . . . . 8 |- ( A e. ( ZZ>= ` 2 ) -> ( sqr ` ( ( A ^ 2 ) - 1 ) ) e. ( CC \ QQ ) )
34	33	adantr 481	. . . . . . 7 |- ( ( A e. ( ZZ>= ` 2 ) /\ ( c e. ( NN0 X. ZZ ) /\ d e. ( NN0 X. ZZ ) ) ) -> ( sqr ` ( ( A ^ 2 ) - 1 ) ) e. ( CC \ QQ ) )
35		nn0ssq 11796	. . . . . . . 8 |- NN0 C_ QQ
36		xp1st 7198	. . . . . . . . 9 |- ( c e. ( NN0 X. ZZ ) -> ( 1st ` c ) e. NN0 )
37	36	ad2antrl 764	. . . . . . . 8 |- ( ( A e. ( ZZ>= ` 2 ) /\ ( c e. ( NN0 X. ZZ ) /\ d e. ( NN0 X. ZZ ) ) ) -> ( 1st ` c ) e. NN0 )
38	35, 37	sseldi 3601	. . . . . . 7 |- ( ( A e. ( ZZ>= ` 2 ) /\ ( c e. ( NN0 X. ZZ ) /\ d e. ( NN0 X. ZZ ) ) ) -> ( 1st ` c ) e. QQ )
39		xp2nd 7199	. . . . . . . . 9 |- ( c e. ( NN0 X. ZZ ) -> ( 2nd ` c ) e. ZZ )
40	39	ad2antrl 764	. . . . . . . 8 |- ( ( A e. ( ZZ>= ` 2 ) /\ ( c e. ( NN0 X. ZZ ) /\ d e. ( NN0 X. ZZ ) ) ) -> ( 2nd ` c ) e. ZZ )
41		zq 11794	. . . . . . . 8 |- ( ( 2nd ` c ) e. ZZ -> ( 2nd ` c ) e. QQ )
42	40, 41	syl 17	. . . . . . 7 |- ( ( A e. ( ZZ>= ` 2 ) /\ ( c e. ( NN0 X. ZZ ) /\ d e. ( NN0 X. ZZ ) ) ) -> ( 2nd ` c ) e. QQ )
43		xp1st 7198	. . . . . . . . 9 |- ( d e. ( NN0 X. ZZ ) -> ( 1st ` d ) e. NN0 )
44	43	ad2antll 765	. . . . . . . 8 |- ( ( A e. ( ZZ>= ` 2 ) /\ ( c e. ( NN0 X. ZZ ) /\ d e. ( NN0 X. ZZ ) ) ) -> ( 1st ` d ) e. NN0 )
45	35, 44	sseldi 3601	. . . . . . 7 |- ( ( A e. ( ZZ>= ` 2 ) /\ ( c e. ( NN0 X. ZZ ) /\ d e. ( NN0 X. ZZ ) ) ) -> ( 1st ` d ) e. QQ )
46		xp2nd 7199	. . . . . . . . 9 |- ( d e. ( NN0 X. ZZ ) -> ( 2nd ` d ) e. ZZ )
47	46	ad2antll 765	. . . . . . . 8 |- ( ( A e. ( ZZ>= ` 2 ) /\ ( c e. ( NN0 X. ZZ ) /\ d e. ( NN0 X. ZZ ) ) ) -> ( 2nd ` d ) e. ZZ )
48		zq 11794	. . . . . . . 8 |- ( ( 2nd ` d ) e. ZZ -> ( 2nd ` d ) e. QQ )
49	47, 48	syl 17	. . . . . . 7 |- ( ( A e. ( ZZ>= ` 2 ) /\ ( c e. ( NN0 X. ZZ ) /\ d e. ( NN0 X. ZZ ) ) ) -> ( 2nd ` d ) e. QQ )
50		qirropth 37473	. . . . . . 7 |- ( ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) e. ( CC \ QQ ) /\ ( ( 1st ` c ) e. QQ /\ ( 2nd ` c ) e. QQ ) /\ ( ( 1st ` d ) e. QQ /\ ( 2nd ` d ) e. QQ ) ) -> ( ( ( 1st ` c ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` c ) ) ) = ( ( 1st ` d ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` d ) ) ) <-> ( ( 1st ` c ) = ( 1st ` d ) /\ ( 2nd ` c ) = ( 2nd ` d ) ) ) )
51	34, 38, 42, 45, 49, 50	syl122anc 1335	. . . . . 6 |- ( ( A e. ( ZZ>= ` 2 ) /\ ( c e. ( NN0 X. ZZ ) /\ d e. ( NN0 X. ZZ ) ) ) -> ( ( ( 1st ` c ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` c ) ) ) = ( ( 1st ` d ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` d ) ) ) <-> ( ( 1st ` c ) = ( 1st ` d ) /\ ( 2nd ` c ) = ( 2nd ` d ) ) ) )
