Theorem addlocpr 7816

Description: Locatedness of addition on positive reals. Lemma 11.16 in [BauerTaylor], p. 53. The proof in BauerTaylor relies on signed rationals, so we replace it with another proof which applies prarloc 7783 to both A and B, and uses nqtri3or 7676 rather than prloc 7771 to decide whether q is too big to be in the lower cut of A +P. B (and deduce that if it is, then r must be in the upper cut). What the two proofs have in common is that they take the difference between q and r to determine how tight a range they need around the real numbers. (Contributed by Jim Kingdon, 5-Dec-2019.)

Assertion

Ref	Expression
addlocpr	|- ( ( A e. P. /\ B e. P. ) -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) ) )

Distinct variable groups:   A, q, r   B, q, r

Proof of Theorem addlocpr

Dummy variables d e h p t u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltexnqq 7688	. . . . . 6 |- ( ( q e. Q. /\ r e. Q. ) -> ( q <Q r <-> E. p e. Q. ( q +Q p ) = r ) )
2	1	biimpa 296	. . . . 5 |- ( ( ( q e. Q. /\ r e. Q. ) /\ q <Q r ) -> E. p e. Q. ( q +Q p ) = r )
3	2	3adant1 1042	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) -> E. p e. Q. ( q +Q p ) = r )
4		halfnqq 7690	. . . . . 6 |- ( p e. Q. -> E. h e. Q. ( h +Q h ) = p )
5	4	ad2antrl 490	. . . . 5 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) -> E. h e. Q. ( h +Q h ) = p )
6		prop 7755	. . . . . . . . . 10 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
7		prarloc 7783	. . . . . . . . . 10 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ h e. Q. ) -> E. d e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( d +Q h ) )
8	6, 7	sylan 283	. . . . . . . . 9 |- ( ( A e. P. /\ h e. Q. ) -> E. d e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( d +Q h ) )
9	8	adantlr 477	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. ) /\ h e. Q. ) -> E. d e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( d +Q h ) )
10	9	3ad2antl1 1186	. . . . . . 7 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ h e. Q. ) -> E. d e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( d +Q h ) )
11	10	ad2ant2r 509	. . . . . 6 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) -> E. d e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( d +Q h ) )
12		prop 7755	. . . . . . . . . . . . . 14 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
13		prarloc 7783	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ h e. Q. ) -> E. e e. ( 1st ` B ) E. t e. ( 2nd ` B ) t <Q ( e +Q h ) )
14	12, 13	sylan 283	. . . . . . . . . . . . 13 |- ( ( B e. P. /\ h e. Q. ) -> E. e e. ( 1st ` B ) E. t e. ( 2nd ` B ) t <Q ( e +Q h ) )
15	14	adantll 476	. . . . . . . . . . . 12 |- ( ( ( A e. P. /\ B e. P. ) /\ h e. Q. ) -> E. e e. ( 1st ` B ) E. t e. ( 2nd ` B ) t <Q ( e +Q h ) )
16	15	3ad2antl1 1186	. . . . . . . . . . 11 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ h e. Q. ) -> E. e e. ( 1st ` B ) E. t e. ( 2nd ` B ) t <Q ( e +Q h ) )
17	16	ad2ant2r 509	. . . . . . . . . 10 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) -> E. e e. ( 1st ` B ) E. t e. ( 2nd ` B ) t <Q ( e +Q h ) )
18	17	adantr 276	. . . . . . . . 9 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) -> E. e e. ( 1st ` B ) E. t e. ( 2nd ` B ) t <Q ( e +Q h ) )
19		simpll1 1063	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) -> ( A e. P. /\ B e. P. ) )
20	19	ad2antrr 488	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> ( A e. P. /\ B e. P. ) )
21	20	simpld 112	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> A e. P. )
22	20	simprd 114	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> B e. P. )
23		simpll3 1065	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) -> q <Q r )
24	23	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> q <Q r )
25		simplrl 537	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) -> h e. Q. )
26	25	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> h e. Q. )
27		simplrr 538	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) -> ( q +Q p ) = r )
28		oveq2 6036	. . . . . . . . . . . . . . . 16 |- ( ( h +Q h ) = p -> ( q +Q ( h +Q h ) ) = ( q +Q p ) )
29	28	eqeq1d 2240	. . . . . . . . . . . . . . 15 |- ( ( h +Q h ) = p -> ( ( q +Q ( h +Q h ) ) = r <-> ( q +Q p ) = r ) )
30	29	ad2antll 491	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) -> ( ( q +Q ( h +Q h ) ) = r <-> ( q +Q p ) = r ) )
31	27, 30	mpbird 167	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) -> ( q +Q ( h +Q h ) ) = r )
32	31	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> ( q +Q ( h +Q h ) ) = r )
33		simprll 539	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) -> d e. ( 1st ` A ) )
34	33	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> d e. ( 1st ` A ) )
35		simprlr 540	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) -> u e. ( 2nd ` A ) )
36	35	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> u e. ( 2nd ` A ) )
37		simplrr 538	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> u <Q ( d +Q h ) )
38		simprll 539	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> e e. ( 1st ` B ) )
39		simprlr 540	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> t e. ( 2nd ` B ) )
40		simprr 533	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> t <Q ( e +Q h ) )
41	21, 22, 24, 26, 32, 34, 36, 37, 38, 39, 40	addlocprlem 7815	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) )
42	41	expr 375	. . . . . . . . . 10 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) ) -> ( t <Q ( e +Q h ) -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) ) )
43	42	rexlimdvva 2659	. . . . . . . . 9 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) -> ( E. e e. ( 1st ` B ) E. t e. ( 2nd ` B ) t <Q ( e +Q h ) -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) ) )
44	18, 43	mpd 13	. . . . . . . 8 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) )
45	44	expr 375	. . . . . . 7 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) -> ( u <Q ( d +Q h ) -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) ) )
46	45	rexlimdvva 2659	. . . . . 6 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) -> ( E. d e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( d +Q h ) -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) ) )
47	11, 46	mpd 13	. . . . 5 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) )
48	5, 47	rexlimddv 2656	. . . 4 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) )
49	3, 48	rexlimddv 2656	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) )
50	49	3expia 1232	. 2 |- ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) ) -> ( q <Q r -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) ) )
51	50	ralrimivva 2615	1 |- ( ( A e. P. /\ B e. P. ) -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   <.cop 3676   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   1stc1st 6310   2ndc2nd 6311   Q.cnq 7560   +Q cplq 7562   <Q cltq 7565   P.cnp 7571   +P. cpp 7573

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7584  df-pli 7585  df-mi 7586  df-lti 7587  df-plpq 7624  df-mpq 7625  df-enq 7627  df-nqqs 7628  df-plqqs 7629  df-mqqs 7630  df-1nqqs 7631  df-rq 7632  df-ltnqqs 7633  df-enq0 7704  df-nq0 7705  df-0nq0 7706  df-plq0 7707  df-mq0 7708  df-inp 7746  df-iplp 7748

This theorem is referenced by:  addclpr  7817