52	51	biimpd 219	. . . . 5 |- ( ( A e. ( ZZ>= ` 2 ) /\ ( c e. ( NN0 X. ZZ ) /\ d e. ( NN0 X. ZZ ) ) ) -> ( ( ( 1st ` c ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` c ) ) ) = ( ( 1st ` d ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` d ) ) ) -> ( ( 1st ` c ) = ( 1st ` d ) /\ ( 2nd ` c ) = ( 2nd ` d ) ) ) )
53		xpopth 7207	. . . . . 6 |- ( ( c e. ( NN0 X. ZZ ) /\ d e. ( NN0 X. ZZ ) ) -> ( ( ( 1st ` c ) = ( 1st ` d ) /\ ( 2nd ` c ) = ( 2nd ` d ) ) <-> c = d ) )
54	53	adantl 482	. . . . 5 |- ( ( A e. ( ZZ>= ` 2 ) /\ ( c e. ( NN0 X. ZZ ) /\ d e. ( NN0 X. ZZ ) ) ) -> ( ( ( 1st ` c ) = ( 1st ` d ) /\ ( 2nd ` c ) = ( 2nd ` d ) ) <-> c = d ) )
55	52, 54	sylibd 229	. . . 4 |- ( ( A e. ( ZZ>= ` 2 ) /\ ( c e. ( NN0 X. ZZ ) /\ d e. ( NN0 X. ZZ ) ) ) -> ( ( ( 1st ` c ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` c ) ) ) = ( ( 1st ` d ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` d ) ) ) -> c = d ) )
56	32, 55	sylbid 230	. . 3 |- ( ( A e. ( ZZ>= ` 2 ) /\ ( c e. ( NN0 X. ZZ ) /\ d e. ( NN0 X. ZZ ) ) ) -> ( ( ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` c ) = ( ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` d ) -> c = d ) )
57	56	ralrimivva 2971	. 2 |- ( A e. ( ZZ>= ` 2 ) -> A. c e. ( NN0 X. ZZ ) A. d e. ( NN0 X. ZZ ) ( ( ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` c ) = ( ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` d ) -> c = d ) )
58		dff1o6 6531	. 2 |- ( ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) : ( NN0 X. ZZ ) -1-1-onto-> { a | E. c e. NN0 E. d e. ZZ a = ( c + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. d ) ) } <-> ( ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) Fn ( NN0 X. ZZ ) /\ ran ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) = { a | E. c e. NN0 E. d e. ZZ a = ( c + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. d ) ) } /\ A. c e. ( NN0 X. ZZ ) A. d e. ( NN0 X. ZZ ) ( ( ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` c ) = ( ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` d ) -> c = d ) ) )
59	4, 17, 57, 58	syl3anbrc 1246	1 |- ( A e. ( ZZ>= ` 2 ) -> ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) : ( NN0 X. ZZ ) -1-1-onto-> { a | E. c e. NN0 E. d e. ZZ a = ( c + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. d ) ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   \ cdif 3571   <.cop 4183   |-> cmpt 4729   X. cxp 5112   ran crn 5115   Fn wfn 5883   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   QQcq 11788   ^cexp 12860   sqrcsqrt 13973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-numer 15443  df-denom 15444

This theorem is referenced by:  rmxyelxp  37477  rmxyval  37480